├── .gitignore ├── LICENSE ├── MO_Problem_Journal.pdf ├── PSets ├── 2020imocamp_m1.tex ├── Geo_Angle_Chase Pset.tex ├── Geo_Asif_vai.pdf ├── Geo_Asif_vai.tex ├── ISL_Alg.tex ├── ISL_Geo.tex ├── NEMO.pdf ├── Special Online Camp 2020 Graph Theory.pdf ├── igo_prep.tex ├── jun16.tex ├── nat2020exam1.tex ├── nat2020exam2.tex ├── nat_2019_probs.tex ├── pracPSet.tex ├── pset_june_16.pdf ├── req_pset_1.tex ├── special_camp_gt.tex ├── who_knows.pdf └── who_knows.tex ├── Pics ├── 2.pdf ├── 2.pdf_tex ├── 2.svg ├── APMO 2010 P4.ggb ├── APMO2014P5.jpg ├── APMO2014P5.png ├── APMO_2015_P1.ggb ├── AoPS1.ggb ├── AoPS2.jpg ├── AoPS2.png ├── AoPS3.jpg ├── AoPS3.png ├── AoPS4.jpg ├── AoPS4.png ├── AoPS5.jpg ├── AoPS5.png ├── AoPS6.1.pdf ├── AoPS6_1.ggb ├── AoPS6_1.pdf ├── AoPS6_1.pdf_tex ├── AoPS6_1.svg ├── AoPS6_2.ggb ├── AoPS6_2.pdf ├── AoPS6_2.pdf_tex ├── AoPS6_2.svg ├── AoPS_c284651h2018537_cute_problem_on_radical_axis_1.ggb ├── AoPS_c284651h2018537_cute_problem_on_radical_axis_1.pdf ├── AoPS_c284651h2018537_cute_problem_on_radical_axis_1.pdf_tex ├── AoPS_c284651h2018537_cute_problem_on_radical_axis_1.svg ├── AoPS_c6h2014023.ggb ├── AoPS_c6h2015540.ggb ├── Archer_EChen_M1P2.ggb ├── Archer_EChen_M1P2.pdf ├── Archer_EChen_M1P2.pdf_tex ├── Archer_EChen_M1P2.svg ├── Archer_EChen_M1P3.ggb ├── Archer_EChen_M1P3.pdf ├── Archer_EChen_M1P3.pdf_tex ├── Archer_EChen_M1P3.svg ├── BC_parallel_through_I.ggb ├── BC_parallel_through_I.pdf ├── BC_parallel_through_I.pdf_tex ├── BC_parallel_through_I.svg ├── Balkan_MO_2017_P3.ggb ├── Balkan_MO_2017_P3.pdf ├── Balkan_MO_2017_P3.pdf_tex ├── Balkan_MO_2017_P3.svg ├── Banach_non_fixed_graph.pdf ├── Banach_non_fixed_graph.pdf_tex ├── Banach_non_fixed_graph.svg ├── Bewarish_1.ggb ├── Bewarish_1.pdf ├── Bewarish_1.pdf_tex ├── Bewarish_1.svg ├── BrocardPoints.jpg ├── BrocardPoints.png ├── CHKMO_2014_p4.ggb ├── CHKMO_2014_p4.pdf ├── CHKMO_2014_p4.pdf_tex ├── CHKMO_2014_p4.svg ├── CHKMO_2014_p4_lem_1.ggb ├── CHKMO_2014_p4_lem_1.pdf ├── CHKMO_2014_p4_lem_1.pdf_tex ├── CHKMO_2014_p4_lem_1.svg ├── ChinAaTST2018T1P3.ggb ├── China Southeast 2020 P2.ggb ├── ChinaTST2018T1P3.ggb ├── ChinaTST2018T1P3.pdf ├── ChinaTST2018T1P3.pdf_tex ├── ChinaTST2018T1P3.svg ├── ChinaTST2018T1P3_lemm.ggb ├── ChinaTST2018T1P3_lemm.pdf ├── ChinaTST2018T1P3_lemm.pdf_tex ├── ChinaTST2018T1P3_lemm.svg ├── Complete_quad_anglebisector.jpg ├── Complete_quad_anglebisector.png ├── ERIQ_Lemma.png ├── Equilateral Triangles Intercepted by Oriented Parallelians.ggb ├── Fuad_3.ggb ├── I3rd2010D3P6.ggb ├── IGO2014P10.ggb ├── IGO2018A5.ggb ├── IGO2018A5.pdf ├── IGO2018A5.pdf_tex ├── IGO2018A5.svg ├── IGO2019G5.ggb ├── IGO2019G5.pdf ├── IGO2019G5.pdf_tex ├── IGO2019G5.svg ├── IGO_2017_A4.asy ├── IGO_2017_A4.ggb ├── IGO_2017_A4.pdf ├── IGO_2017_A4.pdf_tex ├── IGO_2017_A4.svg ├── IGO_2017_A4_1.pdf ├── IGO_2017_Advanced_P3.ggb ├── IGO_2017_Advanced_P3.pdf ├── IGO_2017_Advanced_P3.pdf_tex ├── IGO_2017_Advanced_P3.svg ├── IGO_2018_A4.ggb ├── IGO_2018_A4.pdf ├── IGO_2018_A4.pdf_tex ├── IGO_2018_A4.svg ├── IRAN3rd2016G6.png ├── ISL2006G9.ggb ├── ISL2006G9.pdf ├── ISL2006G9.pdf_tex ├── ISL2006G9.svg ├── ISL2007G5.pdf ├── ISL2007G5.png ├── ISL2009G6_lemma.ggb ├── ISL2009G6_lemma.pdf ├── ISL2009G6_lemma.pdf_tex ├── ISL2009G6_lemma.svg ├── ISL2009G6_lemmaa.pdf ├── ISL2009G6_lemmaa.pdf_tex ├── ISL2009G6_lemmaa.svg ├── ISL2010G5.ggb ├── ISL2010G5.pdf ├── ISL2010G5.pdf_tex ├── ISL2010G5.svg ├── ISL2011G5.ggb ├── ISL2011G7.ggb ├── ISL2011G8.ggb ├── ISL2012G2.ggb ├── ISL2013G3.ggb ├── ISL2014G5.ggb ├── ISL2014G5.pdf ├── ISL2014G5.pdf_tex ├── ISL2014G5.svg ├── ISL2014G5_lemma1.ggb ├── ISL2014G5_lemma1.pdf ├── ISL2014G5_lemma1.pdf_tex ├── ISL2014G5_lemma1.svg ├── ISL2014G6.ggb ├── ISL2014G7.pdf ├── ISL2015G5.ggb ├── ISL2015G5.pdf ├── ISL2015G5.pdf_tex ├── ISL2015G5.svg ├── ISL2015G5_1.ggb ├── ISL2015G5_2.ggb ├── ISL2015G5_2.svg ├── ISL2015G6.ggb ├── ISL2015G6.pdf ├── ISL2015G6.pdf_tex ├── ISL2015G6.svg ├── ISL2019G6.ggb ├── ISL2019G6.pdf ├── ISL2019G6.pdf_tex ├── ISL2019G6.svg ├── ISL2019G6_lem.pdf ├── ISL2019G6_lem.pdf_tex ├── ISL2019G6_lem.svg ├── ISL_2011_G8.ggb ├── ISL_2011_G8.pdf ├── ISL_2011_G8.pdf_tex ├── ISL_2011_G8.svg ├── ISL_GEO │ ├── 1_1_3.ggb │ ├── 1_1_3.pdf │ ├── 1_1_4.ggb │ ├── 1_1_4.pdf │ ├── 1_1_5.ggb │ ├── 1_1_6.ggb │ ├── 1_1_6.pdf │ ├── 1_5_18.ggb │ ├── 1_5_18_1.ggb │ ├── 5_6_12.ggb │ ├── 5_6_12.png │ ├── ISL2007G5.ggb │ ├── ISL2007G7.ggb │ ├── ISL2007G8.ggb │ └── main.ggb ├── ITST2014T1P6.ggb ├── ITST2015D3P2.png ├── ITST2015D3P3.ggb ├── ITST2015D3P3i.png ├── ITST2015D3P3ii.png ├── ITST2017T3P6.ggb ├── ITST2017T3P6.pdf ├── ITST2017T3P6.pdf_tex ├── ITST2017T3P6.svg ├── InExLemma1.ggb ├── InExLemma2.ggb ├── InExLemma3.ggb ├── InExLemma3.pdf ├── InExLemma3.pdf_tex ├── InExLemma3.png ├── InExLemma3.svg ├── InExLemma4.ggb ├── InExLemma4.pdf ├── InExLemma4.pdf_tex ├── InExLemma4.png ├── InExLemma4.svg ├── India_Postals_2015_Set_2.ggb ├── Iran3rd2014G5.ggb ├── IranTST2008P12.ggb ├── IranTST2008P12_Lemma_1.ggb ├── IranTST2011P6.ggb ├── IranTST2014Plast.ggb ├── IranTST2016G3.ggb ├── Iran_TST_2015_D3_P3.ggb ├── IsoConjuCirclesLemma1.png ├── Isodynamic_Points.ggb ├── Isodynamic_Points.pdf ├── Isodynamic_Points.pdf_tex ├── Isodynamic_Points.svg ├── Isogonal_Lemma.ggb ├── Isogonal_Lemma.pdf ├── Isogonal_Lemma.pdf_tex ├── Isogonal_Lemma.svg ├── Isotonic_Lemma.png ├── JMO__.ggb ├── JMO_dunno.ggb ├── MannheimTheorem.png ├── Matematika 2009.ggb ├── Mixt1.png ├── Mixt2.png ├── Mixtilinear.ggb ├── Mixtilinear.pdf ├── Mixtilinear.pdf_tex ├── Mixtilinear.svg ├── Mixtilinear_Gene.ggb ├── Mixtilinear_Gene.pdf ├── Mixtilinear_Gene.pdf_tex ├── Mixtilinear_Gene.svg ├── Orthocenter-Circumcenter-9PCenter.ggb ├── P-HaggeCircle.png ├── P3.ggb ├── PG1_OctagonalPascal.ggb ├── PG1_OctagonalPascal.pdf ├── PG1_OctagonalPascal.pdf_tex ├── PG1_OctagonalPascal.svg ├── PG1_OctagonalPascal_1.png ├── PG1_OctagonalPascal_check.ggb ├── PseudoMiquel.png ├── SATST2015proposed_by_bura │ ├── derakynay1134-1.ggb │ ├── derakynay1134-1.pdf │ ├── derakynay1134-1.pdf_tex │ ├── derakynay1134-1.png │ ├── derakynay1134-1.svg │ ├── derakynay1134-2.ggb │ ├── derakynay1134-2.pdf │ ├── derakynay1134-2.pdf_tex │ ├── derakynay1134-2.png │ ├── derakynay1134-2.svg │ ├── derakynay1134-3.png │ ├── derakynay1134-4.ggb │ ├── derakynay1134-4.pdf │ ├── derakynay1134-4.pdf_tex │ ├── derakynay1134-4.png │ ├── derakynay1134-4.svg │ ├── derakynay1134-5.ggb │ ├── derakynay1134-5.pdf │ ├── derakynay1134-5.pdf_tex │ ├── derakynay1134-5.png │ ├── derakynay1134-5.svg │ ├── derakynay1134-6.ggb │ ├── derakynay1134-6.pdf │ ├── derakynay1134-6.pdf_tex │ ├── derakynay1134-6.png │ ├── derakynay1134-6.svg │ ├── derakynay1134-7.ggb │ ├── derakynay1134-7.pdf │ ├── derakynay1134-7.pdf_tex │ ├── derakynay1134-7.png │ ├── derakynay1134-7.svg │ └── derakynay1134-8.png ├── SimsonLineLemma1.ggb ├── SimsonLineLemma1.pdf ├── SimsonLineLemma1.pdf_tex ├── SimsonLineLemma1.png ├── SimsonLineLemma1.svg ├── SimsonLineLemma2.ggb ├── SimsonLineLemma2.pdf ├── SimsonLineLemma2.pdf_tex ├── SimsonLineLemma2.png ├── SimsonLineLemma2.svg ├── SpiralSimiLemma1.png ├── Symmedian.ggb ├── Symmedian_Lemma_1.ggb ├── Symmedian_Lemma_1.pdf ├── Symmedian_Lemma_1.pdf_tex ├── Symmedian_Lemma_1.svg ├── Symmedian_Point.png ├── TST2020_T5P2.ggb ├── Turkey_TST_2018_p4.ggb ├── Turkey_TST_2018_p4.pdf ├── Turkey_TST_2018_p4.pdf_tex ├── Turkey_TST_2018_p4.svg ├── Two_Symmedian_Points.ggb ├── Two_Symmedian_Points.pdf ├── Two_Symmedian_Points.pdf_tex ├── Two_Symmedian_Points.png ├── USAJMO_2018_p3.ggb ├── USAJMO_2018_p3.pdf ├── USAJMO_2018_p3.pdf_tex ├── USAJMO_2018_p3.svg ├── USAMO_2008_P2.ggb ├── USAMO_2008_P2.pdf ├── USAMO_2008_P2.pdf_tex ├── USAMO_2008_P2.svg ├── USAMO_2014_P5.ggb ├── USAMO_2014_P5.pdf ├── USAMO_2014_P5.pdf_tex ├── USAMO_2014_P5.svg ├── USATST2000P2.ggb ├── USATST2000P2.png ├── USATST2000P2LemmaIgot.ggb ├── USATST2006P6.ggb ├── USATST2006P6.pdf ├── USATST2006P6.pdf_tex ├── USATST2006P6.svg ├── USATST2019P1.ggb ├── USATST2019P1.pdf ├── USATST2019P1_1.ggb ├── USATST2019P1_1.pdf ├── USATST2019P1_1.pdf_tex ├── USATST2019P1_1.svg ├── USATST2019P1_new.ggb ├── USATST2019P1_new.pdf ├── USATST2019P1_new.pdf_tex ├── USATST2019P1_new.svg ├── USATST2019P6.ggb ├── USA_Winter_TST_2020_P2.ggb ├── USA_Winter_TST_2020_P2.pdf ├── USA_Winter_TST_2020_P2.pdf_tex ├── USA_Winter_TST_2020_P2.svg ├── USA_Winter_TST_2020_P2_generalization.ggb ├── USA_Winter_TST_2020_P2_generalization.pdf ├── USA_Winter_TST_2020_P2_generalization.pdf_tex ├── USA_Winter_TST_2020_P2_generalization.svg ├── VNTST2018P6a_Gene.ggb ├── Vietnamese_TST_2017_P3.ggb ├── Vietnamese_TST_2018_P6_a.ggb ├── Vietnamese_TST_2018_P6_a.pdf ├── Vietnamese_TST_2018_P6_a.pdf_tex ├── Vietnamese_TST_2018_P6_a.svg ├── Vietnamese_TST_2018_P6_a_gene.ggb ├── Vietnamese_TST_2018_P6_a_gene.pdf ├── Vietnamese_TST_2018_P6_a_gene.pdf_tex ├── Vietnamese_TST_2018_P6_a_gene.svg ├── Vietnamese_TST_2018_P6_a_inv.ggb ├── Vietnamese_TST_2018_P6_a_inv.pdf ├── Vietnamese_TST_2018_P6_a_inv.pdf_tex ├── Vietnamese_TST_2018_P6_a_inv.svg ├── Vietnamese_TST_2018_P6_a_modified.ggb ├── Vietnamese_TST_2018_P6_a_modified.pdf ├── Vietnamese_TST_2018_P6_a_modified.pdf_tex ├── Vietnamese_TST_2018_P6_a_modified.svg ├── Vietnamese_TST_2018_P6_a_problem.pdf ├── Vietnamese_TST_2018_P6_a_problem.pdf_tex ├── Vietnamese_TST_2018_P6_a_problem.svg ├── XmL_trial_1.ggb ├── antipode_incenter.ggb ├── antipode_incenter.pdf ├── antipode_incenter.pdf_tex ├── antipode_incenter.svg ├── apmo3.ggb ├── apmo3.pdf ├── apmo3.pdf_tex ├── apmo3.svg ├── apmo3_inv.ggb ├── apmo3_inv.pdf ├── apmo3_inv.pdf_tex ├── apmo3_inv.svg ├── apmo3_inv_1.ggb ├── apmo3_inv_1.jpg ├── apmo3_inv_1.pdf ├── apmo3_inv_1.png ├── apmo_2019_p3_and_IGO_2019_p4.ggb ├── apmo_2019_p3_and_IGO_2019_p4.pdf ├── apmo_2019_p3_and_IGO_2019_p4.pdf_tex ├── apmo_2019_p3_and_IGO_2019_p4.svg ├── apmo_2019_p3_and_IGO_2019_p4_0.ggb ├── apmo_2019_p3_and_IGO_2019_p4_0.pdf ├── apmo_2019_p3_and_IGO_2019_p4_0.pdf_tex ├── apmo_2019_p3_and_IGO_2019_p4_0.svg ├── apmo_2019_p3_and_IGO_2019_p4_1.ggb ├── apmo_2019_p3_and_IGO_2019_p4_1.pdf ├── apmo_2019_p3_and_IGO_2019_p4_1.pdf_tex ├── apmo_2019_p3_and_IGO_2019_p4_1.svg ├── apmo_p3.tex ├── basics_lem_1.ggb ├── basics_lem_1.pdf ├── basics_lem_1.pdf_tex ├── basics_lem_1.svg ├── basics_lem_2.ggb ├── basics_lem_2.pdf ├── basics_lem_2.pdf_tex ├── basics_lem_2.svg ├── basics_lem_3.ggb ├── basics_lem_3.pdf ├── basics_lem_3.pdf_tex ├── basics_lem_3.svg ├── bdmo_forum_geo_1.ggb ├── bdmo_forum_geo_1.pdf ├── bdmo_forum_geo_1.pdf_tex ├── bdmo_forum_geo_1.svg ├── bps_1.ggb ├── brocard_points.ggb ├── c374081h2046456.ggb ├── c6t45786f6h1618685.ggb ├── c6t45786f6h1618685_a_1.jpg ├── c6t45786f6h1618685_a_1.png ├── c6t48f6h1508805.ggb ├── c6t48f6h1508805.pdf ├── c6t48f6h1508805.pdf_tex ├── c6t48f6h1508805.svg ├── china tst 2018 p3.ggb ├── circles_through_FeuerbachPoint.ggb ├── circles_through_FeuerbachPoint.pdf ├── circles_through_FeuerbachPoint.pdf_tex ├── circles_through_FeuerbachPoint.svg ├── circles_with_arc_midpoints.ggb ├── circles_with_arc_midpoints.pdf ├── circles_with_arc_midpoints.pdf_tex ├── circles_with_arc_midpoints.svg ├── class2.ggb ├── class2_p2.pdf ├── col_psl_3.5.ggb ├── combi │ ├── algo │ │ ├── BinaryHeap.jpg │ │ ├── BinaryHeap.png │ │ ├── CF744B.pdf │ │ ├── CF744B.png │ │ ├── CF744B.svg │ │ ├── Convex_hull_trick1.png │ │ ├── Max-Heap.pdf │ │ ├── Max-Heap.svg │ │ ├── Minimum_spanning_tree.pdf │ │ ├── Minimum_spanning_tree.pdf_tex │ │ ├── Minimum_spanning_tree.svg │ │ └── TaiwanTST2015R3D1P1.png │ ├── coloring │ │ └── ISL2014C4.pdf │ ├── config │ │ ├── USA_TST_2018_P3_1.pdf │ │ ├── USA_TST_2018_P3_1.svg │ │ ├── USA_TST_2018_P3_2.pdf │ │ ├── USA_TST_2018_P3_3.pdf │ │ └── USA_TST_2018_P3_4.pdf │ ├── geo │ │ ├── .svg │ │ ├── 2007c2.pdf │ │ ├── 2007c2.png │ │ ├── 2007c2.svg │ │ ├── MEMO2015T4.pdf │ │ ├── MEMO2015T4.png │ │ ├── MEMO2015T4.svg │ │ ├── RMM2019P4.ggb │ │ ├── RMM2019P4.pdf │ │ ├── RMM2019P4.pdf_tex │ │ ├── RMM2019P4.png │ │ ├── RMM2019P4.svg │ │ ├── RMM2019P4_1.png │ │ ├── lol.asy │ │ ├── lol.pdf │ │ ├── queen_1212.svg │ │ ├── queen_1414.pdf │ │ ├── queen_1414.svg │ │ ├── queen_1515.pdf │ │ ├── queen_1515.svg │ │ ├── queen_88.pdf │ │ ├── queen_88.svg │ │ └── rook_33.svg │ └── graph │ │ ├── ARO_2005_P10_8.pdf │ │ ├── ARO_2005_P10_8.svg │ │ ├── China_TST_2015_T1_D2_P3.svg │ │ ├── China_TST_2015_T1_D2_P3_1.pdf │ │ ├── China_TST_2015_T1_D2_P3_1.svg │ │ ├── China_TST_2015_T1_D2_P3_2.pdf │ │ ├── China_TST_2015_T1_D2_P3_2.svg │ │ ├── Max-cut.pdf │ │ ├── Max-cut.svg │ │ ├── Min-cut.pdf │ │ ├── Min-cut.pdf_tex │ │ ├── Min-cut.svg │ │ ├── Turan_13-4.pdf │ │ ├── Turan_13-4.svg │ │ ├── prufer_code_example.pdf │ │ ├── prufer_code_example.pdf_tex │ │ ├── prufer_code_example.svg │ │ └── turan_graph.png ├── concurrency_in_prallel_lines_with_BC.png ├── concurrent_lines_in_incenter.ggb ├── concurrent_lines_in_incenter.pdf ├── concurrent_lines_in_incenter.pdf_tex ├── concurrent_lines_in_incenter.png ├── concurrent_lines_in_incenter.svg ├── concurrent_segments_almost_pascal.ggb ├── concurrent_segments_almost_pascal.pdf ├── concurrent_segments_almost_pascal.pdf_tex ├── concurrent_segments_almost_pascal.svg ├── conic_touching_conic.ggb ├── conic_touching_conic.pdf ├── conic_touching_conic.pdf_tex ├── conic_touching_conic.svg ├── conic_touching_quad.ggb ├── conic_touching_quad.pdf ├── conic_touching_quad.pdf_tex ├── conic_touching_quad.svg ├── constructing_second_intersection_with_conic.ggb ├── constructing_second_intersection_with_conic.pdf ├── constructing_second_intersection_with_conic.pdf_tex ├── constructing_second_intersection_with_conic.svg ├── cross_ratio_joydip_1.ggb ├── cross_ratio_joydip_1.pdf ├── cross_ratio_joydip_1.pdf_tex ├── cross_ratio_joydip_1.svg ├── cross_tests.ggb ├── curvi_lem_1.ggb ├── curvi_lem_1.pdf ├── curvi_lem_1.pdf_tex ├── curvi_lem_1.png ├── curvi_lem_1.svg ├── curvilinear_fjGZLv.ggb ├── curvilinear_fjGZLv_2.ggb ├── default.ggb ├── desargues_invo_theorem.ggb ├── desargues_invo_theorem.pdf ├── desargues_invo_theorem.pdf_tex ├── desargues_invo_theorem.svg ├── desargues_invo_theorem_2points.png ├── desargues_perspectivity_theorem.ggb ├── dfs.png ├── doing1.ggb ├── dual_desargues.ggb ├── dual_desargues.pdf ├── dual_desargues.pdf_tex ├── dual_desargues.svg ├── dwip_excenter_orthocenter.ggb ├── elmo_2019_P3.ggb ├── elmo_2019_P3.pdf ├── elmo_2019_P3.pdf_tex ├── elmo_2019_P3.svg ├── eriqp2.ggb ├── excenter_touchpoint_bigarc-midpoints.ggb ├── excenter_touchpoint_bigarc-midpoints.pdf ├── excenter_touchpoint_bigarc-midpoints.pdf_tex ├── excenter_touchpoint_bigarc-midpoints.png ├── excenter_touchpoint_bigarc-midpoints.svg ├── feurbach.ggb ├── feurbach.pdf ├── feurbach.pdf_tex ├── feurbach.svg ├── feurbach_construction.ggb ├── feurbach_construction.pdf ├── feurbach_construction.pdf_tex ├── feurbach_construction.svg ├── feurbach_inversion.ggb ├── feurbach_inversion.pdf ├── feurbach_inversion.pdf_tex ├── feurbach_inversion.svg ├── fuad_2.ggb ├── fuhrmann_circle.pdf ├── fuhrmann_circle.pdf_tex ├── fuhrmann_circle.png ├── furhmann_circle-eps-converted-to.pdf ├── furhmann_circle.ggb ├── geodip_ggbs │ ├── '.ggb │ ├── ,,,,,,,,j,jjb.ggb │ ├── ,k.ggb │ ├── ,lm,.ggb │ ├── 07 g5 pascal pappu.ggb │ ├── 07 g7.ggb │ ├── 1 (2).ggb │ ├── 1 (3).ggb │ ├── 1 (4).ggb │ ├── 1.ggb │ ├── 10.ggb │ ├── 11.ggb │ ├── 111.ggb │ ├── 14 g4.ggb │ ├── 14.ggb │ ├── 2 (2).ggb │ ├── 2.ggb │ ├── 3.ggb │ ├── 3lemma.ggb │ ├── 54654.ggb │ ├── 55555555555555.ggb │ ├── 56.ggb │ ├── 565.ggb │ ├── 5wju77.ggb │ ├── 6e5.ggb │ ├── 6yu.ggb │ ├── 7.ggb │ ├── 8.ggb │ ├── 87.ggb │ ├── 9.ggb │ ├── ;..ggb │ ├── ;ij;p.ggb │ ├── ;l.ggb │ ├── ;okj.ggb │ ├── _.ggb │ ├── __.ggb │ ├── a big lesson 14 g5.ggb │ ├── a generialization of conic.ggb │ └── a nice problem by me.ggb ├── geodip_lemma.ggb ├── ggb │ ├── 1.ggb │ ├── 11.ggb │ ├── Fontene’s first theorem.ggb │ ├── G9.ggb │ ├── Mixtilinear Excircles Perpendicular.ggb │ ├── Sharygin18 8.8.ggb │ ├── aops.ggb │ ├── c6h407514p2276387.ggb │ ├── camp P3.ggb │ ├── camp p3 (Case Conflict).ggb │ ├── chinap5.ggb │ ├── excenters.ggb │ ├── gash.ggb │ ├── geogeogeo.ggb │ ├── involution.ggb │ ├── isl.ggb │ ├── isl09g6.ggb │ ├── j.ggb │ ├── mixtilinear incircle.ggb │ ├── q2h514684p2892003.ggb │ ├── raigeomara.ggb │ ├── test.ggb │ ├── tst g2 P4.ggb │ └── wenhua.ggb ├── gonju_1.ggb ├── gonju_1.pdf ├── gonju_1.pdf_tex ├── gonju_1.svg ├── gonju_1_s.ggb ├── gonju_1_s.pdf ├── gonju_1_s.pdf_tex ├── gonju_1_s.svg ├── haha.ggb ├── homothety+reflection.ggb ├── homothety+reflection.png ├── igonemo4.ggb ├── imo2018p6.ggb ├── imo2018p6.pdf ├── imo2018p6.pdf_tex ├── imo2018p6.svg ├── incenter_main.ggb ├── incenter_main.pdf ├── incenter_main.pdf_tex ├── incenter_main.svg ├── incenters_cyclic_with_mixtilinear_touchpoint.ggb ├── incenters_cyclic_with_mixtilinear_touchpoint.pdf ├── incenters_cyclic_with_mixtilinear_touchpoint.pdf_tex ├── incenters_cyclic_with_mixtilinear_touchpoint.png ├── incenters_cyclic_with_mixtilinear_touchpoint.svg ├── incircle_lemma.ggb ├── incircle_lemma_1.ggb ├── incircle_touchpoint_and_cevian.ggb ├── incircle_touchpoint_and_cevian.pdf ├── incircle_touchpoint_and_cevian.pdf_tex ├── incircle_touchpoint_and_cevian.svg ├── insimilicenter.ggb ├── insimilicenter.pdf ├── insimilicenter.pdf_tex ├── insimilicenter.svg ├── involution_conic.ggb ├── involution_conic.pdf ├── involution_conic.pdf_tex ├── involution_conic.svg ├── iran2009g3.ggb ├── irantst2004P4.ggb ├── irantst2004P4.pdf ├── irantst2004P4.pdf_tex ├── irantst2004P4.svg ├── isl2002G8.ggb ├── isogonal_in_quad.ggb ├── isogonal_in_quad.pdf ├── isogonal_in_quad.pdf_tex ├── isogonal_in_quad.svg ├── itti_same_incircle.ggb ├── itti_same_incircle.pdf ├── itti_same_incircle.pdf_tex ├── ittiprob2.ggb ├── ittiprob20180225.ggb ├── linethroughO_1.png ├── linethroughO_2.png ├── main.ggb ├── mine(!)_1.ggb ├── mine(!)_1.pdf ├── mine(!)_1.pdf_tex ├── mine(!)_1.svg ├── mine_(maybe)_1.ggb ├── mine_(maybe)_1.pdf ├── mine_(maybe)_1.pdf_tex ├── mine_(maybe)_1.svg ├── mine_(maybe)_2.ggb ├── mixti_11.ggb ├── mixti_11.pdf ├── mixti_11.pdf_tex ├── mixti_11.svg ├── mixti_symmedian.ggb ├── mixti_symmedian.pdf ├── mixti_symmedian.pdf_tex ├── mixti_symmedian.svg ├── mixti_touch_with_BC.ggb ├── mixti_touch_with_BC.pdf ├── mixti_touch_with_BC.pdf_tex ├── mixti_touch_with_BC.svg ├── mixtilinear_root_bc.ggb ├── mixtilinear_root_bc.pdf ├── mixtilinear_root_bc.pdf_tex ├── mixtilinear_root_bc.svg ├── nat_pset_16.ggb ├── nat_pset_16.pdf ├── nat_pset_16.pdf_tex ├── nat_pset_16.svg ├── nice_prob_by_geodip.ggb ├── nice_prob_by_geodip.pdf ├── nice_prob_by_geodip.pdf_tex ├── nice_prob_by_geodip.svg ├── o.png ├── orthocenters_on_conic.ggb ├── orthocenters_on_conic.pdf ├── orthocenters_on_conic.pdf_tex ├── orthocenters_on_conic.svg ├── orthogonal_hyperbola.ggb ├── orthogonal_hyperbola.pdf ├── orthogonal_hyperbola.pdf_tex ├── orthogonal_hyperbola.svg ├── pappu_cross.ggb ├── pappu_cross.pdf ├── pappu_cross.pdf_tex ├── pappu_cross.svg ├── pappu_cross_4.ggb ├── pappu_cross_4.pdf ├── pappu_cross_4.pdf_tex ├── pappu_cross_4.svg ├── parabola_foci_construction.ggb ├── parabola_foci_construction.pdf ├── parabola_foci_construction.pdf_tex ├── parabola_foci_construction.svg ├── perp_to_euler_line.ggb ├── perp_to_euler_line.pdf ├── perp_to_euler_line.pdf_tex ├── perp_to_euler_line.svg ├── photo_2020-09-07_09-05-33.jpg ├── phy_2.ggb ├── pole_polar.ggb ├── power_of_point_on_a_midline_of_a_self-polar_triangle.png ├── pset3_4.ggb ├── pton_rahole_haha_lol_xD.ggb ├── quad-same_sides-diff_area.ggb ├── quad-same_sides-diff_area.png ├── rando_1.ggb ├── rando_1.pdf ├── rando_1.pdf_tex ├── rando_1.svg ├── random_circle_pic_1.ggb ├── random_circle_pic_1.pdf ├── random_circle_pic_1.pdf_tex ├── random_circle_pic_1.svg ├── random_circle_pic_2.ggb ├── reconstruct_ABC.ggb ├── reconstruct_ABC.pdf ├── reconstruct_ABC.pdf_tex ├── reconstruct_ABC.png ├── reconstruct_ABC.svg ├── reconstruct_ABC_1.ggb ├── reconstruct_ABC_1.pdf ├── reconstruct_ABC_1.pdf_tex ├── reconstruct_ABC_1.svg ├── reconstruct_ABC_from_A'B'C'.ggb ├── rmm_anu.ggb ├── second_intersection_of_ID_1_with_excircle.ggb ├── sharygin │ ├── 2010_3.ggb │ ├── 2012_21.ggb │ ├── 2012_21.png │ ├── 2012_22.ggb │ ├── 2012_22.png │ └── 2020_p16.ggb ├── sharygin_2009.ggb ├── sharygin_2010_3.ggb ├── sharygin_2010_3.png ├── sharygin_2012_21.png ├── sharygin_2012_22.png ├── sharygin_2012_p21.png ├── sharygin_2012_p22.png ├── some_g4.ggb ├── some_g4.pdf ├── some_g4.pdf_tex ├── some_g4.svg ├── spiral simi.ggb ├── tangent_line_tricc.ggb ├── tangent_line_tricc.pdf ├── tangent_line_tricc.pdf_tex ├── tangent_line_tricc.svg ├── tangential_quad_four_incenters.ggb ├── tangential_quad_four_incenters.pdf ├── tangential_quad_four_incenters.pdf_tex ├── tangential_quad_incenters.ggb ├── tangential_quad_incenters.png ├── tg_telvcohl_2.ggb ├── tools.ggt ├── usamo_2009_p5.ggb ├── wtf.ggb └── wtf.png ├── README.md ├── about.tex ├── alg ├── sec1_5_selected_probs.tex ├── sec1_fe.log ├── sec1_fe.tex ├── sec2_1_irreducibility.tex ├── sec2_poly.log ├── sec2_poly.tex ├── sec3_ineq.log ├── sec3_ineq.tex ├── sec4_calculus.tex ├── sec5_ad_hoc.log ├── sec5_ad_hoc.tex └── sec6_tricks.tex ├── beamer ├── 1.pdf ├── 1.svg ├── Bijection.pdf ├── Bijection.svg ├── Kalpurush.ttf ├── Monotype Corsiva.ttf ├── SiyamRupali.ttf ├── beamerthemeNord.pdf ├── bijection.nav ├── bijection.snm ├── bijection.tex ├── bijection.toc ├── bijection.xdv ├── combi.pdf ├── nat2020exam1.pdf ├── orders.nav ├── orders.pdf ├── orders.snm ├── orders.tex ├── orders.toc ├── orders.xdv ├── problem_1.pdf ├── problem_1.svg ├── trig_grid_1.pdf ├── trig_grid_1.svg ├── trig_grid_2.pdf ├── trig_grid_2.svg ├── trig_grid_3.pdf └── trig_grid_3.svg ├── camp ├── IMO camp p set.tex ├── beamerthemeNord.nav ├── beamerthemeNord.snm ├── beamerthemeNord.toc ├── beamerthemeNord.vrb ├── beamerthemeNord.xdv ├── combi.tex ├── listofprobs.tex ├── listoftheos.tex ├── nemo.tex └── nemo_2.tex ├── cat.idx ├── cat.ind ├── categories.tex ├── combi ├── Game Theory.zip ├── Game Theory │ ├── 1.htm │ ├── 1_files │ │ ├── arrow1.gif │ │ ├── arrow2.gif │ │ ├── board.gif │ │ ├── cross.gif │ │ ├── goback.gif │ │ ├── isleline.gif │ │ ├── options.gif │ │ └── smiley.gif │ ├── 2.htm │ ├── 2_files │ │ ├── arrow1.gif │ │ ├── arrow2.gif │ │ ├── cchess.gif │ │ ├── goback.gif │ │ ├── greenball.gif │ │ ├── isleline2.gif │ │ ├── purpleball.gif │ │ ├── redball.gif │ │ └── star.gif │ ├── 3.htm │ ├── 3_files │ │ ├── 1stick.gif │ │ ├── 1stickg.gif │ │ ├── 1stickr.gif │ │ ├── 2stickg.gif │ │ ├── 2stickr.gif │ │ ├── arrow1.gif │ │ ├── arrow2.gif │ │ ├── dawsonchess.gif │ │ ├── goback.gif │ │ ├── isleline2.gif │ │ ├── map.gif │ │ ├── nostick.gif │ │ ├── redx.gif │ │ └── sumboard.gif │ ├── 4.htm │ ├── 4_files │ │ ├── arrow1.gif │ │ ├── arrow2.gif │ │ ├── goback.gif │ │ ├── isleline2.gif │ │ ├── map2.gif │ │ ├── map2a.gif │ │ ├── map2b.gif │ │ └── map2c.gif │ ├── 5.htm │ └── 5_files │ │ ├── addladd.gif │ │ ├── goback.gif │ │ ├── horsefly.gif │ │ ├── horsefly2.gif │ │ ├── horsefly2a.gif │ │ ├── horsefly2b.gif │ │ └── treeline.gif ├── GameTheoryPset.pdf ├── combi.tex ├── new_1_local.tex ├── sec10_algebraic.tex ├── sec11_bounding.tex ├── sec12_not_sorted.tex ├── sec13_cp_algo.tex ├── sec13_permutation.tex ├── sec1_tricks.tex ├── sec2_binom_iden.tex ├── sec3_sets.tex ├── sec4_algo (M Ahsan Al Mahir's conflicted copy 2020-07-28).tex ├── sec4_algo.tex ├── sec5_1_counting_in_graph.tex ├── sec5_2_graph_alogrithm.tex ├── sec5_graph.tex ├── sec6_game.tex ├── sec7_combi_geo.tex ├── sec8_1_Recursion.tex ├── sec8_sequences.tex ├── sec9_1_coloring_problems.tex ├── sec9_configs.tex └── subsec_1_2n_points.tex ├── count_figs.sty ├── def.idx ├── def.ind ├── dump ├── APMO_2015_P3.tex ├── ISL2009G6.tex ├── USATST_W_2018_P3.tex ├── document.pdf ├── isl2004n1.tex └── isl2004n4.tex ├── ext_doc.tex ├── for_sharing.tex ├── geo ├── basics.tex ├── bigpics │ └── 1_parallels_to_median_and_stuffs.tex ├── sec10_length.tex ├── sec11_pedal.tex ├── sec12_pending.tex ├── sec13_problems.tex ├── sec14_research4later.tex ├── sec1_circum-9p-circle.tex ├── sec2_cevian.tex ├── sec3_in-ex-circle.tex ├── sec4_conjugates.tex ├── sec5_mixti_curvi-linear.tex ├── sec6_circle.tex ├── sec7_comp-quad.tex ├── sec8_5_projective_construction.tex ├── sec8_projective.tex └── sec9_parallel.tex ├── indStyle.ist ├── index.pdf ├── main.idx ├── main.ind ├── main.pdf ├── main.tex ├── main.toc ├── main.xdv ├── nt ├── NUMBER_THEORY.tex ├── sec0_unsorted.tex ├── sec1_orders_mod_prime.tex ├── sec2_primes.tex ├── sec3_fe_poly.tex ├── sec4_dioph_eq.tex ├── sec5_divisibility.tex └── sec6_modular_arithmatic.tex ├── pdfs ├── GEOMETRY.docx ├── GEOMETRY_word.pdf ├── MO_Problem_Journal.pdf ├── alg.pdf ├── bcs_2019.pdf ├── bcs_full_2019.pdf ├── geo.pdf ├── main.pdf ├── nat2020exam1.pdf ├── nat2020exam2.pdf └── nat_2019_probs.pdf ├── preamble.sty ├── prob.idx ├── prob.ind ├── strat.idx ├── strat.ind ├── thm.idx └── thm.ind /.gitignore: -------------------------------------------------------------------------------- 1 | *.out 2 | *.aux 3 | *.fls 4 | *.log 5 | *.fdb_latexmk 6 | *.synctex.gz 7 | *.ilg 8 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | MIT License 2 | 3 | Copyright (c) 2020 M Ahsan Al Mahir 4 | 5 | Permission is hereby granted, free of charge, to any person obtaining a copy 6 | of this software and associated documentation files (the "Software"), to deal 7 | in the Software without restriction, including without limitation the rights 8 | to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 9 | copies of the Software, and to permit persons to whom the Software is 10 | furnished to do so, subject to the following conditions: 11 | 12 | The above copyright notice and this permission notice shall be included in all 13 | copies or substantial portions of the Software. 14 | 15 | THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 16 | IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 17 | FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 18 | AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 19 | LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 20 | OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE 21 | SOFTWARE. 22 | -------------------------------------------------------------------------------- /MO_Problem_Journal.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/MO_Problem_Journal.pdf -------------------------------------------------------------------------------- /PSets/Geo_Asif_vai.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/PSets/Geo_Asif_vai.pdf -------------------------------------------------------------------------------- /PSets/NEMO.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/PSets/NEMO.pdf -------------------------------------------------------------------------------- /PSets/Special Online Camp 2020 Graph Theory.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/PSets/Special Online Camp 2020 Graph Theory.pdf -------------------------------------------------------------------------------- /PSets/igo_prep.tex: -------------------------------------------------------------------------------- 1 | \section*{IGO Preparation 2020} 2 | 3 | \prob{} 4 | {}{}{ 5 | Let $ABC$ be an arbitrary triangle,$P$ is the intersection point of the altitude from 6 | $C$ and the tangent line from $A$ to the circumcircle. The bisector of angle $A$ 7 | intersects $BC$ at $D$ . $PD$ intersects $AB$ at $K$, if $H$ is the orthocenter then 8 | prove : $HK\perp AD$ 9 | } 10 | 11 | \prob{} 12 | {}{}{ 13 | We have $4$ circles in plane such that any two of them are tangent to each other. we 14 | connect the tangency point of two circles to the tangency point of two other circles. 15 | Prove that these three lines are concurrent. 16 | } 17 | 18 | \prob{} 19 | {}{}{ 20 | In an acute triangle $ABC$, points $D,E,F$ are the feet of the altitudes from $A,B,C$, 21 | respectively. A line through $D$ parallel to $EF$ meets $AC$ at $Q$ and $AB$ at $R$. 22 | Lines $BC$ and $EF$ intersect at $P$. Prove that the circumcircle of triangle $PQR$ 23 | passes through the midpoint of $BC$. 24 | } 25 | 26 | 27 | \prob{} 28 | {}{}{ 29 | Given $\triangle ABC$ inscribed in $(O)$ an let $I$ and $I_a$ be it's incenter and 30 | $A$-excenter, respectively. 31 | 32 | Tangent lines to $(O)$ at $C,B$ intersect the angle bisector of $A$ at $M,N$, 33 | respectively. Second tangent lines through $M,N$ intersect $(O)$ at $X,Y$. 34 | 35 | Prove that $XYII_a$ is cyclic. 36 | } 37 | 38 | \prob{} 39 | {}{}{ 40 | Given triangle $ABC$, $D$ is the foot of the external angle bisector of $A$, $I$ its 41 | incenter and $I_a$ its $A$-excenter. Perpendicular from $I$ to $DI_a$ intersects the 42 | circumcircle of triangle in $A'$. Define $B'$ and $C'$ similarly. Prove that $AA',BB'$ 43 | and $CC'$ are concurrent. 44 | } 45 | -------------------------------------------------------------------------------- /PSets/jun16.tex: -------------------------------------------------------------------------------- 1 | \thispagestyle{empty} 2 | 3 | \begin{center} 4 | \LARGE math in the times of corona, exam 1\\[1em] 5 | \normalsize June 16, 2020 6 | \end{center} 7 | 8 | \vspace*{2em} 9 | 10 | \begin{problem_0}%[Romania TST 2013] 11 | Suppose that $a$ and $b$ are two distinct positive real numbers such that $\lfloor na\rfloor$ divides $\lfloor nb\rfloor$ for any positive integer $n$. Prove that $a$ and $b$ are positive integers. 12 | \end{problem_0} 13 | 14 | \begin{problem_0}%[2014 Iran TST P3] 15 | We call a $n*n$ table \emph{selfish} if we number the row and column with $0,1,2,3, \dots n-1$ (from left to right an from up to down), and for every $i,j\in\{0,1,2,...,n-1\}$ the value of cell $(i,j)$ is equal to the number of $i$ in row $j$. 16 | 17 | For example we have a \emph{selfish} table for $n=5$: 18 | \begin{center} 19 | \begin{tabular}{ccccc} 20 | 1 & 0 & 3 & 3 & 4 \\ 21 | 1 & 3 & 2 & 1 & 1 \\ 22 | 0 & 1 & 0 & 1 & 0 \\ 23 | 2 & 1 & 0 & 0 & 0 \\ 24 | 1 & 0 & 0 & 0 & 0 \\ 25 | \end{tabular} 26 | \end{center} 27 | Prove that for $n>5$ there is no \emph{selfish} table. 28 | \end{problem_0} 29 | 30 | \begin{problem_0}%[ARO 2016 11.5] 31 | Let $n$ be a positive integer and let $k_0,k_1, \dots,k_{2n}$ be nonzero integers such that $k_0+k_1 +\dots+k_{2n}\neq 0$. Is it always possible to find a permutation $(a_0,a_1,\dots,a_{2n})$ of $(k_0,k_1,\dots,k_{2n})$ so that the equation 32 | \[a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_0=0\] 33 | has no integer roots? 34 | \end{problem_0} 35 | 36 | \begin{problem_0}%[ISL 1998 C4] 37 | Let $U=\{1,2,\ldots ,n\}$, where $n\geq 3$. A subset $S$ of $U$ is said to be split by a permutation of the elements of $U$ if an element not in $S$ occurs in the permutation somewhere between two elements of $S$. 38 | 39 | For example, $ 13542 $ splits $\{1,2,3\}$ and $ \{1, 5\} $ but not $\{3,4,5\}$. 40 | 41 | Prove that for any $n-2$ subsets of $U$, each containing at least 2 and at most $n-1$ elements, there is an permutation of the elements of $U$ which splits all of them. 42 | \end{problem_0} 43 | 44 | -------------------------------------------------------------------------------- /PSets/pset_june_16.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/PSets/pset_june_16.pdf -------------------------------------------------------------------------------- /PSets/req_pset_1.tex: -------------------------------------------------------------------------------- 1 | \begin{center} 2 | \LARGE 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Olympiad Problems 2 | A journal of theorems, lemmas and problems for Mathematical Olympiads. 3 | 4 | At the moment of compiling, this journal has 682 randomly ordered problems, 210 theorems 5 | and lemmas, and 174 figures, mostly geometric, drawn in geogebra and inkscape. 6 | 7 | This journal is divided into chapter/section/subsection manner. Each subsection starts 8 | with a list of useful theorems of lemmas related to that topic, followed by problems and 9 | solutions or hints. At the end of the journal, there are four indices: Problems, Theorems, 10 | Definitios and Strategies, that alphabetically list all the items in here. You can use 11 | them to quickly search for a problem or a theorem in this journal. 12 | 13 | I intended this journal to be just a list of tricks when I began working on it. Over the years, this file has grown in size and has become massive. Instead of using it as a book, it is advised to use it as an extra large problem set and a resources file. 14 | 15 | To compile you need to have Computer Modern, Share and Titillium fonts, and the sty files 16 | can be found at https://github.com/AnglyPascal/sty 17 | -------------------------------------------------------------------------------- /alg/sec2_poly.log: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/alg/sec2_poly.log -------------------------------------------------------------------------------- /alg/sec3_ineq.log: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/alg/sec3_ineq.log -------------------------------------------------------------------------------- /alg/sec4_calculus.tex: -------------------------------------------------------------------------------- 1 | \newpage\section{Calculus} 2 | 3 | 4 | 5 | \lem{}{ \begin{align*} 6 | (f(x)+g(x))' = f'(x)+g'(x) \\ \\ 7 | (f(x)g(x)) = f'(x)g(x) + f(x)g'(x) \\ \\ 8 | (\frac{f(x)}{g(x)})' = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} \\ \\ 9 | (f(g(x)))' = f'(g(x)) + g'(x) 10 | \end{align*} } 11 | 12 | 13 | \lem{The Derivative of an Odd function is always an Even function, and vice versa.} 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | -------------------------------------------------------------------------------- /alg/sec6_tricks.tex: -------------------------------------------------------------------------------- 1 | \newpage\section{Tricks and Lemmas} 2 | 3 | \theo{https://en.wikipedia.org/wiki/Minkowski's_theorem}{Minkowski's theorem}{Any convex set in $\R^n$, which is symmetric with respect to the origin and with volume greater than $2^n d(L)$ contains a non-zero lattice point.} 4 | 5 | 6 | 7 | \subsection{Ad Hocs} 8 | 9 | \begin{enumerate} 10 | 11 | \item $x^2+1 = (x+i)(x-i)$ [USAMO 2014 P1] 12 | \item Add. Everything. Up. 13 | \item Send SHIT to the infinity. 14 | 15 | \end{enumerate} 16 | -------------------------------------------------------------------------------- /beamer/1.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/beamer/1.pdf -------------------------------------------------------------------------------- /beamer/Bijection.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/beamer/Bijection.pdf -------------------------------------------------------------------------------- /beamer/Kalpurush.ttf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/beamer/Kalpurush.ttf 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{2}{Orders}{19}{0}{2} 7 | \beamer@subsectionintoc {2}{1}{Why do we care?}{20}{0}{2} 8 | \beamer@subsectionintoc {2}{2}{So what is ``Order''?}{27}{0}{2} 9 | \beamer@subsectionintoc {2}{3}{Important Properties of Order}{32}{0}{2} 10 | \beamer@subsectionintoc {2}{4}{Usage in Problems}{50}{0}{2} 11 | \beamer@subsectionintoc {2}{5}{Something more fundamental}{83}{0}{2} 12 | \beamer@subsectionintoc {2}{6}{Conclusion}{94}{0}{2} 13 | -------------------------------------------------------------------------------- /beamer/orders.xdv: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/beamer/orders.xdv -------------------------------------------------------------------------------- /beamer/problem_1.pdf: -------------------------------------------------------------------------------- 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/camp/beamerthemeNord.vrb: -------------------------------------------------------------------------------- 1 | \frametitle{Recommended Free Fonts} 2 | \begin{description}[Selected Fonts] 3 | \item[Selected Fonts] recommended for this theme\\ 4 | \begin{lstlisting}[basicstyle = \ttfamily\small] 5 | \setmainfont{Yanone Kaffeesatz} 6 | \setsansfont{Andika New Basic} 7 | \setmonofont{Share Tech Mono} 8 | \end{lstlisting} 9 | \item[Download] {\small \url{https://www.fontsquirrel.com/}} 10 | \item[Install Fonts] {\small \url{https://www.google.com/get/noto/help/install/}} 11 | \item[Compilation] compile with \XeLaTeX~to use system-wide fonts 12 | \end{description} 13 | 14 | -------------------------------------------------------------------------------- /camp/beamerthemeNord.xdv: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/camp/beamerthemeNord.xdv 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12 | \begin{problem} 13 | $ 2N $ students take a quiz in which the possible scores are $ 0, 1\dots 10 $. It is given that each of these scores appeared at least once, and the average of their scores is $ 7.4 $. Prove that the students can be divided into two sets of $ N $ student with both sets having an average score of $ 7.4 $. 14 | \end{problem} 15 | 16 | \bigskip 17 | 18 | 19 | 20 | \begin{problem} 21 | Triangle $ABC$ circumscribed $(O)$ has $A$-excircle $(J)$ that touches $AB,\ BC,\ AC$ at $F,\ D,\ E$, resp. 22 | 23 | \begin{enumerate} 24 | \item $L$ is the midpoint of $BC$. Circle with diameter $LJ$ cuts $DE,\ DF$ at $K,\ H$. Prove that $(BDK),\ (CDH)$ has an intersecting point on $(J)$. 25 | \item Let $EF\cap BC =\{G\}$ and $GJ$ cuts $AB,\ AC$ at $M,\ N$, resp. $P\in JB$ and $Q\in JC$ such that 26 | $$\angle PAB=\angle QAC=90{}^\circ .$$$PM\cap QN=\{T\}$ and $S$ is the midpoint of the larger $BC$-arc of $(O)$. $(I)$ is the incircle of $ABC$. Prove that $SI\cap AT\in (O)$. 27 | \end{enumerate} 28 | \end{problem} 29 | 30 | \bigskip 31 | 32 | 33 | 34 | \begin{problem} 35 | Let $p_n$ be the $n^{\mbox{th}}$ prime counting from the smallest prime $2$ in increasing order. For example, $p_1=2, p_2=3, p_3 =5, \cdots$ 36 | 37 | \begin{enumerate} 38 | \item For a given $n \ge 10$, let $r$ be the smallest integer satisfying 39 | \[2\le r \le n-2, \quad n-r+1 < p_r\] 40 | and define $N_s=(sp_1p_2\cdots p_{r-1})-1$ for $s=1,2,\ldots, p_r$. Prove that there exists $j, 1\le j \le p_r$, such that none of $p_1,p_2,\cdots, p_n$ divides $N_j$. 41 | 42 | \item Using the result of (3.1), find all positive integers $m$ for which 43 | \[p_{m+1}^2 < p_1p_2\cdots p_m\] 44 | \end{enumerate} 45 | \end{problem} 46 | 47 | \end{document} -------------------------------------------------------------------------------- /camp/nemo_2.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | \RequirePackage[left=4cm, right=4cm, top=2.5cm, bottom=2cm, headheight=4pt, headsep=20pt]{geometry} 3 | 4 | \newtheorem{problem}{Problem} 5 | 6 | \title{\Huge{No Extra Math Olympiad}} 7 | 8 | \begin{document} 9 | \Large 10 | \maketitle 11 | 12 | \begin{problem} 13 | Are there integers $ a, b $ such that both $ a^5b+3 $ and $ ab^5 +3 $ are perfect cubes of integers. 14 | \end{problem} 15 | 16 | \begin{problem} 17 | Consider the following sequence $ 1, 3, 5, 7, 9\dots $ of all positive integers consisting only of odd digits. Find the $ 2017^{th} $ term of the above sequence. 18 | \end{problem} 19 | 20 | \begin{problem} 21 | Five distinct points $ A, B, C, D $ and $ E $ lie in this order on a circle of radius $ r $ and satisfy $ AC=BD=CE=r $. Prove that the orthocenters of the triangles $ ACD, BCD $ and $ BCE $ are the vertices of a right angled triangle. 22 | \end{problem} 23 | 24 | \end{document} -------------------------------------------------------------------------------- /cat.idx: -------------------------------------------------------------------------------- 1 | \indexentry{Sets!Max-set with property!USA TST 2005 P1|hyperpage}{11} 2 | \indexentry{Sets!Local!Iran TST 2008 D3P1|hyperpage}{11} 3 | \indexentry{Sets!Local!Romanian TST 2016 D1P2|hyperpage}{11} 4 | \indexentry{Sets!Subsets of $[N]$!American Mathematical Monthly E2309|hyperpage}{12} 5 | \indexentry{Sets!Subsets of $[N]$!CGMO 2010 P1|hyperpage}{12} 6 | \indexentry{Sets!Local!ISL 2002 C5|hyperpage}{12} 7 | \indexentry{Sets!Subsets of $[N]$!ISL 1988 P10|hyperpage}{12} 8 | \indexentry{Sets!Max-set with property!Iran TST 2013 D1P2|hyperpage}{13} 9 | \indexentry{Permutations!USAMO 2017 P5|hyperpage}{120} 10 | \indexentry{Permutations!ISL 2008 C2|hyperpage}{120} 11 | -------------------------------------------------------------------------------- /cat.ind: -------------------------------------------------------------------------------- 1 | \begin{theindex} 2 | 3 | \large\Titillium% 4 | \texttt{P}\par\nopagebreak 5 | 6 | \item \small Permutations 7 | \subitem ISL 2008 C2 \dotfill \hyperpage{120} 8 | \subitem USAMO 2017 P5 \dotfill \hyperpage{120} 9 | 10 | \indexspace 11 | 12 | \large\Titillium% 13 | \texttt{S}\par\nopagebreak 14 | 15 | \item \small Sets 16 | \subitem Local 17 | \subsubitem Iran TST 2008 D3P1 \dotfill \hyperpage{11} 18 | \subsubitem ISL 2002 C5 \dotfill \hyperpage{12} 19 | \subsubitem Romanian TST 2016 D1P2 \dotfill \hyperpage{11} 20 | \subitem Max-set with property 21 | \subsubitem Iran TST 2013 D1P2 \dotfill \hyperpage{13} 22 | \subsubitem USA TST 2005 P1 \dotfill \hyperpage{11} 23 | \subitem Subsets of $[N]$ 24 | \subsubitem American Mathematical Monthly E2309 \dotfill 25 | \hyperpage{12} 26 | \subsubitem CGMO 2010 P1 \dotfill \hyperpage{12} 27 | \subsubitem ISL 1988 P10 \dotfill \hyperpage{12} 28 | 29 | \end{theindex} 30 | -------------------------------------------------------------------------------- /categories.tex: -------------------------------------------------------------------------------- 1 | Permutations 2 | -------------------------------------------------------------------------------- /combi/Game Theory.zip: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/combi/Game Theory.zip -------------------------------------------------------------------------------- /combi/Game Theory/1_files/arrow1.gif: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/combi/Game Theory/1_files/arrow1.gif 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-------------------------------------------------------------------------------- /combi/sec8_1_Recursion.tex: -------------------------------------------------------------------------------- 1 | \newpage \subsection{Recurrent Sequences} 2 | 3 | 4 | \begin{myitemize} 5 | \item \href{https://ufile.io/ywbniil2}{WOOT 2010-11 Recursion} 6 | \end{myitemize} 7 | 8 | 9 | \begin{BoxedTheorem}{Sum of Geometric Sequences}{}\label{theorem:Sum of Geometric Sequences} 10 | Every recurrent sequence can be written as a sum of some geometric sequences. Given a recurrent sequence, 11 | \[x_n = a_1x_{n-1} + a_2x_{n-2} +\dots + a_kx_{n-k}\] 12 | Then $ x_n $ can be written as 13 | \[x_n = c_1r_1^n + c_2r_2^n +\dots + c_lr_l^n\] 14 | For all $ c_i $ if $ r_i $ are the roots of the \emph{characteristic polynomial} of the recursion. Which is: 15 | \[\tcboxmath[colback=white, colframe=white]{x^k - a_1x^{k-1} - a_2x^{k-2} \dots - a_k = 0}\] 16 | If there are double roots, say $ r_1 = r_2 = r_3 $, then we instead have, 17 | \[x_n = \tcboxmath[colback=white, colframe=white]{c_1r_1^n + c_2n\ r_2^n + c_3n^2\ r^n} \dots + c_lr_l^n\] 18 | 19 | Reversely, we can say that a sequence defined by a sum of geometric recurrent series is a recursion. 20 | \end{BoxedTheorem} 21 | 22 | \lem{}{Let $ F_n $ be the $ n $th Fibonacci number. Then the following holds: 23 | \[F_{n}^2 + F_{n+1}^2 = F_{2n+1}\]} 24 | 25 | \proof{Expanding the general form of the terms, and showing that $ a_n = F_{n}^2 + F_{n+1}^2 - F_{2n+1} $ is a recursion by \autoref{theorem:Sum of Geometric Sequences}.} 26 | 27 | \begin{BoxedTheorem}[title=Repertoire Method]{}{} 28 | Given a recurrent function defined by 29 | \[f(n) = A(n)\a + B(n)\beta + C(n)\gamma\] 30 | We plug in different values for $ f(n) $, for example, $ f(n) = 1, n, 2n $ etc. for which the values are known from the recursion, and then solve for $ A, B, C $. 31 | \end{BoxedTheorem} -------------------------------------------------------------------------------- /count_figs.sty: -------------------------------------------------------------------------------- 1 | \counterwithout{figure}{chapter} 2 | \newcounter{totfigures} 3 | \newcounter{tottables} 4 | 5 | \providecommand\totfig{} 6 | \providecommand\tottab{} 7 | 8 | \makeatletter 9 | \AtEndDocument{% 10 | \addtocounter{totfigures}{\value{figure}}% 11 | \addtocounter{tottables}{\value{table}}% 12 | \immediate\write\@mainaux{% 13 | \string\gdef\string\totfig{\number\value{totfigures}}% 14 | \string\gdef\string\tottab{\number\value{tottables}}% 15 | }% 16 | } 17 | \makeatother 18 | 19 | \pretocmd{\chapter}{\addtocounter{totfigures}{\value{figure}}\setcounter{figure}{0}}{}{} 20 | \pretocmd{\chapter}{\addtocounter{tottables}{\value{table}}\setcounter{table}{0}}{}{} 21 | 22 | % There are \totfig\ figures and \tottab\ tables in this document. 23 | -------------------------------------------------------------------------------- /def.idx: -------------------------------------------------------------------------------- 1 | \indexentry{Cut|hyperpage}{11} 2 | -------------------------------------------------------------------------------- /def.ind: -------------------------------------------------------------------------------- 1 | \begin{theindex} 2 | 3 | \item Cut, \hyperpage{11} 4 | 5 | \end{theindex} 6 | -------------------------------------------------------------------------------- /dump/APMO_2015_P3.tex: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/dump/APMO_2015_P3.tex -------------------------------------------------------------------------------- /dump/ISL2009G6.tex: -------------------------------------------------------------------------------- 1 | \section{haha} 2 | 3 | \lem{}{Let two rays $ h, i $ from point $ P $ and a fixed line $ l $. $ H_2 $ on $ h $ and $ E_2 $ be another point such that $ E_2\in i $ and $ H_2E_2\parallel l$. $ H_1 $ be another point on the plain. $ Z $ is a point on $ H_1H_2 $ such that $ E_2Z\parallel PH_1 $. Prove that as $ H_2 $ moves on $ h $, $ Z $ moves on a line parallel to $ h $} 4 | 5 | \proof{Let $ N\in PH_1 $ such that $ ZN\parallel h $. We will prove that as $ H_2 $ moves, $ \dfrac{PN}{NZ} $ stays constant. 6 | 7 | \figdf{.5}{ISL2009G6_lemma}{} 8 | 9 | If we take $ M = E_2Z\cap h $, we see that, 10 | \[\frac{PN}{NH_1}=\frac{H_2Z}{ZH_1}=\frac{H_2M}{MP}\] 11 | which is constant since $ PH_1 $ and $ l $ is constant. \hfill \qed} 12 | 13 | 14 | \bigskip 15 | 16 | 17 | Now looking at our main problem, we see that if we move $ C $ and $ D $ on $ H_1A $ and $ H_1B $, keeping $ CD $ parallel to a fixed line, we see that if for some $ C $ the problem is true, then it's true for all other $ C $. Because, moving $ C $, moves $ H_2 $, $ E_2 $ along two fixed lines going though $ P $, and the perpendicular from $ E_2 $ to $ AB $ is parallel to $ PH_1 $. So the intersection of the perpendicular from $ E_2 $ to $ AB $ with $ H_1, H_2 $ lies on a line perpendicular to $ CD $. 18 | 19 | \bigskip 20 | 21 | 22 | So we are done if we can show the result for $ C=A $. 23 | 24 | 25 | 26 | 27 | -------------------------------------------------------------------------------- /dump/document.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/dump/document.pdf -------------------------------------------------------------------------------- /dump/isl2004n1.tex: -------------------------------------------------------------------------------- 1 | if $p\not|n$, $\tau(pn)=\tau(p)\tau(n)= 2\tau(n)=n$ which is true for $n$ 2 | 3 | let's try $p^k$, and $p^t||n$, 4 | \[\tau(p^{k+t}m)=(k+t+1)\tau(m)=p^tm\] 5 | 6 | for $k+1 < p$, if $t=0$, we don't really have an obvious contradiction. 7 | 8 | So what if $k+1=p$, then, $t=0, 1$ 9 | 10 | if $t=0$, $\ p\tau(m)=m$ but $(p, m)=1$ 11 | 12 | and if $t=1$, \[(p+1)\tau(m)=pm\] but trivial bounding shows that $m<=4$, for which there is no solution by case check 13 | 14 | -------------------------------------------------------------------------------- /dump/isl2004n4.tex: -------------------------------------------------------------------------------- 1 | Let's play with some integer $ n $ and Fibonacci sequences $ \mod\ n $. What should we take the value of $ n $? As $ 11=16-5 $, let's take it first. We see that the period of Fibonacci sequences $ \mod\ 11 $ is at most $ 10 $. From here it is natural to make a conjecture that for prime $ n $'s, the period probably is $ n-1 $. We also design a proof that there is a sequence which doesn't contain any of $ n $'s products. \\ 2 | 3 | So let's see if it works for all primes. No it doesn't, breaks at $ 7 $. How is $ 7 $ so different than $ 11 $? The most straightforward guess is that probably $ 7\not| 4k^2-5 $ for any $ k $. And it is true. So what's so special about the primes dividing $ 4k^2-5 $? 4 | Writing it in modular arithmetic manner, $ 4k^2\equiv 5\ (\mod\ p) $. Wait, $ 5 $ is a quadratic residue $ \mod\ p $? But isn't $ \sqrt{5} $ related to Fibonacci sequences? What's the general formula for a Fibonacci sequence starting with $ a, b $? Wait, now that explains why the period of the Fibonacci sequences $ \mod $ these primes is $ p-1 $. -------------------------------------------------------------------------------- /ext_doc.tex: -------------------------------------------------------------------------------- 1 | \externaldocument{combi/sec1_tricks} 2 | \externaldocument{combi/sec2_binom_iden} 3 | \externaldocument{combi/sec3_sets} 4 | \externaldocument{combi/sec4_algo} 5 | \externaldocument{combi/sec5_graph} 6 | \externaldocument{combi/sec6_game} 7 | \externaldocument{combi/sec7_combi_geo} 8 | \externaldocument{combi/sec8_sequences} 9 | \externaldocument{combi/sec9_configs} 10 | \externaldocument{combi/sec10_not_sorted} 11 | 12 | \externaldocument{alg/sec1_fe} 13 | \externaldocument{alg/sec2_poly} 14 | \externaldocument{alg/sec3_ineq} 15 | \externaldocument{alg/sec4_calculus} 16 | \externaldocument{alg/sec5_ad_hoc} 17 | \externaldocument{alg/sec6_tricks} 18 | 19 | \externaldocument{geo/sec1_circum-9p-circle} 20 | \externaldocument{geo/sec2_cevian} 21 | \externaldocument{geo/sec3_in-ex-circle} 22 | \externaldocument{geo/sec4_conjugates} 23 | \externaldocument{geo/sec5_mixtilinear} 24 | \externaldocument{geo/sec6_circle} 25 | \externaldocument{geo/sec7_comp-quad} 26 | \externaldocument{geo/sec8_projective} 27 | \externaldocument{geo/sec9_parallel} 28 | \externaldocument{geo/sec10_length} 29 | \externaldocument{geo/sec11_pedal} 30 | \externaldocument{geo/sec12_pending} 31 | \externaldocument{geo/sec13_problems} 32 | \externaldocument{geo/sec14_research4later} 33 | 34 | \externaldocument{nt/NUMBER_THEORY} 35 | \externaldocument{nt/sec1_orders_mod_prime} 36 | \externaldocument{nt/sec2_primes} 37 | \externaldocument{nt/sec3_fe_poly} 38 | \externaldocument{nt/sec4_dioph_eq} 39 | \externaldocument{nt/sec0_unsorted} 40 | 41 | \externaldocument{geo/ISL_Geo} 42 | \externaldocument{alg/ISL_Alg} -------------------------------------------------------------------------------- /for_sharing.tex: -------------------------------------------------------------------------------- 1 | \section{Problems to try before some fixed day} 2 | 3 | \prob{} 4 | {Thue's Note}{}{ 5 | Let $S$ be a set of all positive integers which can be represented as 6 | $a^2+ 5b^2$ for some coprime integers $a,b$. Let $p$ be a prime number 7 | such that $p= 4n+ 3$ for some integer $n$. Show that if for some positive 8 | integer $k$ the numberk $p$ is in $S$, then $2p$ is in $S$ as well. 9 | } 10 | 11 | Konig's theorem, Dilworth theorem, Max flow Min cut 12 | -------------------------------------------------------------------------------- /geo/sec11_pedal.tex: -------------------------------------------------------------------------------- 1 | \newpage\section{Pedal Triangles} 2 | 3 | 4 | \den{Pedal Triangles}{Let $ P $ be an arbitrary point, let $ \triangle A_1B_1C_1 $ be its pedal triangle wrt $ \triangle ABC $. Let $ A', B', C' $ and $ A_0, B_0, C_0 $ be the feet of the altitudes and the midpoints of $ \triangle ABC $. \\ 5 | 6 | \[ B_1C_1\cap B_0C_0=A_2,\ C_1A_1\cap C_0A_0 = B_2,\ A_1B_1\cap A_0B_0 = C_2 \] 7 | \[ B'C'\cap B_0C_0=A_3,\ C'A'\cap C_0A_0=B_3,\ A'B'\cap A_0B_0=C_3 \]} 8 | 9 | 10 | 11 | 12 | \theo{https://www.awesomemath.org/wp-pdf-files/math-reflections/mr-2015-02/article_1_bocanu.pdf}{Fontene's First Theorem}{$ A_1A_2, B_1B_2, C_1C_2 $ are concurrent at the intersection of $ \odot A_1B_1C_1 $ and $ \odot A_0B_0C_0 $} 13 | 14 | 15 | 16 | 17 | \lem{}{$ A'A_3, B'B_3, C'C_3 $ and $ A_0A_3, B_0B_3, C_0C_3 $ concur at the nine point circle of $ \triangle ABC. $} 18 | 19 | 20 | 21 | \theo{https://www.awesomemath.org/wp-pdf-files/math-reflections/mr-2015-02/article_1_bocanu.pdf}{Fontene's Second Theorem}{Let the concurrency point in the first theorem be $ Q $. Then, if the line $ OP $ is fixed and $ P $ moves along that line, $ Q $ will stay fixed.} 22 | 23 | The previous result leads to another beautiful result: 24 | 25 | 26 | 27 | 28 | 29 | \lem{}{Suppose a varying point $ P $ is chosen on the Euler Line of $ \triangle ABC $. Then the pedal circle of $ P $ wrt $ \triangle ABC $ intersects the 9p circle at a fixed point which is the Euler Reflection Point of the median triangle.} 30 | 31 | 32 | 33 | 34 | 35 | -------------------------------------------------------------------------------- /geo/sec14_research4later.tex: -------------------------------------------------------------------------------- 1 | \newpage\section{Research Stuffs for later} 2 | 3 | 4 | 5 | 6 | 7 | \prob{https://artofproblemsolving.com/community/c6h1499703p8903397}{AoPS}{}{Let $ABC$ be a triangle with incenter $I.$ $L_a,$ $L_b,$ $L_c$ are symmedian points of triangles $IBC,$ $ICA,$ $IAB.$ Let $X,Y,Z$ be the reflections of $I$ through $L_a,$ $L_b,$ $L_c.$ 8 | 9 | \begin{itemize} 10 | 11 | \item Prove that $AX,$ $BY,$ $CZ$ and $OI$ are concurrent. 12 | 13 | \item Let $I_a,$ $I_b,$ $I_c$ be the excenters of $ABC.$ Prove that $I_aX,$ $I_bY,$ $I_cZ$ are concurrent at a point $P$ and isogonal conjugate of $P$ with respect to triangle $I_aI_bI_c$ lies on Euler line of $ABC.$ 14 | 15 | \end{itemize}} 16 | 17 | 18 | \prob{https://artofproblemsolving.com/community/c6h1180223}{buratinogigle Tough P1}{}{Let $ABC$ be a triangle inscribed in circle $(O)$ with $A$-excircle $(J)$. Circle passing through $A,B$ touches $(J)$ at $M$. Circle passing through $A,C$ touches $(J)$ at $N$. $BM$ cuts $CN$ at $P$. Prove that $AP$ passes through tangent point of $A$-mixtilinear incircle with $(O)$.} 19 | 20 | 21 | 22 | 23 | 24 | -------------------------------------------------------------------------------- /geo/sec9_parallel.tex: -------------------------------------------------------------------------------- 1 | \newpage\section{Parallelogram Stuff} 2 | 3 | 4 | \theo{http://forumgeom.fau.edu/FG2005volume5/FG200510.pdf}{Maximality of the Area of a Cyclic Quadrilateral}{Among all quadrilaterals with given side lengths, the cyclic one has maximal area. 5 | \fig{.5}{quad-same_sides-diff_area}{The cyclic quad has the maximal area} 6 | } 7 | 8 | 9 | 10 | \prob{https://artofproblemsolving.com/community/c6h1508225_simple_parallelogram_geo}{IOM 2017 P1}{E}{Let $ABCD$ be a parallelogram in which angle at $B$ is obtuse and $AD>AB$. Points $K$ and $L$ on $AC$ such that $\angle ADL=\angle KBA$(the points $A, K, C, L$ are all different, with $K$ between $A$ and $L$). The line $BK$ intersects the circumcircle $\omega$ of $ABC$ at points $B$ and $E$, and the line $EL$ intersects $\omega$ at points $E$ and $F$. Prove that $BF\parallel AC$.} 11 | 12 | \hl{Simplify}: Make the diagram easier to draw. 13 | 14 | 15 | \prob{https://artofproblemsolving.com/community/c6h148830p841255}{USA TST 2006 P6}{E}{Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of triangle $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle{BAP}= \angle{CAQ}$. Segments $BQ$ and $CP$ meet at $R$. Let $O$ be the circumcenter of triangle $BCR$. Prove that $AO \perp PQ.$} 16 | 17 | \figdf{.5}{USATST2006P6}{USA TST 2006 P6, That is a prallelogram} -------------------------------------------------------------------------------- /indStyle.ist: -------------------------------------------------------------------------------- 1 | headings_flag 1 2 | 3 | heading_prefix "\n\\large\\Titillium% 4 | \\noindent\\texttt{"heading_suffix "}\\par\\nopagebreak\n" 5 | 6 | item_0 "\n \\item \\small " 7 | 8 | delim_0 " \\dotfill " 9 | delim_1 " \\dotfill " 10 | delim_2 " \\dotfill " -------------------------------------------------------------------------------- /index.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/index.pdf -------------------------------------------------------------------------------- /main.idx: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/main.idx -------------------------------------------------------------------------------- /main.ind: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/main.ind -------------------------------------------------------------------------------- /main.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/main.pdf -------------------------------------------------------------------------------- /main.xdv: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/AnglyPascal/MO-Problem-Journal/83ff9b542999386ea182863e4f25f0b488d3984f/main.xdv -------------------------------------------------------------------------------- /nt/sec4_dioph_eq.tex: -------------------------------------------------------------------------------- 1 | \section{Diophantine Equations} 2 | 3 | 4 | \prob{https://artofproblemsolving.com/community/c6h79786p456605}{APMO 1999 P4}{EM}{Determine all pairs $(a,b)$ of integers with the property that the numbers $a^2+4b$ and $b^2+4a$ are both perfect squares.} 5 | 6 | \solu{Easy case work assuming positive or negative values for $ a, b $.} 7 | 8 | 9 | 10 | \prob{https://artofproblemsolving.com/community/c6t177f6h1437607_infinitely_many_pairs}{All Squares}{E}{Prove that there are infinitely many pairs of positive integers $(x, y)$ satisfying that $x+y, x-y, xy+1$ are all perfect squares.}\label{problem:construction_1} 11 | 12 | \solu{Just work it out.} 13 | 14 | 15 | 16 | \prob{https://artofproblemsolving.com/community/c6h418641p2362006}{ISL 2010 N3}{E}{Find the smallest number $n$ such that there exist polynomials $f_1, f_2, \ldots , f_n$ with rational coefficients satisfying \[x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.\]} 17 | 18 | \solu{Find the obvious answer, which is very small, so we can probably case work it out. We find the case for $ 3 $ a bit challenging. But we plan to show that $ 7a^2 $ can't be written as a sum of $ 3 $ squares. Which numbers can be written as the sum of $ 3 $ squares? Investigate...} 19 | 20 | 21 | \prob{https://artofproblemsolving.com/community/c6h1113199p5083571}{ISL 2014 N5}{E}{Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.} 22 | 23 | \solu{The old school trick, replacement. Assume $ x