$hide=mobile

[Solutions] Sharygin Geometry Mathematical Olympiad 2015 (Final Round)

Grade 8

  1. In trapezoid $ABCD$ angles $A$ and $B$ are right, $AB = AD$, $CD = BC + AD$, $BC < AD$. Prove that $\widehat{ADC} = 2\widehat{ABE}$, where $E$ is the midpoint of segment $D$.
  2. A circle passing through $A$, $B$ and the orthocenter of triangle $ABC$ meets sides $AC$, $BC$ at their inner points. Prove that $60^0 < \widehat{C} < 90^0$.
  3. In triangle $ABC$ we have $AB = BC$, $\widehat{B} = 20^0$. Point $M$ on $AC$ is such that $AM : MC = 1 : 2$, point $H$ is the projection of $C$ to $BM$. Find angle $\widehat{AHB}$.
  4. Prove that an arbitrary convex quadrilateral can be divided into five polygons having symmetry axes.
  5. Two equal hard triangles are given. One of their angles is equal to $\alpha$ (these angles are marked). Dispose these triangles on the plane in such a way that the angle formed by some three vertices would be equal to $\alpha/2$. (No instruments are allowed, even a pencil.)
  6. Lines $b$ and $c$ passing through vertices $B$ and $C$ of triangle $ABC$ are perpendicular to sideline $BC$. The perpendicular bisectors to $AC$ and $AB$ meet $b$ and $c$ at points $P$ and $Q$ respectively. Prove that line $PQ$ is perpendicular to median $AM$ of triangle $ABC$.
  7. Point $M$ on side $AB$ of quadrilateral $ABCD$ is such that quadrilaterals $AMCD$ and $BMDC$ are circumscribed around circles centered at $O_1$ and $O_2$ respectively. Line $O_1O_2$ cuts an isosceles triangle with vertex $M$ from angle $CMD$. Prove that $ABCD$ is a cyclic quadrilateral.
  8. Points $C_1$, $B_1$ on sides $AB$, $AC$ respectively of triangle $ABC$ are such that $BB_1 \perp CC_1$. Point $X$ lying inside the triangle is such that $\widehat{XBC} = \widehat{B_1BA}$, $\widehat{XCB} = \widehat{C_1CA}$. Prove that $\widehat{B_1XC_1} = 90^0 − \widehat{A}$.

Grade 9

  1.  Circles $\alpha$ and $\beta$ pass through point $C$. The tangent to $\alpha$ at this point meets $\beta$ at point $B$, and the tangent to $\beta$ at $C$ meets $\alpha$ at point $A$ so that $A$ and $B$ are distinct from $C$ and angle $ACB$ is obtuse. Line $AB$ meets $\alpha$ and $\beta$ for the second time at points $N$ and $M$ respectively. Prove that $2MN < AB$.
  2. A convex quadrilateral is given. Using a compass and a ruler construct a point such that its projections to the sidelines of this quadrilateral are the vertices of a parallelogram.
  3. Let $100$ discs lie on the plane in such a way that each two of them have a common point. Prove that there exists a point lying inside at least $15$ of these discs.
  4. A fixed triangle $ABC$ is given. Point $P$ moves on its circumcircle so that segments $BC$ and $AP$ intersect. Line $AP$ divides triangle $BPC$ into two triangles with incenters $I_1$ and $I_2$. Line $I_1I_2$ meets $BC$ at point $Z$. Prove that all lines $ZP$ pass through a fixed point.
  5. Let $BM$ be a median of nonisosceles right-angled triangle $ABC$ ($\widehat B = 90^0$), and $H_a$, $H_c$ be the orthocenters of triangles $ABM$, $CBM$ respectively. Prove that lines $AH_c$ and $CH_a$ meet on the medial line of triangle $ABC$.
  6. The diagonals of convex quadrilateral $ABCD$ are perpendicular. Points $A_0$, $B_0$, $C_0$, $D_0$ are the circumcenters of triangles $ABD$, $BCA$, $CDB$, $DAC$ respectively. Prove that lines $AA_0$, $BB_0$, $CC_0$, $DD_0$ concur.
  7. Let $ABC$ be an acute-angled, nonisosceles triangle. Altitudes $AA_0$ and $BB_0$ meet at point $H$, and the medians of triangle $AHB$ meet at point $M$. Line $CM$ bisects segment $A_0B_0$. Find angle $C$.
  8. A perpendicular bisector to side $BC$ of triangle $ABC$ meets lines $AB$ and $AC$ at points $A_B$ and $A_C$ respectively. Let $O_a$ be the circumcenter of triangle $AA_BA_C$. Points $O_b$ and $O_c$ are defined similarly. Prove that the circumcircle of triangle $O_aO_bO_c$ touches the circumcircle of the original triangle.

Grade 10

  1. Let $K$ be an arbitrary point on side $BC$ of triangle $ABC$, and $KN$ be a bisector of triangle $AKC$. Lines $BN$ and $AK$ meet at point $F$, and lines $CF$ and $AB$ meet at point $D$. Prove that $KD$ is a bisector of triangle $AKB$.
  2. Prove that an arbitrary triangle with area $1$ can be covered by an isosceles triangle with area less than \sqrt{2}.
  3. Let $A_1$, $B_1$ and $C_1$ be the midpoints of sides $BC$, $CA$ and $AB$ of triangle $ABC$. Points $B_2$ and $C_2$ are the midpoints of segments $BA_1$ and $CA_1$ respectively. Point $B_3$ is symmetric to $C_1$ wrt $B$, and $C_3$ is symmetric to $B_1$ wrt $C$. Prove that one of common points of circles $BB_2B_3$ and $CC_2C_3$ lies on the circumcircle of triangle $ABC$.
  4. Let $AA_1$, $BB_1$, $CC_1$ be the altitudes of an acute-angled, nonisosceles triangle $ABC$, and $A_2$, $B_2$, $C_2$ be the touching points of sides $BC$, $CA$, $AB$ with the correspondent excircles. It is known that line $B_1C_1$ touches the incircle of $ABC$. Prove that $A_1$ lies on the circumcircle of $A_2B_2C_2$.
  5. Let $BM$ be a median of right-angled nonisosceles triangle $ABC$ ($\widehat B = 90^0$), and $H_a$, $H_c$ be the orthocenters of triangles $ABM$, $CBM$ respectively. Lines $AH_c$ and $CH_a$ meet at point $K$. Prove that $\widehat{MBK}$ = 90^0$.
  6. Let $H$ and $O$ be the orthocenter and the circumcenter of triangle $ABC$. The circumcircle of triangle $AOH$ meets the perpendicular bisector to $BC$ at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that lines $AA_1$, $BB_1$ and $CC_1$ concur.
  7. Let $SABCD$ be an inscribed pyramid, and $AA_1$, $BB_1$, $CC_1$, $DD_1$ be the perpendiculars from $A$, $B$, $C$, $D$ to lines $SC$, $SD$, $SA$, $SB$ respectively. Points $S$, $A_1$, $B_1$, $C_1$, $D_1$ are distinct and lie on a sphere. Prove that points $A_1$, $B_1$, $C_1$ and $D_1$ are complanar.
  8. Does there exist a rectangle which can be divided into a regular hexagon with sidelength $1$ and several equal right-angled triangles with legs $1$ and $\sqrt 3$?

Post a Comment


$hide=home

$type=three$count=6$sr=random$t=oot$h=1$l=0$meta=hide$rm=hide$sn=0

$hide=post$type=three$count=6$sr=random$t=oot$h=1$l=0$meta=hide$rm=hide$sn=0

$hide=home

Kỷ Yếu$cl=violet$type=three$count=6$sr=random$t=oot$h=1$l=0$meta=hide$rm=hide$sn=0

Journals$cl=green$type=three$count=6$sr=random$t=oot$h=1$l=0$meta=hide$rm=hide$sn=0

Name

Ả-rập Xê-út,1,Abel,5,Albania,2,AMM,2,Amsterdam,5,Ấn Độ,1,An Giang,22,Andrew Wiles,1,Anh,2,Áo,1,APMO,19,Ba Đình,2,Ba Lan,1,Bà Rịa Vũng Tàu,52,Bắc Giang,49,Bắc Kạn,1,Bạc Liêu,9,Bắc Ninh,47,Bắc Trung Bộ,7,Bài Toán Hay,5,Balkan,37,Baltic Way,30,BAMO,1,Bất Đẳng Thức,66,Bến Tre,46,Benelux,13,Bình Định,44,Bình Dương,21,Bình Phước,38,Bình Thuận,34,Birch,1,Booklet,11,Bosnia Herzegovina,3,BoxMath,3,Brazil,2,Bùi Đắc Hiên,1,Bùi Thị Thiện Mỹ,1,Bùi Văn Tuyên,1,Bùi Xuân Diệu,1,Bulgaria,5,Buôn Ma Thuột,1,BxMO,12,Cà Mau,13,Cần Thơ,14,Canada,39,Cao Bằng,6,Cao Quang Minh,1,Câu Chuyện Toán Học,36,Caucasus,2,CGMO,10,China,10,Chọn Đội Tuyển,347,Chu Tuấn Anh,1,Chuyên Đề,124,Chuyên Sư Phạm,31,Chuyên Trần Hưng Đạo,3,Collection,8,College Mathematic,1,Concours,1,Cono Sur,1,Contest,610,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,25,Đà Nẵng,39,Đa Thức,2,Đại Số,20,Đắk Lắk,54,Đắk Nông,7,Đan Phượng,1,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi HSG,1643,Đề Thi JMO,1,Điện Biên,8,Định Lý,1,Định Lý Beaty,1,Đỗ Hữu Đức Thịnh,1,Do Thái,3,Doãn Quang Tiến,4,Đoàn Quỳnh,1,Đoàn Văn Trung,1,Đống Đa,4,Đồng Nai,49,Đồng Tháp,51,Du Hiền Vinh,1,Đức,1,Duyên Hải Bắc Bộ,25,E-Book,33,EGMO,16,ELMO,19,EMC,8,Epsilon,1,Estonian,5,Euler,1,Evan Chen,1,Fermat,3,Finland,4,Forum Of Geometry,2,Furstenberg,1,G. Polya,3,Gặp Gỡ Toán Học,26,Gauss,1,GDTX,3,Geometry,12,Gia Lai,25,Gia Viễn,2,Giải Tích Hàm,1,Giảng Võ,1,Giới hạn,2,Goldbach,1,Hà Giang,2,Hà Lan,1,Hà Nam,29,Hà Nội,231,Hà Tĩnh,72,Hà Trung Kiên,1,Hải Dương,49,Hải Phòng,42,Hàn Quốc,5,Hậu Giang,4,Hậu Lộc,1,Hilbert,1,Hình Học,33,HKUST,7,Hòa Bình,13,Hoài Nhơn,1,Hoàng Bá Minh,1,Hoàng Minh Quân,1,Hodge,1,Hojoo Lee,2,HOMC,5,HongKong,8,HSG 10,100,HSG 11,87,HSG 12,581,HSG 9,402,HSG Cấp Trường,78,HSG Quốc Gia,99,HSG Quốc Tế,16,Hứa Lâm Phong,1,Hứa Thuần Phỏng,1,Hùng Vương,2,Hưng Yên,32,Hương Sơn,2,Huỳnh Kim Linh,1,Hy Lạp,1,IMC,25,IMO,54,India,45,Inequality,13,InMC,1,International,307,Iran,11,Jakob,1,JBMO,41,Jewish,1,Journal,20,Junior,38,K2pi,1,Kazakhstan,1,Khánh Hòa,16,KHTN,53,Kiên Giang,64,Kim Liên,1,Kon Tum,18,Korea,5,Kvant,2,Kỷ Yếu,42,Lai Châu,4,Lâm Đồng,33,Lạng Sơn,21,Langlands,1,Lào Cai,16,Lê Hải Châu,1,Lê Hải Khôi,1,Lê Hoành Phò,4,Lê Khánh Sỹ,3,Lê Minh Cường,1,Lê Phúc Lữ,1,Lê Phương,1,Lê Quý Đôn,1,Lê Viết Hải,1,Lê Việt Hưng,1,Leibniz,1,Long An,42,Lớp 10,10,Lớp 10 Chuyên,452,Lớp 10 Không Chuyên,230,Lớp 11,1,Lục Ngạn,1,Lượng giác,1,Lương Tài,1,Lưu Giang Nam,2,Lý Thánh Tông,1,Macedonian,1,Malaysia,1,Margulis,2,Mark Levi,1,Mathematical Excalibur,1,Mathematical Reflections,1,Mathematics Magazine,1,Mathematics Today,1,Mathley,1,MathLinks,1,MathProblems Journal,1,Mathscope,8,MathsVN,5,MathVN,1,MEMO,10,Metropolises,4,Mexico,1,MIC,1,Michael Guillen,1,Mochizuki,1,Moldova,1,Moscow,1,Mỹ,9,MYTS,4,Nam Định,32,Nam Phi,1,Nam Trung Bộ,1,National,249,Nesbitt,1,Newton,4,Nghệ An,50,Ngô Bảo Châu,2,Ngô Việt Hải,1,Ngọc Huyền,2,Nguyễn Anh Tuyến,1,Nguyễn Bá Đang,1,Nguyễn Đình Thi,1,Nguyễn Đức Tấn,1,Nguyễn Đức Thắng,1,Nguyễn Duy Khương,1,Nguyễn Duy Tùng,1,Nguyễn Hữu Điển,3,Nguyễn Mình Hà,1,Nguyễn Minh Tuấn,8,Nguyễn Phan Tài Vương,1,Nguyễn Phú Khánh,1,Nguyễn Phúc Tăng,1,Nguyễn Quản Bá Hồng,1,Nguyễn Quang Sơn,1,Nguyễn Tài Chung,5,Nguyễn Tăng Vũ,1,Nguyễn Tất Thu,1,Nguyễn Thúc Vũ Hoàng,1,Nguyễn Trung Tuấn,8,Nguyễn Tuấn Anh,2,Nguyễn Văn Huyện,3,Nguyễn Văn Mậu,25,Nguyễn Văn Nho,1,Nguyễn Văn Quý,2,Nguyễn Văn Thông,1,Nguyễn Việt Anh,1,Nguyễn Vũ Lương,2,Nhật Bản,3,Nhóm $\LaTeX$,4,Nhóm Toán,1,Ninh Bình,41,Ninh Thuận,15,Nội Suy Lagrange,2,Nội Suy Newton,1,Nordic,19,Olympiad Corner,1,Olympiad Preliminary,2,Olympic 10,98,Olympic 10/3,5,Olympic 11,89,Olympic 12,30,Olympic 24/3,6,Olympic 27/4,20,Olympic 30/4,66,Olympic KHTN,6,Olympic Sinh Viên,73,Olympic Tháng 4,12,Olympic Toán,300,Olympic Toán Sơ Cấp,3,PAMO,1,Phạm Đình Đồng,1,Phạm Đức Tài,1,Phạm Huy Hoàng,1,Pham Kim Hung,3,Phạm Quốc Sang,2,Phan Huy Khải,1,Phan Thành Nam,1,Pháp,2,Philippines,8,Phú Thọ,30,Phú Yên,26,Phùng Hồ Hải,1,Phương Trình Hàm,11,Phương Trình Pythagoras,1,Pi,1,Polish,32,Problems,1,PT-HPT,14,PTNK,45,Putnam,25,Quảng Bình,44,Quảng Nam,31,Quảng Ngãi,33,Quảng Ninh,43,Quảng Trị,26,Quỹ Tích,1,Riemann,1,RMM,12,RMO,24,Romania,36,Romanian Mathematical,1,Russia,1,Sách Thường Thức Toán,7,Sách Toán,69,Sách Toán Cao Học,1,Sách Toán THCS,7,Saudi Arabia,7,Scholze,1,Serbia,17,Sharygin,24,Shortlists,56,Simon Singh,1,Singapore,1,Số Học - Tổ Hợp,27,Sóc Trăng,28,Sơn La,11,Spain,8,Star Education,5,Stars of Mathematics,11,Swinnerton-Dyer,1,Talent Search,1,Tăng Hải Tuân,2,Tạp Chí,14,Tập San,6,Tây Ban Nha,1,Tây Ninh,29,Thạch Hà,1,Thái Bình,39,Thái Nguyên,49,Thái Vân,2,Thanh Hóa,57,THCS,2,Thổ Nhĩ Kỳ,5,Thomas J. Mildorf,1,THPT Chuyên Lê Quý Đôn,1,THPTQG,15,THTT,6,Thừa Thiên Huế,35,Tiền Giang,19,Tin Tức Toán Học,1,Titu Andreescu,2,Toán 12,7,Toán Cao Cấp,3,Toán Chuyên,2,Toán Rời Rạc,5,Toán Tuổi Thơ,3,Tôn Ngọc Minh Quân,2,TOT,1,TPHCM,126,Trà Vinh,5,Trắc Nghiệm,1,Trắc Nghiệm Toán,2,Trại Hè,34,Trại Hè Hùng Vương,25,Trại Hè Phương Nam,5,Trần Đăng Phúc,1,Trần Minh Hiền,2,Trần Nam Dũng,9,Trần Phương,1,Trần Quang Hùng,1,Trần Quốc Anh,2,Trần Quốc Luật,1,Trần Quốc Nghĩa,1,Trần Tiến Tự,1,Trịnh Đào Chiến,2,Trung Quốc,12,Trường Đông,19,Trường Hè,7,Trường Thu,1,Trường Xuân,2,TST,55,Tuyên Quang,6,Tuyển Sinh,3,Tuyển Tập,44,Tuymaada,4,Undergraduate,66,USA,44,USAJMO,10,USATST,7,Uzbekistan,1,Vasile Cîrtoaje,4,Vật Lý,1,Viện Toán Học,2,Vietnam,4,Viktor Prasolov,1,VIMF,1,Vinh,27,Vĩnh Long,20,Vĩnh Phúc,63,Virginia Tech,1,VLTT,1,VMEO,4,VMF,12,VMO,46,VNTST,22,Võ Anh Khoa,1,Võ Quốc Bá Cẩn,26,Võ Thành Văn,1,Vojtěch Jarník,6,Vũ Hữu Bình,7,Vương Trung Dũng,1,WFNMC Journal,1,Wiles,1,Yên Bái,17,Yên Định,1,Yên Thành,1,Zhautykov,11,Zhou Yuan Zhe,1,
ltr
item
MOlympiad: [Solutions] Sharygin Geometry Mathematical Olympiad 2015 (Final Round)
[Solutions] Sharygin Geometry Mathematical Olympiad 2015 (Final Round)
MOlympiad
https://www.molympiad.net/2017/11/sharygin-geometry-olympiad-2015-solutions.html
https://www.molympiad.net/
https://www.molympiad.net/
https://www.molympiad.net/2017/11/sharygin-geometry-olympiad-2015-solutions.html
true
2506595080985176441
UTF-8
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS PREMIUM CONTENT IS LOCKED Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy