$hide=mobile

[Solutions] Sharygin Geometry Mathematical Olympiad 2010 (Correspondence Round)

  1. Does there exist a triangle, whose side is equal to some of its altitudes, another side is equal to some of its bisectors, and the third is equal to some of its medians? 
  2. Bisectors $AA_1$ and $BB_1$ of a right triangle $ABC \ (\angle C=90^\circ )$ meet at a point $I.$ Let $O$ be the circumcenter of triangle $CA_1B_1.$ Prove that $OI \perp AB.$ 
  3. Points $A'$, $B'$, $C'$ lie on sides $BC$, $CA$, $AB$ of triangle $ABC.$ for a point $X$ one has $$\angle AXB =\angle A'C'B' + \angle ACB,\quad \angle BXC = \angle B'A'C' +\angle BAC.$$ Prove that the quadrilateral $XA'BC'$ is cyclic. 
  4. The diagonals of a cyclic quadrilateral $ABCD$ meet in a point $N.$ The circumcircles of triangles $ANB$ and $CND$ intersect the sidelines $BC$ and $AD$ for the second time in points $A_1$, $B_1$, $C_1$, $D_1.$ Prove that the quadrilateral $A_1B_1C_1D_1$ is inscribed in a circle centered at $N.$ 
  5. A point $E$ lies on the altitude $BD$ of triangle $ABC$, and $\angle AEC=90^\circ.$ Points $O_1$ and $O_2$ are the circumcenters of triangles $AEB$ and $CEB$; points $F, L$ are the midpoints of the segments $AC$ and $O_1O_2.$ Prove that the points $L$, $E$, $F$ are collinear. 
  6. Points $M$ and $N$ lie on the side $BC$ of the regular triangle $ABC$ ($M$ is between $B$ and $N$), and $\angle MAN=30^\circ.$ The circumcircles of triangles $AMC$ and $ANB$ meet at a point $K.$ Prove that the line $AK$ passes through the circumcenter of triangle $AMN.$ 
  7. The line passing through the vertex $B$ of a triangle $ABC$ and perpendicular to its median $BM$ intersects the altitudes dropped from $A$ and $C$ (or their extensions) in points $K$ and $N.$ Points $O_1$ and $O_2$ are the circumcenters of the triangles $ABK$ and $CBN$ respectively. Prove that $O_1M=O_2M.$ 
  8. Let $AH$ be the altitude of a given triangle $ABC.$ The points $I_b$ and $I_c$ are the incenters of the triangles $ABH$ and $ACH$ respectively. $BC$ touches the incircle of the triangle $ABC$ at a point $L.$ Find $\angle LI_bI_c.$ 
  9. A point inside a triangle is called "good" if three cevians passing through it are equal. Assume for an isosceles triangle $ABC \ (AB=BC)$ the total number of "good" points is odd. Find all possible values of this number. 
  10. Let three lines forming a triangle $ABC$ be given. Using a two-sided ruler and drawing at most eight lines construct a point $D$ on the side $AB$ such that $\frac{AD}{BD}=\frac{BC}{AC}.$ 
  11. A convex $n-$gon is split into three convex polygons. One of them has $n$ sides, the second one has more than $n$ sides, the third one has less than $n$ sides. Find all possible values of $n.$ 
  12. Let $AC$ be the greatest leg of a right triangle $ABC,$ and $CH$ be the altitude to its hypotenuse. The circle of radius $CH$ centered at $H$ intersects $AC$ in point $M.$ Let a point $B'$ be the reflection of $B$ with respect to the point $H.$ The perpendicular to $AB$ erected at $B'$ meets the circle in a point $K$. Prove that
    a) $B'M \parallel BC$.
    b) $AK$ is tangent to the circle. 
  13. Let us have a convex quadrilateral $ABCD$ such that $AB=BC.$ A point $K$ lies on the diagonal $BD,$ and $\angle AKB+\angle BKC=\angle A + \angle C.$ Prove that $AK \cdot CD = KC \cdot AD.$ 
  14. We have a convex quadrilateral $ABCD$ and a point $M$ on its side $AD$ such that $CM$ and $BM$ are parallel to $AB$ and $CD$ respectively. Prove that $S_{ABCD} \geq 3 S_{BCM}.$ ($S$ denotes the area function.)
  15. Let $AA_1$, $BB_1$ and $CC_1$ be the altitudes of an acute-angled triangle $ABC.$ $AA_1$ meets $B_1C_1$ in a point $K.$ The circumcircles of triangles $A_1KC_1$ and $A_1KB_1$ intersect the lines $AB$ and $AC$ for the second time at points $N$ and $L$ respectively. Prove that
    a) The sum of diameters of these two circles is equal to $BC,$
    b) $\dfrac{A_1N}{BB_1} + \dfrac{A_1L}{CC_1}=1.$
  16. A circle touches the sides of an angle with vertex $A$ at points $B$ and $C.$ A line passing through $A$ intersects this circle in points $D$ and $E.$ A chord $BX$ is parallel to $DE.$ Prove that $XC$ passes through the midpoint of the segment $DE.$ 
  17. Construct a triangle, if the lengths of the bisectrix and of the altitude from one vertex, and of the median from another vertex are given. 
  18. A point $B$ lies on a chord $AC$ of circle $\omega.$ Segments $AB$ and $BC$ are diameters of circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ respectively. These circles intersect $\omega$ for the second time in points $D$ and $E$ respectively. The rays $O_1D$ and $O_2E$ meet in a point $F,$ and the rays $AD$ and $CE$ do in a point $G.$ Prove that the line $FG$ passes through the midpoint of the segment $AC.$ 
  19. A quadrilateral $ABCD$ is inscribed into a circle with center $O.$ Points $P$ and $Q$ are opposite to $C$ and $D$ respectively. Two tangents drawn to that circle at these points meet the line $AB$ in points $E$ and $F.$ ($A$ is between $E$ and $B$, $B$ is between $A$ and $F$). The line $EO$ meets $AC$ and $BC$ in points $X$ and $Y$ respectively, and the line $FO$ meets $AD$ and $BD$ in points $U$ and $V$ respectively. Prove that $XV=YU.$ 
  20. The incircle of an acute-angled triangle $ABC$ touches $AB$, $BC$, $CA$ at points $C_1$, $A_1$, $B_1$ respectively. Points $A_2$, $B_2$ are the midpoints of the segments $B_1C_1$, $A_1C_1$ respectively. Let $P$ be a common point of the incircle and the line $CO$, where $O$ is the circumcenter of triangle $ABC.$ Let also $A'$ and $B'$ be the second common points of $PA_2$ and $PB_2$ with the incircle. Prove that a common point of $AA'$ and $BB'$ lies on the altitude of the triangle dropped from the vertex $C.$ 
  21. A given convex quadrilateral $ABCD$ is such that $$\angle ABD + \angle ACD > \angle BAC + \angle BDC.$$ Prove that \[S_{ABD}+S_{ACD} > S_{BAC}+S_{BDC}.\]
  22. A circle centered at a point $F$ and a parabola with focus $F$ have two common points. Prove that there exist four points $A$, $B$, $C$, $D$ on the circle such that the lines $AB$, $BC$, $CD$ and $DA$ touch the parabola. 
  23. A cyclic hexagon $ABCDEF$ is such that $$AB \cdot CF= 2BC \cdot FA, CD \cdot EB = 2 DE \cdot BC$$ and $EF \cdot AD = 2FA \cdot DE.$ Prove that the lines $AD$, $BE$ and $CF$ are concurrent. 
  24. Let us have a line $\ell$ in the space and a point $A$ not lying on $\ell.$ For an arbitrary line $\ell'$ passing through $A$, $XY$ ($Y$ is on $\ell'$) is a common perpendicular to the lines $\ell$ and $\ell'.$ Find the locus of points $Y.$ 
  25. For two different regular icosahedrons it is known that some six of their vertices are vertices of a regular octahedron. Find the ratio of the edges of these icosahedrons.

Post a Comment


$hide=mobile

$hide=mobile

$hide=mobile

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

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

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

Name

Ả-rập Xê-út,1,Abel,5,Albania,2,AMM,3,Amsterdam,5,Ấn Độ,2,An Giang,23,Andrew Wiles,1,Anh,2,Áo,1,APMO,19,Ba Đình,2,Ba Lan,1,Bà Rịa Vũng Tàu,52,Bắc Giang,50,Bắc Kạn,1,Bạc Liêu,9,Bắc Ninh,48,Bắc Trung Bộ,7,Bài Toán Hay,5,Balkan,38,Baltic Way,30,BAMO,1,Bất Đẳng Thức,66,Bến Tre,46,Benelux,14,Bình Định,45,Bình Dương,23,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,6,Buôn Ma Thuột,1,BxMO,13,Cà Mau,14,Cần Thơ,14,Canada,40,Cao Bằng,7,Cao Quang Minh,1,Câu Chuyện Toán Học,36,Caucasus,2,CGMO,10,China,10,Chọn Đội Tuyển,353,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,618,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,26,Đà Nẵng,39,Đa Thức,2,Đại Số,20,Đắk Lắk,56,Đắk Nông,7,Đan Phượng,1,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1,Đề Thi HSG,1770,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,52,Du Hiền Vinh,1,Đức,1,Duyên Hải Bắc Bộ,25,E-Book,33,EGMO,17,ELMO,19,EMC,9,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,26,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,232,Hà Tĩnh,73,Hà Trung Kiên,1,Hải Dương,50,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 10,101,HSG 11,91,HSG 12,587,HSG 9,425,HSG Cấp Trường,78,HSG Quốc Gia,106,HSG Quốc Tế,16,Hứa Lâm Phong,1,Hứa Thuần Phỏng,1,Hùng Vương,2,Hưng Yên,33,Hương Sơn,2,Huỳnh Kim Linh,1,Hy Lạp,1,IMC,26,IMO,56,IMT,1,India,45,Inequality,13,InMC,1,International,315,Iran,11,Jakob,1,JBMO,41,Jewish,1,Journal,20,Junior,38,K2pi,1,Kazakhstan,1,Khánh Hòa,17,KHTN,54,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,17,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,455,Lớp 10 Không Chuyên,229,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,11,Metropolises,4,Mexico,1,MIC,1,Michael Guillen,1,Mochizuki,1,Moldova,1,Moscow,1,Mỹ,10,MYM,227,MYTS,4,Nam Định,33,Nam Phi,1,Nam Trung Bộ,1,National,249,Nesbitt,1,Newton,4,Nghệ An,52,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,4,Nhóm $\LaTeX$,4,Nhóm Toán,1,Ninh Bình,43,Ninh Thuận,15,Nội Suy Lagrange,2,Nội Suy Newton,1,Nordic,19,Olympiad Corner,1,Olympiad Preliminary,2,Olympic 10,99,Olympic 10/3,5,Olympic 11,92,Olympic 12,30,Olympic 24/3,7,Olympic 27/4,20,Olympic 30/4,69,Olympic KHTN,6,Olympic Sinh Viên,73,Olympic Tháng 4,12,Olympic Toán,304,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,29,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,32,Quảng Ngãi,34,Quảng Ninh,43,Quảng Trị,27,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,12,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,62,THCS,2,Thổ Nhĩ Kỳ,5,Thomas J. Mildorf,1,THPT Chuyên Lê Quý Đôn,1,THPTQG,15,THTT,7,Thừa Thiên Huế,36,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,6,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,14,Trường Đông,19,Trường Hè,7,Trường Thu,1,Trường Xuân,2,TST,56,Tuyên Quang,6,Tuyển Sinh,3,Tuyển Sinh 10,680,Tuyển Tập,44,Tuymaada,4,Undergraduate,67,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,21,Vĩnh Phúc,64,Virginia Tech,1,VLTT,1,VMEO,4,VMF,12,VMO,47,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,20,Yên Định,1,Yên Thành,1,Zhautykov,11,Zhou Yuan Zhe,1,
ltr
item
MOlympiad: [Solutions] Sharygin Geometry Mathematical Olympiad 2010 (Correspondence Round)
[Solutions] Sharygin Geometry Mathematical Olympiad 2010 (Correspondence Round)
MOlympiad
https://www.molympiad.net/2018/08/sharygin-geometry-mathematical-olympiad-2010-correspondence.html
https://www.molympiad.net/
https://www.molympiad.net/
https://www.molympiad.net/2018/08/sharygin-geometry-mathematical-olympiad-2010-correspondence.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