Olympiad problems

1991 Flanders Math Olympiad, 3

Given $\Delta ABC$ equilateral, with $X\in[A,B]$. Then we define unique points Y,Z so that $Y\in[B,C]$, $Z\in[A,C]$, $\Delta XYZ$ equilateral. If $Area\left(\Delta ABC\right) = 2 \cdot Area\left(\Delta XYZ\right)$, find the ratio of $\frac{AX}{XB},\frac{BY}{YC},\frac{CZ}{ZA}$.

2013 239 Open Mathematical Olympiad, 7

Tags: geometry , geometric inequalities

Point $M$ is the midpoint of side $BC$ of convex quadrilateral $ABCD$. If $\angle{AMD} < 120^{\circ}$. Prove that $$(AB+AM)^2 + (CD+DM)^2 > AD \cdot BC + 2AB \cdot CD.$$

2018 Bosnia and Herzegovina Junior BMO TST, 3

Tags: geometry

Let $\Gamma$ be circumscribed circle of triangle $ABC $ $(AB \neq AC)$. Let $O$ be circumcenter of the triangle $ABC$. Let $M$ be a point where angle bisector of angle $BAC$ intersects $\Gamma$. Let $D$ $(D \neq M)$ be a point where circumscribed circle of the triangle $BOM$ intersects line segment $AM$ and let $E$ $(E \neq M)$ be a point where circumscribed circle of triangle $COM$ intersects line segment $AM$. Prove that $BD+CE=AM$.

2006 Sharygin Geometry Olympiad, 9.3

Tags: geometry , similar triangles , equal ratio , collinear

Triangles $ABC$ and $A_1B_1C_1$ are similar and differently oriented. On the segment $AA_1$, a point $A'$ is taken such that $AA' / A_1A'= BC / B_1C_1$. We similarly construct $B'$ and $C'$. Prove that $A', B',C'$ lie on one straight line.

ICMC 6, 2

Tags: geometry , 3d geometry , tetrahedron

Show that if the distance between opposite edges of a tetrahedron is at least $1$, then its volume is at least $\frac{1}{3}$. [i]Proposed by Simeon Kiflie[/i]

2024 Nigerian MO Round 3, Problem 3

Tags: circumcircle , trigonometry , geometry

Let $ABC$ be a triangle, and let $O$ be its circumcenter. Let $\overline{CO}\cap AB\equiv D$. Let $\angle BAC=\alpha$, and $\angle CBA=\beta$. Prove that $$\dfrac{OD}{OC}=\Bigg|\dfrac{\cos(\alpha+\beta)}{\cos(\alpha-\beta)}\Bigg|$$\\ For clarification, $\overline{CO}$ represents the line $CO$, and $AC$ represents the segment $AC$. Cases in which $D$ doesn't exist should be ignored.

2015 Czech-Polish-Slovak Match, 1

Tags: geometry

On a circle of radius $r$, the distinct points $A$, $B$, $C$, $D$, and $E$ lie in this order, satisfying $AB=CD=DE>r$. Show that the triangle with vertices lying in the centroids of the triangles $ABD$, $BCD$, and $ADE$ is obtuse. [i]Proposed by Tomáš Jurík, Slovakia[/i]

2004 India IMO Training Camp, 3

Tags: combinatorics , circles , geometry , coordinate geometry , coordinates , triangle

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

2023 Romania EGMO TST, P3

Tags: cyclic quadrilateral , circumcircle , geometry

In a cyclic quadrilateral $ABCD$ with $AB=AD$ points $M$,$N$ lie on the sides $BC$ and $CD$ respectively so that $MN=BM+DN$ . Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$ .

2004 Iran MO (3rd Round), 12

Tags: algorithm , geometric transformation , polynomial , induction , number theory , algebra , geometry

$\mathbb{N}_{10}$ is generalization of $\mathbb{N}$ that every hypernumber in $\mathbb{N}_{10}$ is something like: $\overline{...a_2a_1a_0}$ with $a_i \in {0,1..9}$ (Notice that $\overline {...000} \in \mathbb{N}_{10}$) Also we easily have $+,*$ in $\mathbb{N}_{10}$. first $k$ number of $a*b$= first $k$ nubmer of (first $k$ number of a * first $k$ number of b) first $k$ number of $a+b$= first $k$ nubmer of (first $k$ number of a + first $k$ number of b) Fore example $\overline {...999}+ \overline {...0001}= \overline {...000}$ Prove that every monic polynomial in $\mathbb{N}_{10}[x]$ with degree $d$ has at most $d^2$ roots.

2024 Tuymaada Olympiad, 4

Tags: midpoint , geometry , tuymadaa , excircle

A triangle $ABC$ is given. The segment connecting the points where the excircles touch $AB$ and $AC$ meets the bisector of angle $C$ at $X$. The segment connecting the points where the excircles touch $BC$ and $AC$ meets the bisector of angle $A$ at $Y$. Prove that the midpoint of $XY$ is equidistant from $A$ and $C$.

2013 Stanford Mathematics Tournament, 5

Tags: 3d geometry , prism , geometry

A polygonal prism is made from a flexible material such that the two bases are regular $2^n$-gons $(n>1)$ of the same size. The prism is bent to join the two bases together without twisting, giving a figure with $2^n$ faces. The prism is then repeatedly twisted so that each edge of one base becomes aligned with each edge of the base exactly once. For an arbitrary $n$, what is the sum of the number of faces over all of these configurations (including the non-twisted case)?

1985 IMO Shortlist, 21

Tags: circumcircle , trigonometry , geometry , triangle

The tangents at $B$ and $C$ to the circumcircle of the acute-angled triangle $ABC$ meet at $X$. Let $M$ be the midpoint of $BC$. Prove that [i](a)[/i] $\angle BAM = \angle CAX$, and [i](b)[/i] $\frac{AM}{AX} = \cos\angle BAC.$

2006 Singapore Junior Math Olympiad, 4

Tags: angle bisector , geometry , incenter , angle chasing

In $\vartriangle ABC$, the bisector of $\angle B$ meets $AC$ at $D$ and the bisector of $\angle C$ meets $AB$ at $E$. These bisectors intersect at $O$ and $OD = OE$. If $AD \ne AE$, prove that $\angle A = 60^o$.

2016 Sharygin Geometry Olympiad, 6

Tags: circumcircle , geometry , collinearity

A triangle $ABC$ is given. The point $K$ is the base of the external bisector of angle $A$. The point $M$ is the midpoint of the arc $AC$ of the circumcircle. The point $N$ on the bisector of angle $C$ is such that $AN \parallel BM$. Prove that the points $M,N,K$ are collinear. [i](Proposed by Ilya Bogdanov)[/i]

2005 China Western Mathematical Olympiad, 2

Tags: circumcircle , geometry , similar triangles

Given three points $P$, $A$, $B$ and a circle such that the lines $PA$ and $PB$ are tangent to the circle at the points $A$ and $B$, respectively. A line through the point $P$ intersects that circle at two points $C$ and $D$. Through the point $B$, draw a line parallel to $PA$; let this line intersect the lines $AC$ and $AD$ at the points $E$ and $F$, respectively. Prove that $BE = BF$.

1993 Italy TST, 3

Tags: geometry , inradius , exradius , altitude

Let $ABC$ be an isosceles triangle with base $AB$ and $D$ be a point on side $AB$ such that the incircle of triangle $ACD$ is congruent to the excircle of triangle $DCB$ across $C$. Prove that the diameter of each of these circles equals half the altitude of $\vartriangle ABC$ from $A$

2021 Taiwan TST Round 2, G

Tags: circumcircle , reflection , geometric transformation , geometry

Let $ABC$ be a triangle with circumcircle $\Gamma$, and points $E$ and $F$ are chosen from sides $CA$, $AB$, respectively. Let the circumcircle of triangle $AEF$ and $\Gamma$ intersect again at point $X$. Let the circumcircles of triangle $ABE$ and $ACF$ intersect again at point $K$. Line $AK$ intersect with $\Gamma$ again at point $M$ other than $A$, and $N$ be the reflection point of $M$ with respect to line $BC$. Let $XN$ intersect with $\Gamma$ again at point $S$ other that $X$. Prove that $SM$ is parallel to $BC$. [i] Proposed by Ming Hsiao[/i]

2015 India National Olympiad, 1

Tags: trigonometry , circumcircle , incenter , geometry

Let $ABC$ be a right-angled triangle with $\angle{B}=90^{\circ}$. Let $BD$ is the altitude from $B$ on $AC$. Let $P,Q$ and $I $be the incenters of triangles $ABD,CBD$ and $ABC$ respectively.Show that circumcenter of triangle $PIQ$ lie on the hypotenuse $AC$.

2006 Denmark MO - Mohr Contest, 1

Tags: area , geometry

The star shown is symmetric with respect to each of the six diagonals shown. All segments connecting the points $A_1, A_2, . . . , A_6$ with the centre of the star have the length $1$, and all the angles at $B_1, B_2, . . . , B_6$ indicated in the figure are right angles. Calculate the area of the star. [img]https://1.bp.blogspot.com/-Rso2aWGUq_k/XzcAm4BkAvI/AAAAAAAAMW0/277afcqTfCgZOHshf_6ce2XpinWWR4SZACLcBGAsYHQ/s0/2006%2BMohr%2Bp1.png[/img]

2022 Switzerland - Final Round, 1

Tags: ratio , geometry

Let $k$ be a circle with centre $M$ and let $AB$ be a diameter of $k$. Furthermore, let $C$ be a point on $k$ such that $AC = AM$. Let $D$ be the point on the line $AC$ such that $CD = AB$ and $C$ lies between $A$ and $D$. Let $E$ be the second intersection of the circumcircle of $BCD$ with line $AB$ and $F$ be the intersection of the lines $ED$ and $BC$. The line $AF$ cuts the segment $BD$ in $X$. Determine the ratio $BX/XD$.

2018 Czech-Polish-Slovak Junior Match, 2

Tags: area , ratio , geometry , symmetry , right triangle

Given a right triangle $ABC$ with the hypotenuse $AB$. Let $K$ be any interior point of triangle $ABC$ and points $L, M$ are symmetric of point $K$ wrt lines $BC, AC$ respectively. Specify all possible values for $S_{ABLM} / S_{ABC}$, where $S_{XY ... Z}$ indicates the area of the polygon $XY...Z$ .

2010 Serbia National Math Olympiad, 2

Tags: ratio , circumcircle , geometric transformation , geometry , reflection

In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$. [i]Proposed by Marko Djikic[/i]

1983 All Soviet Union Mathematical Olympiad, 368

Tags: geometry , inequalities , geometric inequality

The points $D,E,F$ belong to the sides $(AB), (BC)$ and $(CA)$ of the triangle $ABC$ respectively (but they are not vertices). Let us denote with $d_0, d_1, d_2$, and $d_3$ the maximal side length of the triangles $DEF$, $DEA$, $DBF$, $CEF$, respectively. Prove that $$d_0 \ge \frac{\sqrt3}{2} min\{d_1, d_2, d_3\}$$ When the equality takes place?

2016 Bangladesh Mathematical Olympiad, 3

Tags: geometry , angle chasing

$\triangle ABC$ is isosceles $AB = AC$. $P$ is a point inside $\triangle ABC$ such that $\angle BCP = 30$ and $\angle APB = 150$ and $\angle CAP = 39$. Find $\angle BAP$.