Olympiad problems

2012 Purple Comet Problems, 7

Two convex polygons have a total of 33 sides and 243 diagonals. Find the number of diagonals in the polygon with the greater number of sides.

2000 India National Olympiad, 4

Tags: pythagorean theorem , geometry , trigonometry

In a convex quadrilateral $PQRS$, $PQ =RS$, $(\sqrt{3} +1 )QR = SP$ and $\angle RSP - \angle SQP = 30^{\circ}$. Prove that $\angle PQR - \angle QRS = 90^{\circ}.$

2025 Malaysian IMO Training Camp, 1

Tags: geometry

Let $ABC$ be a triangle with $AB<AC$ and with its incircle touching the sides $AB$ and $BC$ at $M$ and $J$ respectively. A point $D$ lies on the extension of $AB$ beyond $B$ such that $AD=AC$. Let $O$ be the midpoint of $CD$. Prove that the points $J$, $O$, $M$ are collinear. [i](Proposed by Tan Rui Xuen)[/i]

MOAA Team Rounds, 2019.8

Tags: team , algebra

Suppose that $$\frac{(\sqrt2)^5 + 1}{\sqrt2 + 1} \times \frac{2^5 + 1}{2 + 1} \times \frac{4^5 + 1}{4 + 1} \times \frac{16^5 + 1}{16 + 1} =\frac{m}{7 + 3\sqrt2}$$ for some integer $m$. How many $0$’s are in the binary representation of $m$? (For example, the number $20 = 10100_2$ has three $0$’s in its binary representation.)

Estonia Open Senior - geometry, 2013.2.3

Tags: concyclic , geometry , circles

Circles $c_1, c_2$ with centers $O_1, O_2$, respectively, intersect at points $P$ and $Q$ and touch circle c internally at points $A_1$ and $A_2$, respectively. Line $PQ$ intersects circle c at points $B$ and $D$. Lines $A_1B$ and $A_1D$ intersect circle $c_1$ the second time at points $E_1$ and $F_1$, respectively, and lines $A_2B$ and $A_2D$ intersect circle $c_2$ the second time at points $ E_2$ and $F_2$, respectively. Prove that $E_1, E_2, F_1, F_2$ lie on a circle whose center coincides with the midpoint of line segment $O_1O_2$.

2019 Costa Rica - Final Round, G2

Tags: geometry , angle , circumcenter , orthocenter

Let $H$ be the orthocenter and $O$ the circumcenter of the acute triangle $\vartriangle ABC$. The circle with center $H$ and radius $HA$ intersects the lines $AC$ and $AB$ at points $P$ and $Q$, respectively. Let point $O$ be the orthocenter of triangle $\vartriangle APQ$, determine the measure of $\angle BAC$.

2013 AMC 10, 15

Tags: geometry , ratio

A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$? ${ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac{\sqrt{6}}{2} \qquad\textbf{(C)}\ \sqrt{3}\qquad\textbf{(D}}\ 2 \qquad\textbf{(E)}\ \frac{3\sqrt{2}}{2} $

1951 Poland - Second Round, 5

Tags: geometry , trigonometry

Prove that if the relationship between the sides and opposite angles $ A $ and $ B $ of the triangle $ ABC $ is $$ (a^2 + b^2) \sin (A - B) = (a^2 - b^2) \sin (A + B)$$ then such a triangle is right-angled or isosceles.

2018-2019 Winter SDPC, 4

Tags:

Tom is chasing Jerry on the coordinate plane. Tom starts at $(x, y)$ and Jerry starts at $(0, 0)$. Jerry moves to the right at $1$ unit per second. At each positive integer time $t$, if Tom is within $1$ unit of Jerry, he hops to Jerry’s location and catches him. Otherwise, Tom hops to the midpoint of his and Jerry’s location. [i]Example. If Tom starts at $(3, 2)$, then at time $t = 1$ Tom will be at $(2, 1)$ and Jerry will be at $(1, 0)$. At $t = 2$ Tom will catch Jerry.[/i] Assume that Tom catches Jerry at some integer time $n$. (a) Show that $x \geq 0$. (b) Find the maximum possible value of $\frac{y}{x+1}$.

2000 National Olympiad First Round, 15

Tags: probability

$A,B,C$ are playing backgammon tournament. At first, $A$ plays with $B$. Then the winner plays with $C$. As the tournament goes on, the last winner plays with the player who did not play in the previous game. When a player wins two successive games, he will win the tournament. If each player has equal chance to win a game, what is the probability that $C$ wins the tournament? $ \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac13 \qquad\textbf{(C)}\ \frac3{14} \qquad\textbf{(D)}\ \frac 17 \qquad\textbf{(E)}\ \text{None} $

2020 Vietnam Team Selection Test, 4

Tags: combinatorics

Let $n$ be a positive integer. In a $(2n+1)\times (2n+1)$ board, each grid is dyed white or black. In each row and each column, if the number of white grids is smaller than the number of black grids, then we mark all white grids. If the number of white grids is bigger than the number of black grids, then we mark all black grids. Let $a$ be the number of black grids, and $b$ be the number of white grids, $c$ is the number of marked grids. In this example of $3\times 3$ table, $a=3$, $b=6$, $c=4$. (forget about my watermark) Proof that no matter how is the dyeing situation in the beginning, there is always $c\geq\frac{1}{2}\min\{a,b\}$.

2024 IFYM, Sozopol, 3

Tags: sequence , number theory

The sequence $ (a_n)_{n\geq 1} $ of positive integers is such that $ a_1 = 1 $ and $ a_{m+n} $ divides $ a_m + a_n $ for any positive integers $ m $ and $ n $. a) Prove that if the sequence is unbounded, then $ a_n = n $ for all $ n $. b) Does there exist a non-constant bounded sequence with the above properties? (A sequence $ (a_n)_{n\geq 1} $ of positive integers is bounded if there exists a positive integer $ A $ such that $ a_n \leq A $ for all $ n $, and unbounded otherwise.)

2014 Thailand TSTST, 3

Tags: geometry , tangential , orthocenter , collinear , incenter

Let $O$ be the incenter of a tangential quadrilateral $ABCD$. Prove that the orthocenters of $\vartriangle AOB$, $\vartriangle BOC$, $\vartriangle COD$, $\vartriangle DOA$ lie on a line.

2016 Saudi Arabia GMO TST, 4

Tags: game , combinatorics

There are totally $16$ teams participating in a football tournament, each team playing with every other exactly $1$ time. In each match, the winner gains $3$ points, the loser gains $0$ point and each teams gain $1$ point for the tie match. Suppose that at the end of the tournament, each team gains the same number of points. Prove that there are at least $4$ teams that have the same number of winning matches, the same number of losing matches and the same number of tie matches.

2003 May Olympiad, 2

Tags: right triangle , area , geometry , area of a triangle

The triangle $ABC$ is right in $A$ and $R$ is the midpoint of the hypotenuse $BC$ . On the major leg $AB$ the point $P$ is marked such that $CP = BP$ and on the segment $BP$ the point $Q$ is marked such that the triangle $PQR$ is equilateral. If the area of triangle $ABC$ is $27$, calculate the area of triangle $PQR$ .

KoMaL A Problems 2022/2023, A. 834

Tags: octagon , conic section , geometry

Let $A_1A_2\ldots A_8$ be a convex cyclic octagon, and for $i=1,2\ldots,8$ let $B_i=A_iA_{i+3}\cap A_{i+1}A_{i+4}$ (indices are meant modulo 8). Prove that points $B_1,\ldots, B_8$ lie on the same conic section.

2022 Novosibirsk Oral Olympiad in Geometry, 2

Tags: reflection , geometry , rectangle

A ball was launched on a rectangular billiard table at an angle of $45^o$ to one of the sides. Reflected from all sides (the angle of incidence is equal to the angle of reflection), he returned to his original position . It is known that one of the sides of the table has a length of one meter. Find the length of the second side. [img]https://cdn.artofproblemsolving.com/attachments/3/d/e0310ea910c7e3272396cd034421d1f3e88228.png[/img]

1992 ITAMO, 1

Tags: geometry , 3d geometry , cube , plane , combinatorial geometry

A cube is divided into $27$ equal smaller cubes. A plane intersects the cube. Find the maximum possible number of smaller cubes the plane can intersect.

MathLinks Contest 3rd, 1

Tags: algebra , functional

Find all functions $f : (0, +\infty) \to (0, +\infty)$ which are increasing on $[1, +\infty)$ and for all positive reals $a, b, c$ they fulfill the following relation $f(ab)f(bc)f(ca)=f(a^2b^2c^2)+f(a^2)+f(b^2)+f(c^2)$.

2010 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: altitude , euler circle , geometry

Let $AA_1$, $BB_1$ and $CC_1$ be altitudes of triangle $ABC$ and let $A_1A_2$, $B_1B_2$ and $C_1C_2$ be diameters of Euler circle of triangle $ABC$. Prove that lines $AA_2$, $BB_2$ and $CC_2$ are concurrent

2009 Ukraine National Mathematical Olympiad, 4

Tags: geometry , parallelogram , ratio , trigonometry

Let $ABCD$ be a parallelogram with $\angle BAC = 45^\circ,$ and $AC > BD .$ Let $w_1$ and $w_2$ be two circles with diameters $AC$ and $DC,$ respectively. The circle $w_1$ intersects $AB$ at $E$ and the circle $w_2$ intersects $AC$ at $O$ and $C$, and $AD$ at $F.$ Find the ratio of areas of triangles $AOE$ and $COF$ if $AO = a,$ and $FO = b .$