Olympiad problems

2019 Iranian Geometry Olympiad, 1

Tags: geometry

There is a table in the shape of a $8\times 5$ rectangle with four holes on its corners. After shooting a ball from points $A, B$ and $C$ on the shown paths, will the ball fall into any of the holes after 6 reflections? (The ball reflects with the same angle after contacting the table edges.) [img]http://s5.picofile.com/file/8372960750/E01.png[/img] [i]Proposed by Hirad Alipanah[/i]

2010 Princeton University Math Competition, 1

Tags: polynomial , algebra

Find the sum of the coefficients of the polynomial $(63x-61)^4$.

2018 Pan-African Shortlist, N5

Tags: number theory , diophantine equation

Find all quadruplets $(a, b, c, d)$ of positive integers such that \[ \left( 1 + \frac{1}{a} \right) \left( 1 + \frac{1}{b} \right) \left( 1 + \frac{1}{c} \right) \left( 1 + \frac{1}{d} \right) = 4. \]

2003 Gheorghe Vranceanu, 1

Tags: floor function , algebra , equation

Solve in $ \mathbb{R}^2 $ the equation $ \lfloor x/y-y/x \rfloor =x^2/y+y/x^2. $

2017 India Regional Mathematical Olympiad, 5

Tags: geometry , circles

Let $\Omega$ be a circle with a chord $AB$ which is not a diameter. $\Gamma_{1}$ be a circle on one side of $AB$ such that it is tangent to $AB$ at $C$ and internally tangent to $\Omega$ at $D$. Likewise, let $\Gamma_{2}$ be a circle on the other side of $AB$ such that it is tangent to $AB$ at $E$ and internally tangent to $\Omega$ at $F$. Suppose the line $DC$ intersects $\Omega$ at $X \neq D$ and the line $FE$ intersects $\Omega$ at $Y \neq F$. Prove that $XY$ is a diameter of $\Omega$ .

1957 AMC 12/AHSME, 27

Tags:

The sum of the reciprocals of the roots of the equation $ x^2 \plus{} px \plus{} q \equal{} 0$ is: $ \textbf{(A)}\ \minus{}\frac{p}{q} \qquad \textbf{(B)}\ \frac{q}{p}\qquad \textbf{(C)}\ \frac{p}{q}\qquad \textbf{(D)}\ \minus{}\frac{q}{p}\qquad \textbf{(E)}\ pq$

1954 Putnam, A7

Tags: integer , number theory

Prove that there are no integers $x$ and $y$ for which $$x^2 +3xy-2y^2 =122.$$

2024 Kyiv City MO Round 2, Problem 2

Tags: algebra , construction

Find the smallest positive integer $n$ for which one can select $n$ distinct real numbers such that each of them is equal to the sum of some two other selected numbers. [i]Proposed by Anton Trygub[/i]

2023 Saint Petersburg Mathematical Olympiad, 3

Tags: number theory

Find all positive integers $x, y$ and primes $p$, such that $x^5+y^4=pxy$.

2008 Tournament Of Towns, 2

Tags: ratio , combinatorial geometry , combinatorics , segment

There are ten congruent segments on a plane. Each intersection point divides every segment passing through it in the ratio $3:4$. Find the maximum number of intersection points.

2001 Junior Balkan Team Selection Tests - Moldova, 2

Tags: algebra , floor function

Solve in $R$ equation $[x] \cdot \{x\} = 2001 x$, where$ [ .]$ and $\{ .\}$ represent respectively the floor and the integer functions.

2020 Kazakhstan National Olympiad, 4

Tags: geometry , incenter , arc midpoint , incircle

The incircle of the triangle $ ABC $ touches the sides of $ AB, BC, CA $ at points $ C_0, A_0, B_0 $, respectively. Let the point $ M $ be the midpoint of the segment connecting the vertex $ C_0 $ with the intersection point of the altitudes of the triangle $ A_0B_0C_0 $, point $ N $ be the midpoint of the arc $ ACB $ of the circumscribed circle of the triangle $ ABC $. Prove that line $ MN $ passes through the center of incircle of triangle $ ABC $.

2024 Assara - South Russian Girl's MO, 8

Tags: combinatorics

There are $15$ boys and $15$ girls in the class. The first girl is friends with $4$ boys, the second with $5$, the third with $6$, . . . , the $11$th with $14$, and each of the other four girls is friends with all the boys. It turned out that there are exactly $3 \cdot 2^{25}$ ways to split the entire class into pairs, so that each pair has a boy and a girl who are friends. Prove that any of the friends of the first girl are friends with all the other girls too. [i]G.M.Sharafetdinova[/i]

2001 India National Olympiad, 6

Tags: function , algebra

Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(x +y) = f(x) f(y) f(xy)$ for all $x, y \in \mathbb{R}.$

2014 Poland - Second Round, 2.

Tags: geometry , radii

Distinct points $A$, $B$ and $C$ lie on a line in this order. Point $D$ lies on the perpendicular bisector of the segment $BC$. Denote by $M$ the midpoint of the segment $BC$. Let $r$ be the radius of the incircle of the triangle $ABD$ and let $R$ be the radius of the circle with center lying outside the triangle $ACD$, tangent to $CD$, $AC$ and $AD$. Prove that $DM=r+R$.

2015 Peru Cono Sur TST, P6

Tags: combinatorics

Let $n$ be a positive integer. On a $2n\times 2n$ board, the $2n^2$ squares were painted white and the other $2n^2$ squares were painted black. One operation is to choose a $2\times 2$ subtable and mirror its $4$ cells about the vertical or horizontal axis of symmetry of that subtable. For what values of $n$ is it always possible to obtain a chess-like coloring from any initial coloring?

2020 CHMMC Winter (2020-21), 1

Tags: geometry

A unit circle is centered at $(0, 0)$ on the $(x, y)$ plane. A regular hexagon passing through $(1, 0)$ is inscribed in the circle. Two points are randomly selected from the interior of the circle and horizontal lines are drawn through them, dividing the hexagon into at most three pieces. The probability that each piece contains exactly two of the hexagon's original vertices can be written as \[ \frac{2\left(\frac{m\pi}{n}+\frac{\sqrt{p}}{q}\right)^2}{\pi^2} \] for positive integers $m$, $n$, $p$, and $q$ such that $m$ and $n$ are relatively prime and $p$ is squarefree. Find $m+n+p+q$.

1994 IMC, 3

Tags: real analysis , function , calculus , derivative

Let $f$ be a real-valued function with $n+1$ derivatives at each point of $\mathbb R$. Show that for each pair of real numbers $a$, $b$, $a<b$, such that $$\ln\left( \frac{f(b)+f'(b)+\cdots + f^{(n)} (b)}{f(a)+f'(a)+\cdots + f^{(n)}(a)}\right)=b-a$$ there is a number $c$ in the open interval $(a,b)$ for which $$f^{(n+1)}(c)=f(c)$$

PEN N Problems, 15

Tags: more sequences

In the sequence $00$, $01$, $02$, $03$, $\cdots$, $99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19$, $39$, or $28$, but not by $30$ or $20$). What is the maximal number of terms that could remain on their places?

2017 Israel Oral Olympiad, 2

Tags: algebra , simplify , fraction

Simplify the fraction: $\frac{(1^4+4)\cdot (5^4+4)\cdot (9^4+4)\cdot ... (69^4+4)\cdot(73^4+4)}{(3^4+4)\cdot (7^4+4)\cdot (11^4+4)\cdot ... (71^4+4)\cdot(75^4+4)}$.

2023 Indonesia Regional, 5

Tags: geometry , regional olympiad

Given $\triangle ABC$ and points $D$ and $E$ at the line $BC$, furthermore there are points $X$ and $Y$ inside $\triangle ABC$. Let $P$ be the intersection of line $AD$ and $XE$, and $Q$ be the intersection of line $AE$ and $YD$. If there exist a circle that passes through $X, Y, D, E$, and $$\angle BXE + \angle BCA = \angle CYD + \angle CBA = 180^{\circ}$$ Prove that the line $BP$, $CQ$, and the perpendicular bisector of $BC$ intersect at one point.

2023 HMNT, 6

Tags:

There are five people in a room. They each simultaneously pick two of the other people in the room independently and uniformly at random and point at them. Compute the probability that there exists a group of three people such that each of them is pointing at the other two in the group.

2009 Ukraine National Mathematical Olympiad, 2

Tags:

On the party every boy gave $1$ candy to every girl and every girl gave $1$ candy to every boy. Then every boy ate $2$ candies and every girl ate $3$ candies. It is known that $\frac 14$ of all candies was eaten. Find the greatest possible number of children on the party.

2012 Princeton University Math Competition, A2 / B4

Tags: algebra

If $x, y$, and $z$ are real numbers with $\frac{x - y}{z}+\frac{y - z}{x}+\frac{z - x}{y}= 36$, find $2012 +\frac{x - y}{z}\cdot \frac{y - z}{x}\cdot\frac{z - x}{y}$ .

1998 Romania National Olympiad, 1

Tags: inequalities , algebra , perfect square , number theory

Let $n$ be a positive integer and $x_1,x_2,...,x_n$ be integer numbers such that $$x_1^2+x_2^2+...+x_n^2+ n^3 \le (2n - 1)(x_1+x_2+...+x_n ) + n^2$$ . Show that : a) $x_1,x_2,...,x_n$ are non-negative integers b) the number $x_1+x_2+...+x_n+n+1$ is not a perfect square.