Olympiad problems

2011 SEEMOUS, Problem 4

Let $f:[0,1]\to\mathbb R$ be a twice continuously differentiable increasing function. Define the sequences given by $L_n=\frac1n\sum_{k=0}^{n-1}f\left(\frac kn\right)$ and $U_n=\frac1n\sum_{k=0}^nf\left(\frac kn\right)$, $n\ge1$. 1. The interval $[L_n,U_n]$ is divided into three equal segments. Prove that, for large enough $n$, the number $I=\int^1_0f(x)\text dx$ belongs to the middle one of these three segments.

2009 Tournament Of Towns, 1

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Is it possible to cut a square into nine squares and colour one of them white, three of them grey and ve of them black, such that squares of the same colour have the same size and squares of different colours will have different sizes? [i](3 points)[/i]

2019 All-Russian Olympiad, 6

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There is point $D$ on edge $AC$ isosceles triangle $ABC$ with base $BC$. There is point $K$ on the smallest arc $CD$ of circumcircle of triangle $BCD$. Ray $CK$ intersects line parallel to line $BC$ through $A$ at point $T$. Let $M$ be midpoint of segment $DT$. Prove that $\angle AKT=\angle CAM$.

2024 LMT Fall, 32

Tags: guts

Let $a$ and $b$ be positive integers such that\[a^2+(a+1)^2=b^4.\]Find the least possible value of $a+b$.

2004 Greece Junior Math Olympiad, 2

Tags: geometry , trapezoid , area , isosceles

Let $ABCD$ be a rectangle. Let $K,L$ be the midpoints of $BC, AD$ respectively. From point $B$ the perpendicular line on $AK$, intersects $AK$ at point $E$ and $CL$ at point $Z$. a) Prove that the quadrilateral $AKZL$ is an isosceles trapezoid b) Prove that $2S_{ABKZ}=S_{ABCD}$ c) If quadrilateral $ABCD$ is a square of side $a$, calculate the area of the isosceles trapezoid $AKZL$ in terms of side $BC=a$

1984 Putnam, A1

Tags: geometry , 3d geometry , prism

Let $A$ be a solid $a\times b\times c$ rectangular brick, where $a,b,c>0$. Let $B$ be the set of all points which are a distance of at most one from some point of $A$. Express the volume of $B$ as a polynomial in $a,b,$ and $c$.

2011 Kazakhstan National Olympiad, 1

Tags: ratio , trigonometry , geometry

Inscribed in a triangle $ABC$ with the center of the circle $I$ touch the sides $AB$ and $AC$ at points $C_{1}$ and $B_{1}$, respectively. The point $M$ divides the segment $C_{1}B_{1}$ in a 3:1 ratio, measured from $C_{1}$. $N$ - the midpoint of $AC$. Prove that the points $I, M, B_{1}, N$ lie on a circle, if you know that $AC = 3 (BC-AB)$.

2016 AMC 12/AHSME, 7

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Which of these describes the graph of $x^2(x+y+1)=y^2(x+y+1)$ ? $\textbf{(A)}\ \text{two parallel lines}$\\ $\textbf{(B)}\ \text{two intersecting lines}$\\$\textbf{(C)}\ \text{three lines that all pass through a common point}$\\ $\textbf{(D)}\ \text{three lines that do not all pass through a common point}$\\$\textbf{(E)}\ \text{a line and a parabola}$

2023 China Team Selection Test, P12

Tags: combinatorial geometry , combinatorics , analytic geometry , lattice

Prove that there exists some positive real number $\lambda$ such that for any $D_{>1}\in\mathbb{R}$, one can always find an acute triangle $\triangle ABC$ in the Cartesian plane such that [list] [*] $A, B, C$ lie on lattice points; [*] $AB, BC, CA>D$; [*] $S_{\triangle ABC}<\frac{\sqrt 3}{4}D^2+\lambda\cdot D^{4/5}$.

2009 India Regional Mathematical Olympiad, 5

Tags: geometry , function , induction , circumcircle , conic , inequalities , ellipse

A convex polygon is such that the distance between any two vertices does not exceed $ 1$. $ (i)$ Prove that the distance between any two points on the boundary of the polygon does not exceed $ 1$. $ (ii)$ If $ X$ and $ Y$ are two distinct points inside the polygon, prove that there exists a point $ Z$ on the boundary of the polygon such that $ XZ \plus{} YZ\le1$.

1989 Vietnam National Olympiad, 3

Tags: geometry

Let be given a parallelepiped $ ABCD.A'B'C'D'$. Show that if a line $ \Delta$ intersects three of the lines $ AB'$, $ BC'$, $ CD'$, $ DA'$, then it intersects also the fourth line.

2008 India National Olympiad, 5

Tags: angle bisector , geometry , incenter , circumcircle , geometric transformation , homothety

Let $ ABC$ be a triangle; $ \Gamma_A,\Gamma_B,\Gamma_C$ be three equal, disjoint circles inside $ ABC$ such that $ \Gamma_A$ touches $ AB$ and $ AC$; $ \Gamma_B$ touches $ AB$ and $ BC$; and $ \Gamma_C$ touches $ BC$ and $ CA$. Let $ \Gamma$ be a circle touching circles $ \Gamma_A, \Gamma_B, \Gamma_C$ externally. Prove that the line joining the circum-centre $ O$ and the in-centre $ I$ of triangle $ ABC$ passes through the centre of $ \Gamma$.

2019 Latvia Baltic Way TST, 2

Tags: functional equation , algebra , function

Let $\mathbb R$ be set of real numbers. Determine all functions $f:\mathbb R\to \mathbb R$ such that $$f(y^2 - f(x)) = yf(x)^2+f(x^2y+y)$$ holds for all real numbers $x; y$

2019 Switzerland - Final Round, 3

Tags: algebra

Find all periodic sequences $x_1,x_2,\dots$ of strictly positive real numbers such that $\forall n \geq 1$ we have $$x_{n+2}=\frac{1}{2} \left( \frac{1}{x_{n+1}}+x_n \right)$$

1972 Putnam, B4

Tags: polynomial , integer

Show that for $n > 1$ we can find a polynomial $P(a, b, c)$ with integer coefficients such that $$P(x^{n},x^{n+1},x+x^{n+2})=x.$$

1993 Tournament Of Towns, (386) 4

Tags: combinatorics

Diagonals of a $1$ by $1$ square are arranged in an $8$ by $8$ table (one in each $1$ by $1 $ square). Consider the union $W$ of all $64$ diagonals drawn. The set $W$ consists of several connected pieces (two points belong to the same piece if and only if W contains a path between them). Can the number of the pieces be greater than (a) $15$, (b) $20$? (NB Vassiliev)

2011 May Olympiad, 4

Tags: combinatorics

Given $n$ points in a circle, Arnaldo write 0 or 1 in all the points. Bernado can do a operation, he can chosse some point and change its number and the numbers of the points on the right and left side of it. Arnaldo wins if Bernado can´t change all the numbers in the circle to 0, and Bernado wins if he can a) Show that Bernado can win if $n=101$ b) Show that Arnaldo wins if $n=102$

2021 South East Mathematical Olympiad, 3

Tags: algebra , number theory

Let $p$ be an odd prime and $\{u_i\}_{i\ge 0}$be an integer sequence. Let $v_n=\sum_{i=0}^{n} C_{n}^{i} p^iu_i$ where $C_n^i$ denotes the binomial coefficients. If $v_n=0$ holds for infinitely many $n$ , prove that it holds for every positive integer $n$.

2009 Saint Petersburg Mathematical Olympiad, 6

Tags: algebra

$(x_n)$ is sequence, such that $x_{n+2}=|x_{n+1}|-x_n$. Prove, that it is periodic.

2017 CCA Math Bonanza, I2

Tags: geometry , perimeter , rectangle

A rectangle is inscribed in a circle of area $32\pi$ and the area of the rectangle is $34$. Find its perimeter. [i]2017 CCA Math Bonanza Individual Round #2[/i]

2012 BAMO, 1

Tags: chessboard , maximum , combinatorics

Hugo places a chess piece on the top left square of a $20 \times 20$ chessboard and makes $10$ moves with it. On each of these $10$ moves, he moves the piece either one square horizontally (left or right) or one square vertically (up or down). After the last move, he draws an $X$ on the square that the piece occupies. When Hugo plays the game over and over again, what is the largest possible number of squares that could eventually be marked with an $X$? Prove that your answer is correct.

1982 IMO Longlists, 6

Tags: geometry

On the three distinct lines $a, b$, and $c$ three points $A, B$, and $C$ are given, respectively. Construct three collinear points $X, Y,Z$ on lines $a, b, c$, respectively, such that $\frac{BY}{AX} = 2$ and $ \frac{CZ}{AX} = 3$.

2021 Princeton University Math Competition, B2

Tags: number theory , induction

Let $p$ be an odd prime. Prove that for every integer $k$, there exist integers $a, b$ such that $p|a^2 + b^2 - k$.

2024 Belarus - Iran Friendly Competition, 1.1

Tags: combinatorics

Given a polyhedron $P$. Mikita claims that he can write one integer on each face of $P$ such that not all the written numbers are zeros, and for each vertex $V$ of $P$ the sum of numbers on faces containing $V$ is equals to 0. Matvei claims that he can write one integer in each vertex of $P$ such that not all the written numbers are zeros, and for each face $F$ of $P$ the sum of numbers in vertices belonging to $F$ is equals to 0. Show that if the number of edges of polyhedron $P$ is odd, then at least one of the boys is right.

2013 EGMO, 6

Tags: combinatorics , pigeonhole principle

Snow White and the Seven Dwarves are living in their house in the forest. On each of $16$ consecutive days, some of the dwarves worked in the diamond mine while the remaining dwarves collected berries in the forest. No dwarf performed both types of work on the same day. On any two different (not necessarily consecutive) days, at least three dwarves each performed both types of work. Further, on the first day, all seven dwarves worked in the diamond mine. Prove that, on one of these $16$ days, all seven dwarves were collecting berries.