Olympiad problems

2015 Swedish Mathematical Competition, 5

Given a finite number of points in the plane as well as many different rays starting at the origin. It is always possible to pair the points with the rays so that they parallell displaced rays starting in respective points do not intersect?

2019 India IMO Training Camp, P2

Tags: geometry

Let $ABC$ be an acute-angled scalene triangle with circumcircle $\Gamma$ and circumcenter $O$. Suppose $AB < AC$. Let $H$ be the orthocenter and $I$ be the incenter of triangle $ABC$. Let $F$ be the midpoint of the arc $BC$ of the circumcircle of triangle $BHC$, containing $H$. Let $X$ be a point on the arc $AB$ of $\Gamma$ not containing $C$, such that $\angle AXH = \angle AFH$. Let $K$ be the circumcenter of triangle $XIA$. Prove that the lines $AO$ and $KI$ meet on $\Gamma$. [i]Proposed by Anant Mudgal[/i]

1982 Spain Mathematical Olympiad, 6

Tags: algebra , inequalities

Prove that if $u, v$ are any nonnegative real numbers, and $a,b$ positive real numbers such that $a + b = 1$, then $$u^a v^b \le au + bv.$$

2010 Purple Comet Problems, 11

Tags: probability , number theory , relatively prime

A jar contains one white marble, two blue marbles, three red marbles, and four green marbles. If you select two of these marbles without replacement, the probability that both marbles will be the same color is $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

2020 Peru Cono Sur TST., P2

Tags: functional equation , number theory

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ that satisfy the conditions: $i) f(f(x)) = xf(x) - x^2 + 2,\forall x\in\mathbb{Z}$ $ii) f$ takes all integer values

2011 Vietnam National Olympiad, 3

Tags: polynomial , algebra

Let $n\in\mathbb N$ and define $P(x,y)=x^n+xy+y^n.$ Show that we cannot obtain two non-constant polynomials $G(x,y)$ and $H(x,y)$ with real coefficients such that $P(x,y)=G(x,y)\cdot H(x,y).$

1994 Bulgaria National Olympiad, 6

Tags: combinatorics , subset , set

Let $n$ be a positive integer and $A$ be a family of subsets of the set $\{1,2,...,n\},$ none of which contains another subset from A . Find the largest possible cardinality of $A$ .

2010 Today's Calculation Of Integral, 593

Tags: calculus , integration , trigonometry , function , calculus computations

For a positive integer $m$, prove the following ineqaulity. $0\leq \int_0^1 \left(x+1-\sqrt{x^2+2x\cos \frac{2\pi}{2m+1}+1\right)dx\leq 1.}$ 1996 Osaka University entrance exam

2007 Princeton University Math Competition, 1

Tags:

If \[ \begin {eqnarray*} x + y + z + w = 20 \\ y + 2z - 3w = 28 \\ x - 2y + z = 36 \\ -7x - y + 5z + 3w = 84 \]then what is $(x,y,z,w)$?

2020 Iran RMM TST, 3

Tags: combinatorics

There are n stations $1,2,...,n$ in a broken road (like in Cars) in that order such that the distance between station $i$ and $i+1$ is one unit. The distance betwen two positions of cars is the minimum units needed to be fixed so that every car can go from its place in the first position to its place in the second (two cars can be in the same station in a position). Prove that for every $\alpha<1$ thre exist $n$ and $100^n$ positions such that the distance of every two position is at least $n\alpha$.

I Soros Olympiad 1994-95 (Rus + Ukr), 10.5

Tags: algebra , inequalities

Let $a_1,a_2,...,a_{1994}$ be real numbers in the interval $[-1,1]$, $$S=\frac{a_1+a_2+...+a_{1994}}{1994}.$$ Prove that for an arbitrary natural , $1\le n \le 1994$, holds the inequality $$| a_1+a_2+...+a_n - nS | \le 997.$$

1988 AMC 12/AHSME, 2

Tags: ratio

Triangles $ABC$ and $XYZ$ are similar, with $A$ corresponding to $X$ and $B$ to $Y$. If $AB=3$, $BC=4$, and $XY=5$, then $YZ$ is: $ \textbf{(A)}\ 3\frac 3 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 6\frac 1 4 \qquad \textbf{(D)}\ 6\frac 2 3 \qquad \textbf{(E)}\ 8$

2005 Romania Team Selection Test, 4

Tags: modular arithmetic , number theory

a) Prove that there exists a sequence of digits $\{c_n\}_{n\geq 1}$ such that or each $n\geq 1$ no matter how we interlace $k_n$ digits, $1\leq k_n\leq 9$, between $c_n$ and $c_{n+1}$, the infinite sequence thus obtained does not represent the fractional part of a rational number. b) Prove that for $1\leq k_n\leq 10$ there is no such sequence $\{c_n\}_{n\geq 1}$. [i]Dan Schwartz[/i]

2017 Iranian Geometry Olympiad, 4

Tags: geometry

$P_1,P_2,\ldots,P_{100}$ are $100$ points on the plane, no three of them are collinear. For each three points, call their triangle [b]clockwise[/b] if the increasing order of them is in clockwise order. Can the number of [b]clockwise[/b] triangles be exactly $2017$? [i]Proposed by Morteza Saghafian[/i]

2009 National Olympiad First Round, 3

Tags:

If $ x \equal{} \sqrt [3]{11 \plus{} \sqrt {337}} \plus{} \sqrt [3]{11 \minus{} \sqrt {337}}$, then $ x^3 \plus{} 18x$ = ? $\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 22 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 10$

2022 Czech and Slovak Olympiad III A, 2

Tags: number theory , palindrome

We say that a positive integer $k$ is [i]fair [/i] if the number of $2021$-digit palindromes that are a multiple of $k$ is the same as the number of $2022$-digit palindromes that are a multiple of $k$. Does the set $M = \{1, 2,..,35\}$ contain more numbers that are fair or those that are not fair? (A palindrome is an integer that reads the same forward and backward.) [i](David Hruska)[/i]

2012 Kosovo Team Selection Test, 4

Tags: monovariant , search , combinatorics

Each term in a sequence $1,0,1,0,1,0...$starting with the seventh is the sum of the last 6 terms mod 10 .Prove that the sequence $...,0,1,0,1,0,1...$ never occurs

2012 India IMO Training Camp, 1

Tags: trigonometry , geometry

Let $ABC$ be an isosceles triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD=2DC$. Let $P$ be a point on the segment $AD$ such that $\angle BAC=\angle BPD$. Prove that $\angle BAC=2\angle DPC$.

2008 Polish MO Finals, 3

Tags: trigonometry , geometry

In a convex pentagon $ ABCDE$ in which $ BC\equal{}DE$ following equalities hold: \[ \angle ABE \equal{}\angle CAB \equal{}\angle AED\minus{}90^{\circ},\qquad \angle ACB\equal{}\angle ADE\] Show that $ BCDE$ is a parallelogram.

1959 Kurschak Competition, 1

Tags: number theory , integer

$a, b, c$ are three distinct integers and $n$ is a positive integer. Show that $$\frac{a^n}{(a - b)(a - c)}+\frac{ b^n}{(b - a)(b - c)} +\frac{ c^n}{(c - a)(c - b)}$$ is an integer.

2024 Brazil Team Selection Test, 4

Tags: algebra , functional equation , function

Find all pairs of positive integers $ (a, b) $ such that $ f(x) = x $ is the only function $ f : \mathbb{R} \to \mathbb{R} $ that satisfies \[ f^a(x)f^b(y) + f^b(x)f^a(y) = 2xy \quad \text{for all } x, y \in \mathbb{R}. \] Here, $ f^n(x) $ represents the function obtained by applying $ f $ $ n $ times to $ x $. That is, $ f^1(x) = f(x) $ and $ f^{n+1}(x) = f(f^n(x))$ for all $n \geq 1$.

2023 239 Open Mathematical Olympiad, 1

Tags: combinatorics , board , coloring

Each cell of an $100\times 100$ board is divided into two triangles by drawing some diagonal. What is the smallest number of colors in which it is always possible to paint these triangles so that any two triangles having a common side or vertex have different colors?

2022 Moscow Mathematical Olympiad, 2

Tags: combinatorics

The volleyball championship with $16$ teams was held in one round (each team played with each exactly one times, there are no draws in volleyball). It turned out that some two teams won the same number of matches. Prove there are the three teams that beat each other in a round robin (i.e. A beat B, B beat C, and C beat A).

2025 AIME, 14

Tags: geometry , quadrilateral

Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such \[AK = AL = BK = CL = KL = 14.\] The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$

2013 AMC 12/AHSME, 24

Tags: probability , geometry , inequalities , trigonometry , triangle inequality , trig identities , law of sines

Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? $ \textbf{(A)} \ \frac{553}{715} \qquad \textbf{(B)} \ \frac{443}{572} \qquad \textbf{(C)} \ \frac{111}{143} \qquad \textbf{(D)} \ \frac{81}{104} \qquad \textbf{(E)} \ \frac{223}{286}$