Olympiad problems

1968 Miklós Schweitzer, 7

For every natural number $ r$, the set of $ r$-tuples of natural numbers is partitioned into finitely many classes. Show that if $ f(r)$ is a function such that $ f(r)\geq 1$ and $ \lim _{r\rightarrow \infty} f(r)\equal{}\plus{}\infty$, then there exists an infinite set of natural numbers that, for all $ r$, contains $ r$-triples from at most $ f(r)$ classes. Show that if $ f(r) \not \rightarrow \plus{}\infty$, then there is a family of partitions such that no such infinite set exists. [i]P. Erdos, A. Hajnal[/i]

2001 Balkan MO, 4

Tags: 3d geometry , analytic geometry , combinatorics , geometry

A cube side 3 is divided into 27 unit cubes. The unit cubes are arbitrarily labeled 1 to 27 (each cube is given a different number). A move consists of swapping the cube labeled 27 with one of its 6 neighbours. Is it possible to find a finite sequence of moves at the end of which cube 27 is in its original position, but cube $n$ has moved to the position originally occupied by $27-n$ (for each $n = 1, 2, \ldots , 26$)?

2007 IMAR Test, 2

Tags: geometry , 3d geometry , sphere , combinatorics

Denote by $ \mathcal{C}$ the family of all configurations $ C$ of $ N > 1$ distinct points on the sphere $ S^2,$ and by $ \mathcal{H}$ the family of all closed hemispheres $ H$ of $ S^2.$ Compute: $ \displaystyle\max_{H\in\mathcal{H}}\displaystyle\min_{C\in\mathcal{C}}|H\cap C|$, $ \displaystyle\min_{H\in\mathcal{H}}\displaystyle\max_{C\in\mathcal{C}}|H\cap C|$ $ \displaystyle\max_{C\in\mathcal{C}}\displaystyle\min_{H\in\mathcal{H}}|H\cap C|$ and $ \displaystyle\min_{C\in\mathcal{C}}\displaystyle\max_{H\in\mathcal{H}}|H\cap C|.$

2023 Tuymaada Olympiad, 6

Tags: number theory , euclidean algorithm

An $\textit{Euclidean step}$ transforms a pair $(a, b)$ of positive integers, $a > b$, to the pair $(b, r)$, where $r$ is the remainder when a is divided by $b$. Let us call the $\textit{complexity}$ of a pair $(a, b)$ the number of Euclidean steps needed to transform it to a pair of the form $(s, 0)$. Prove that if $ad - bc = 1$, then the complexities of $(a, b)$ and $(c, d)$ differ at most by $2$.

2000 Tournament Of Towns, 6

Tags: tournament , combinatorics

In the spring round of the Tournament of Towns this year, $6$ problems were posed in the Senior A-Level paper. In a certain country, each problem was solved by exactly $1000$ participants, but no two participants solved all $6$ problems between them. What is the smallest possible number of participants from this country in the spring round Senior A-Level paper? (R Zhenodarov)

2005 Estonia Team Selection Test, 5

Tags: coloring , combinatorics

On a horizontal line, $2005$ points are marked, each of which is either white or black. For every point, one finds the sum of the number of white points on the right of it and the number of black points on the left of it. Among the $2005$ sums, exactly one number occurs an odd number of times. Find all possible values of this number.

2020 Canadian Mathematical Olympiad Qualification, 8

Tags: radical , rational , equation , algebra

Find all pairs $(a, b)$ of positive rational numbers such that $\sqrt[b]{a}= ab$

2013 Today's Calculation Of Integral, 897

Tags: calculus , integration , geometry , geometric transformation , rotation , calculus computations , logarithm

Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.

2019 LIMIT Category C, Problem 2

Tags: inequalities , absolute value

Let $x,y\in[0,\infty)$. Which of the following is true? $\textbf{(A)}~\left|\log\left(1+x^2\right)-\log\left(1+y^2\right)\right|\le|x-y|$ $\textbf{(B)}~\left|\sin^2x-\sin^2y\right|\le|x-y|$ $\textbf{(C)}~\left|\tan^{-1}x-\tan^{-1}y\right|\le|x-y|$ $\textbf{(D)}~\text{None of the above}$

2004 AIME Problems, 2

Tags: absolute value , p2 catalyst , wilkinson catalyst , birch reduction

Set $A$ consists of $m$ consecutive integers whose sum is $2m$, and set $B$ consists of $2m$ consecutive integers whose sum is $m$. The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is $99$. Find $m$.

2000 Polish MO Finals, 2

Tags: geometry , trigonometry , circumcircle , analytic geometry , geometric transformation , reflection , cyclic quadrilateral

Let a triangle $ABC$ satisfy $AC = BC$; in other words, let $ABC$ be an isosceles triangle with base $AB$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$. Denote by $M$ the midpoint of the segment $AB$. Show that $\angle APM + \angle BPC = 180^{\circ}$.

2006 Iran Team Selection Test, 5

Tags: circumcircle , trigonometry , parallelogram , angle bisector , geometry

Let $ABC$ be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to $A$. Suppose that the outer inscribed circle with respect to $A$ touches $BC,AC,AB$ at $M,N,L$. Prove that $O$ (Center of circumcircle) is the orthocenter of $MNL$.

2022 Purple Comet Problems, 15

Tags:

Find the number of rearrangements of the nine letters $\text{AAABBBCCC}$ where no three consecutive letters are the same. For example, count $\text{AABBCCABC}$ and $\text{ACABBCCAB}$ but not $\text{ABABCCCBA}.$

2025 NEPALTST, 1

Tags: geometry

Consider a triangle $\triangle ABC$ and some point $X$ on $BC$. The perpendicular from $X$ to $AB$ intersects the circumcircle of $\triangle AXC$ at $P$ and the perpendicular from $X$ to $AC$ intersects the circumcircle of $\triangle AXB$ at $Q$. Show that the line $PQ$ does not depend on the choice of $X$. [i](Shining Sun, USA)[/i]

2012 National Olympiad First Round, 8

Tags: algebra , binomial theorem

In how many different ways can one select two distinct subsets of the set $\{1,2,3,4,5,6,7\}$, so that one includes the other? $ \textbf{(A)}\ 2059 \qquad \textbf{(B)}\ 2124 \qquad \textbf{(C)}\ 2187 \qquad \textbf{(D)}\ 2315 \qquad \textbf{(E)}\ 2316$

2009 Abels Math Contest (Norwegian MO) Final, 4a

Tags: algebra , inequalities

Show that $\left(\frac{2010}{2009}\right)^{2009}> 2$.

2009 Belarus Team Selection Test, 1

Tags: algebra , binary operation , operation

On R a binary algebraic operation ''*'' is defined which satisfies the following two conditions: i) for all $a,b \in R$, there exists a unique $x \in R$ such that $x *a=b$ (write $x=b/a$) ii) $(a*b)*c= (a*c)* (b*c)$ for all $a,b,c \in R$ a) Is this operation necesarily commutative (i.e. $a*b=b*a$ for all $a,b \in R$) ? b) Prove that $(a/b)/c = (a/c) / (b/c)$ and $(a/b)*c = (a*c) / (b*c)$ for all $a,b,c \in R$. A. Mirotin

2019 USEMO, 3

Tags: chessboard , combinatorics

Consider an infinite grid $\mathcal G$ of unit square cells. A [i]chessboard polygon[/i] is a simple polygon (i.e. not self-intersecting) whose sides lie along the gridlines of $\mathcal G$. Nikolai chooses a chessboard polygon $F$ and challenges you to paint some cells of $\mathcal G$ green, such that any chessboard polygon congruent to $F$ has at least $1$ green cell but at most $2020$ green cells. Can Nikolai choose $F$ to make your job impossible? [i]Nikolai Beluhov[/i]

1985 Tournament Of Towns, (092) T3

Tags: algebra , inequalities

Three real numbers $a, b$ and $c$ are given . It is known that $a + b + c >0 , bc+ ca + ab > 0$ and $abc > 0$ . Prove that $a > 0 , b > 0$ and $c > 0$ .

2016 Saudi Arabia GMO TST, 3

Tags: geometry , incenter , perpendicular , incircle

Let $ABC$ be a triangle with incenter $I$ . Let $CI, BI$ intersect $AB, AC$ at $D, E$ respectively. Denote by $\Delta_b,\Delta_c$ the lines symmetric to the lines $AB, AC$ with respect to $CD, BE$ correspondingly. Suppose that $\Delta_b,\Delta_c$ meet at $K$. a) Prove that $IK \perp BC$. b) If $I \in (K DE)$, prove that $BD + C E = BC$.

1985 Poland - Second Round, 6

Tags: geometry , 3d geometry , right angle

There are various points in space $ A, B, C_0, C_1, C_2 $, with $ |AC_i| = 2 |BC_i| $ for $ i = 0,1,2 $ and $ |C_1C_2|=\frac{4}{3}|AB| $. Prove that the angle $ C_1C_0C_2 $ is right and the points $ A, B, C_1, C_2 $ lie on one plane.

1976 AMC 12/AHSME, 14

Tags:

The measures of the interior angles of a convex polygon are in arithmetic progression. If the smallest angle is $100^\circ$, and the largest is $140^\circ$, then the number of sides the polygon has is $\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad \textbf{(E) }12$

2005 National High School Mathematics League, 12

Tags:

If the sum of all digits of a number is $7$, then we call it [i]lucky number[/i]. Put all [i]lucky numbers[/i] in order (from small to large): $a_1,a_2,\cdots,a_n,\cdots$. If $a_n=2005$, then $a_{5n}=$________.

2003 All-Russian Olympiad Regional Round, 10.8

Tags: combinatorics , weighing

In a set of 17 externally identical coins, two are counterfeit, differing from the rest in weight. It is known that the total weight of two counterfeit coins is twice the weight of a real one.s it always possible to determine the couple of counterfeit coins, having made $5$ weighings on a cup scale without weights? (It is not necessary to determine which of the fakes is heavier.)

2006 BAMO, 3

Tags: centroid , median , geometry

In triangle $ABC$, choose point $A_1$ on side $BC$, point $B_1$ on side $CA$, and point $C_1$ on side $AB$ in such a way that the three segments $AA_1, BB_1$, and $CC_1$ intersect in one point $P$. Prove that $P$ is the centroid of triangle $ABC$ if and only if $P$ is the centroid of triangle $A_1B_1C_1$. Note: A median in a triangle is a segment connecting a vertex of the triangle with the midpoint of the opposite side. The centroid of a triangle is the intersection point of the three medians of the triangle. The centroid of a triangle is also known by the names ”center of mass” and ”medicenter” of the triangle.