Olympiad problems

1952 Miklós Schweitzer, 1

Find all convex polyhedra which have no diagonals (that is, for which every segment connecting two vertices lies on the boundary of the polyhedron).

2005 All-Russian Olympiad Regional Round, 11.7

Tags: inequalities , number theory

11.7 Let $N$ be a number of perfect squares from $\{1,2,...,10^{20}\}$, which 17-th digit from the end is 7, and $M$ be a number of perfect squares from $\{1,2,...,10^{20}\}$, which 17-th digit from the end is 8. Compare $M$ and $N$. ([i]A. Golovanov[/i])

2007 Tournament Of Towns, 3

Tags: number theory

Determine all finite increasing arithmetic progressions in which each term is the reciprocal of a positive integer and the sum of all the terms is $1$.

KoMaL A Problems 2023/2024, A. 881

Tags: combinatorics , chessboard , chess king

We visit all squares exactly once on a $n\times n$ chessboard (colored in the usual way) with a king. Find the smallest number of times we had to switch colors during our walk. [i]Proposed by Dömötör Pálvölgyi, Budapest[/i]

2012 Tournament of Towns, 7

Tags: combinatorics

Konstantin has a pile of $100$ pebbles. In each move, he chooses a pile and splits it into two smaller ones until he gets $100 $ piles each with a single pebble. (a) Prove that at some point, there are $30$ piles containing a total of exactly $60$ pebbles. (b) Prove that at some point, there are $20$ piles containing a total of exactly $60$ pebbles. (c) Prove that Konstantin may proceed in such a way that at no point, there are $19$ piles containing a total of exactly $60$ pebbles.

2006 Estonia Math Open Junior Contests, 8

Tags: geometry

Two non-intersecting circles, not lying inside each other, are drawn in the plane. Two lines pass through a point P which lies outside each circle. The first line intersects the first circle at A and A′ and the second circle at B and B′; here A and B are closer to P than A′ and B′, respectively, and P lies on segment AB. Analogously, the second line intersects the first circle at C and C′ and the second circle at D and D′. Prove that the points A, B, C, D are concyclic if and only if the points A′, B′, C′, D′ are concyclic.

2020/2021 Tournament of Towns, P1

Tags: geometry

Is it possible to select 100 points on a circle so that there are exactly 1000 right triangles with the vertices at selected points? [i]Sergey Dvoryaninov[/i]

2017 Brazil Team Selection Test, 4

Tags: algebra , functional equation

Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

2005 Colombia Team Selection Test, 1

Tags: geometry , 3d geometry , number theory

Let $a,b,c$ be integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$ prove that $abc$ is a perfect cube!

2016 Estonia Team Selection Test, 11

Tags: number theory , perfect square

Find all positive integers $n$ such that $(n^2 + 11n - 4) \cdot n! + 33 \cdot 13^n + 4$ is a perfect square

2008 Bundeswettbewerb Mathematik, 2

Tags: algebra

Represent the number $ 2008$ as a sum of natural number such that the addition of the reciprocals of the summands yield 1.

2021 Romania National Olympiad, 1

Tags: geometry , perpendicular bisector , circles

Let $\mathcal C$ be a circle centered at $O$ and $A\ne O$ be a point in its interior. The perpendicular bisector of the segment $OA$ meets $\mathcal C$ at the points $B$ and $C$, and the lines $AB$ and $AC$ meet $\mathcal C$ again at $D$ and $E$, respectively. Show that the circles $(OBC)$ and $(ADE)$ have the same centre. [i]Ion Pătrașcu, Ion Cotoi[/i]

1992 IMO Longlists, 43

Tags: floor function , number theory , relatively prime , series summation , number theoretic functions

Find the number of positive integers $n$ satisfying $\phi(n) | n$ such that \[\sum_{m=1}^{\infty} \left( \left[ \frac nm \right] - \left[\frac{n-1}{m} \right] \right) = 1992\] What is the largest number among them? As usual, $\phi(n)$ is the number of positive integers less than or equal to $n$ and relatively prime to $n.$

1994 Putnam, 3

Tags: function

Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f^{\prime}(x) > f(x)$ for all $x,$ there is some number $N$ such that $f(x) > e^{kx}$ for all $x > N.$

2007 iTest Tournament of Champions, 4

Tags: inequalities

Find the smallest positive integer $k$ such that \[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\] for some ordered triple of positive integers $(a,b,c)$.

2009 Math Prize For Girls Problems, 3

Tags:

The [i]Fibonacci numbers[/i] are defined recursively by the equation \[ F_n \equal{} F_{n \minus{} 1} \plus{} F_{n \minus{} 2}\] for every integer $ n \ge 2$, with initial values $ F_0 \equal{} 0$ and $ F_1 \equal{} 1$. Let $ G_n \equal{} F_{3n}$ be every third Fibonacci number. There are constants $ a$ and $ b$ such that every integer $ n \ge 2$ satisfies \[ G_n \equal{} a G_{n \minus{} 1} \plus{} b G_{n \minus{} 2}.\] Compute the ordered pair $ (a, b)$.

2025 Serbia Team Selection Test for the IMO 2025, 4

Tags: combinatorics

For a permutation $\pi$ of the set $A = \{1, 2, \ldots, 2025\}$, define its [i]colorfulness [/i]as the greatest natural number $k$ such that: - For all $1 \le i, j \le 2025$, $i \ne j$, if $|i - j| < k$, then $|\pi(i) - \pi(j)| \ge k$. What is the maximum possible colorfulness of a permutation of the set $A$? Determine how many such permutations have maximal colorfulness. [i]Proposed by Pavle Martinović[/i]

2014 ELMO Shortlist, 1

Tags: inequalities , trigonometry , calculus , topology , three variable inequality

In a non-obtuse triangle $ABC$, prove that \[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]

2007 Harvard-MIT Mathematics Tournament, 7

Tags: college , modular arithmetic

A student at Harvard named Kevin Was counting his stones by $11$ He messed up $n$ times And instead counted $9$s And wound up at $2007$. How many values of $n$ could make this limerick true?

2002 Croatia National Olympiad, Problem 3

Tags: geometry , triangle

If two triangles with side lengths $a,b,c$ and $a',b',c'$ and the corresponding angle $\alpha,\beta,\gamma$ and $\alpha',\beta',\gamma'$ satisfy $\alpha+\alpha'=\pi$ and $\beta=\beta'$, prove that $aa'=bb'+cc'$.

1995 Czech And Slovak Olympiad IIIA, 4

Tags: digit , number theory , divisible

Do there exist $10000$ ten-digit numbers divisible by $7$, all of which can be obtained from one another by a reordering of their digits?

1953 Moscow Mathematical Olympiad, 250

Tags: number theory , combinatorics , divisible

Somebody wrote $1953$ digits on a circle. The $1953$-digit number obtained by reading these figures clockwise, beginning at a certain point, is divisible by $27$. Prove that if one begins reading the figures at any other place, one gets another $1953$-digit number also divisible by $27$.

1989 Putnam, B4

Tags: infinity , set theory

Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?

1988 Poland - Second Round, 5

Tags: geometry , rectangle , geometric inequalities , combinatorial geometry , combinatorics

Decide whether any rectangle that can be covered by 25 circles of radius 2 can also be covered by 100 circles of radius 1.

Russian TST 2019, P3

Tags: number theory

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.