Olympiad problems

1995 Belarus National Olympiad, Problem 8

Tags: invariant , combinatorics

Five numbers 1,2,3,4,5 are written on a blackboard. A student may erase any two of the numbers a and b on the board and write the numbers a+b and ab replacing them. If this operation is performed repeatedly, can the numbers 21,27,64,180,540 ever appear on the board?

2020 CHMMC Winter (2020-21), 7

Tags: number theory

For any positive integer $n$, let $f(n)$ denote the sum of the positive integers $k \le n$ such that $k$ and $n$ are relatively prime. Let $S$ be the sum of $\frac{1}{f(m)}$ over all positive integers $m$ that are divisible by at least one of $2$, $3$, or $5$, and whose prime factors are only $2$, $3$, or $5$. Then $S = \frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $p+q$.

2016 JBMO Shortlist, 3

Tags: number theory

Find all positive integers $n$ such that the number $A_n =\frac{ 2^{4n+2}+1}{65}$ is a) an integer, b) a prime.

1985 Canada National Olympiad, 3

Tags: perimeter , trigonometry , calculus , integration , inequalities , geometry

Let $P_1$ and $P_2$ be regular polygons of 1985 sides and perimeters $x$ and $y$ respectively. Each side of $P_1$ is tangent to a given circle of circumference $c$ and this circle passes through each vertex of $P_2$. Prove $x + y \ge 2c$. (You may assume that $\tan \theta \ge \theta$ for $0 \le \theta < \frac{\pi}{2}$.)

2024 Turkey EGMO TST, 3

Tags: graph theory , combinatorics

Initially, all edges of the $K_{2024}$ are painted with $13$ different colors. If there exist $k$ colors such that the subgraph constructed by the edges which are colored with these $k$ colors is connected no matter how the initial coloring was, find the minimum value of $k$.

1953 Moscow Mathematical Olympiad, 257

Tags: recurrence relation , sequence , inequalities , algebra

Let $x_0 = 10^9$, $x_n = \frac{x^2_{n-1}+2}{2x_{n-1}}$ for $n > 0$. Prove that $0 < x_{36} - \sqrt2 < 10^{-9}$.

1985 All Soviet Union Mathematical Olympiad, 396

Tags: sum of digits , sum of squares , sum , number theory , decimal representation , digit

Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?

1982 Swedish Mathematical Competition, 5

Tags: coloring , combinatorial geometry , combinatorics

Each point in a $12 \times 12$ array is colored red, white or blue. Show that it is always possible to find $4$ points of the same color forming a rectangle with sides parallel to the sides of the array.

1978 Bundeswettbewerb Mathematik, 2

Tags: combinatorics , arrangement , line

A set of $n^2$ counters are labeled with $1,2,\ldots, n$, each label appearing $n$ times. Can one arrange the counters on a line in such a way that for all $x \in \{1,2,\ldots, n\}$, between any two successive counters with the label $x$ there are exactly $x$ counters (with labels different from $x$)?

1999 German National Olympiad, 6a

Tags: isosceles right triangle , combinatorial geometry , combinatorics

Suppose that an isosceles right-angled triangle is divided into $m$ acute-angled triangles. Find the smallest possible $m$ for which this is possible.

2006 Spain Mathematical Olympiad, 2

Tags: number theory , product , consecutive , perfect square , perfect cube

Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.

2014 ELMO Shortlist, 9

Tags: inequalities

Let $a$, $b$, $c$ be positive reals. Prove that \[ \sqrt{\frac{a^2(bc+a^2)}{b^2+c^2}}+\sqrt{\frac{b^2(ca+b^2)}{c^2+a^2}}+\sqrt{\frac{c^2(ab+c^2)}{a^2+b^2}}\ge a+b+c. \][i]Proposed by Robin Park[/i]

2011 All-Russian Olympiad Regional Round, 9.4

Tags: inequalities , calculus , three variable inequality

$x$, $y$ and $z$ are positive real numbers. Prove the inequality \[\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x}{y}+\frac{y}{z}+\frac{z}{x}.\] (Authors: A. Khrabrov, B. Trushin)

2011 Middle European Mathematical Olympiad, 5

Tags: symmetry , geometry

Let $ABCDE$ be a convex pentagon with all five sides equal in length. The diagonals $AD$ and $EC$ meet in $S$ with $\angle ASE = 60^\circ$. Prove that $ABCDE$ has a pair of parallel sides.

I Soros Olympiad 1994-95 (Rus + Ukr), 9.8

Tags: algebra , inequalities

Let $f(x) =x^2-2x$. Find all $x$ for which $f(f(x))<3$.

1999 Tournament Of Towns, 2

Tags: ratio , square , geometry , right triangle

$ABC$ is a right-angled triangle. A square $ABDE$ is constructed on the opposite side of the hypothenuse $AB$ from $C$. The bisector of $\angle C$ cuts $DE$ at $F$. If $AC = 1$ and $BC = 3$, compute $\frac{DF}{EF}$. (A Blinkov)

2017 QEDMO 15th, 11

Tags: sum , algebra

Calculate $$\frac{(2^1+3^1)(2^2+3^2)(2^4+3^4)(2^8+3^8)...(2^{2048}+3^{2048})+2^{4096}}{3^{4096}}$$

2016 Hanoi Open Mathematics Competitions, 1

Tags: exponential equation , algebra

If $2016 = 2^5 + 2^6 + ...+ 2^m$ then $m$ is equal to (A): $8$ (B): $9$ (C): $10$ (D): $11$ (E): None of the above.

2003 National Olympiad First Round, 13

Tags: geometry , angle bisector

Let $ABC$ be a triangle such that $|AB|=8$ and $|AC|=2|BC|$. What is the largest value of altitude from side $[AB]$? $ \textbf{(A)}\ 3\sqrt 2 \qquad\textbf{(B)}\ 3\sqrt 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ \dfrac {16}3 \qquad\textbf{(E)}\ 6 $

2009 Postal Coaching, 1

Tags: sequence , power of 2 , recurrence relation , algebra

Let $a_1, a_2, a_3, . . . , a_n, . . . $ be an infinite sequence of natural numbers in which $a_1$ is not divisible by $5$. Suppose $a_{n+1} = a_n + b_n$ where bn is the last digit of $a_n$, for every $n$. Prove that the sequence $\{a_n\}$ contains infinitely many powers of 2.

2013 Saudi Arabia BMO TST, 1

Tags: lattice , combinatorial geometry , combinatorics

The set $G$ is defined by the points $(x,y)$ with integer coordinates, $1 \le x \le 5$ and $1 \le y \le 5$. Determine the number of five-point sequences $(P_1, P_2, P_3, P_4, P_5)$ such that for $1 \le i \le 5$, $P_i = (x_i,i)$ is in $G$ and $|x_1 - x_2| = |x_2 - x_3| = |x_3 - x_4|=|x_4 - x_5| = 1$.

2004 Germany Team Selection Test, 1

Tags: vector , geometric transformation , similar triangles , geometry

Let $ABC$ be an acute triangle, and let $M$ and $N$ be two points on the line $AC$ such that the vectors $MN$ and $AC$ are identical. Let $X$ be the orthogonal projection of $M$ on $BC$, and let $Y$ be the orthogonal projection of $N$ on $AB$. Finally, let $H$ be the orthocenter of triangle $ABC$. Show that the points $B$, $X$, $H$, $Y$ lie on one circle.