Olympiad problems

1953 Moscow Mathematical Olympiad, 257

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.

2022/2023 Tournament of Towns, P5

Tags: number theory , sum of digits

In an infinite arithmetic progression of positive integers there are two integers with the same sum of digits. Will there necessarily be one more integer in the progression with the same sum of digits? [i]Proposed by A. Shapovalov[/i]

2023 Malaysian Squad Selection Test, 7

Tags: number theory

Find all polynomials with integer coefficients $P$ such that for all positive integers $n$, the sequence $$0, P(0), P(P(0)), \cdots$$ is eventually constant modulo $n$. [i]Proposed by Ivan Chan Kai Chin[/i]

1984 IMO Longlists, 23

Tags: combinatorics

A $2\times 2\times 12$ box fixed in space is to be filled with twenty-four $1 \times 1 \times 2$ bricks. In how many ways can this be done?

2023 Korea Summer Program Practice Test, P3

Tags: geometry , incenter , circumcircle

$\triangle ABC$ is a triangle such that $\angle A = 60^{\circ}$. The incenter of $\triangle ABC$ is $I$. $AI$ intersects with $BC$ at $D$, $BI$ intersects with $CA$ at $E$, and $CI$ intersects with $AB$ at $F$, respectively. Also, the circumcircle of $\triangle DEF$ is $\omega$. The tangential line of $\omega$ at $E$ and $F$ intersects at $T$. Show that $\angle BTC \ge 60^{\circ}$