Olympiad problems

2002 Germany Team Selection Test, 2

Tags: perimeter , geometry

Prove: If $x, y, z$ are the lengths of the angle bisectors of a triangle with perimeter 6, than we have: \[\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} \geq 1.\]

1999 Belarusian National Olympiad, 3

Tags: sequence , algebra

A sequence of numbers $a_1,a_2,...,a_{1999}$ is given. In each move it is allowed to choose two of the numbers, say $a_m,a_n$, and replace them by the numbers $$\frac{a_n^2}{a_m^2}-\frac{n}{m}\left(\frac{a_m^2}{a_n}-a_m\right), \frac{a_m^2}{a_n^2}-\frac{m}{n}\left(\frac{a_n^2}{a_m}-a_n\right) $$ respectively. Starting with the sequence $a_i = 1$ for $20 \nmid i$ and $a_i =\frac{1}{5}$ for $20 \mid i$, is it possible to obtain a sequence whose all terms are integers?

1994 All-Russian Olympiad, 4

Tags: search , analytic geometry , vector , probability , combinatorics

Real numbers are written on the squares of an infinite grid. Two figures consisting of finitely many squares are given. They may be translated anywhere on the grid as long as their squares coincide with those of the grid. It is known that wherever the first figure is translated, the sum of numbers it covers is positive. Prove that the second figure can be translated so that the sum of the numbers it covers is also positive.

1983 AIME Problems, 6

Tags: modular arithmetic , function , binomial theorem , number theory , algebra

Let $a_n = 6^n + 8^n$. Determine the remainder on dividing $a_{83}$ by 49.

2008 AMC 10, 7

Tags: algebra , difference of squares , special factorizations

The fraction \[\frac {(3^{2008})^2 - (3^{2006})^2}{(3^{2007})^2 - (3^{2005})^2}\] simplifies to which of the following? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac {9}{4} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac {9}{2} \qquad \textbf{(E)}\ 9$

KoMaL A Problems 2020/2021, A. 783

Tags: combinatorics

A polyomino is a figure which consists of unit squares joined together by their sides. (A polyomino may contain holes.) Let $n\ge3$ be a positive integer. Consider a grid of unit square cells which extends to infinity in all directions. Find, in terms of $n$, the greatest positive integer $C$ which satisfies the following condition: For every colouring of the cells of the grid in $n$ colours, there is some polyomino within the grid which contains at most $n-1$ colours and whose area is at least $C$. Proposed by Nikolai Beluhov, Stara Zagora, Bulgaria and Stefan Gerdjikov, Sofia, Bulgaria

1958 Czech and Slovak Olympiad III A, 2

Tags: construction , triangle , geometry

Construct a triangle $ABC$ given the magnitude of the angle $BCA$ and lengths of height $h_c$ and median $m_c$. Discuss conditions of solvability.

2021 Dutch IMO TST, 4

Tags: triangle inequality , geometry

Determine all positive integers $n$ with the following property: for each triple $(a, b, c)$ of positive real numbers there is a triple $(k, \ell, m)$ of non-negative integer numbers so that $an^k$, $bn^{\ell}$ and $cn^m$ are the lengths of the sides of a (non-degenerate) triangle shapes.

2003 Balkan MO, 3

Tags: function , algebra

Find all functions $f: \mathbb{Q}\to\mathbb{R}$ which fulfill the following conditions: a) $f(1)+1>0$; b) $f(x+y) -xf(y) -yf(x) = f(x)f(y) -x-y +xy$, for all $x,y\in\mathbb{Q}$; c) $f(x) = 2f(x+1) +x+2$, for every $x\in\mathbb{Q}$.

2007 AMC 12/AHSME, 19

Tags: geometry , rhombus , trigonometry

Rhombus $ ABCD$, with a side length $ 6$, is rolled to form a cylinder of volume $ 6$ by taping $ \overline{AB}$ to $ \overline{DC}.$ What is $ \sin(\angle ABC)$? $ \textbf{(A)}\ \frac {\pi}{9} \qquad \textbf{(B)}\ \frac {1}{2} \qquad \textbf{(C)}\ \frac {\pi}{6} \qquad \textbf{(D)}\ \frac {\pi}{4} \qquad \textbf{(E)}\ \frac {\sqrt3}{2}$

2012 Czech And Slovak Olympiad IIIA, 4

Tags: geometry , parallelogram , area , max

Inside the parallelogram $ABCD$ is a point $X$. Make a line that passes through point $X$ and divides the parallelogram into two parts whose areas differ from each other the most.

2015 APMO, 5

Tags: sequence , number theory

Determine all sequences $a_0 , a_1 , a_2 , \ldots$ of positive integers with $a_0 \ge 2015$ such that for all integers $n\ge 1$: (i) $a_{n+2}$ is divisible by $a_n$ ; (ii) $|s_{n+1} - (n + 1)a_n | = 1$, where $s_{n+1} = a_{n+1} - a_n + a_{n-1} - \cdots + (-1)^{n+1} a_0$ . [i]Proposed by Pakawut Jiradilok and Warut Suksompong, Thailand[/i]

IV Soros Olympiad 1997 - 98 (Russia), 10.6

Tags: combinatorics

Is it possible to arrange $n \times n$ in the cells of a square table the numbers $0$,$ 1$ or $2$ so that the sums of the numbers in rows and columns took on all different values from $1$ to $2n$? Consider two cases: a) $n$ is an odd number; b) $n$ is an even number.

1979 IMO Longlists, 14

Tags: combinatorics

Let $S$ be a set of $n^2 + 1$ closed intervals ($n$ a positive integer). Prove that at least one of the following assertions holds: [b](i)[/b] There exists a subset $S'$ of $n+1$ intervals from $S$ such that the intersection of the intervals in $S'$ is nonempty. [b](ii)[/b] There exists a subset $S''$ of $n + 1$ intervals from $S$ such that any two of the intervals in $S''$ are disjoint.

2016 HMNT, 9

Tags: geometry , 3d geometry , sphere

A cylinder with radius $15$ and height $16$ is inscribed in a sphere. Three congruent smaller spheres of radius $x$ are externally tangent to the base of the cylinder, externally tangent to each other, and internally tangent to the large sphere. What is the value of $x$?

2012 BMT Spring, 3

Tags:

Find the largest prime factor of \[ \frac{1}{\sum_{n=1}^\infty \frac{2012}{n(n+1)(n+2)(n+3)\dots(n+2012)}} \]

LMT Team Rounds 2021+, B13

Tags: number theory

Call a $4$-digit number $\overline{a b c d}$ [i]unnoticeable [/i] if $a +c = b +d$ and $\overline{a b c d} +\overline{c d a b}$ is a multiple of $7$. Find the number of unnoticeable numbers. Note: $a$, $b$, $c$, and $d$ are nonzero distinct digits. [i]Proposed by Aditya Rao[/i]

Estonia Open Junior - geometry, 2014.1.5

Tags: midpoint , geometry , concurrency

In a triangle $ABC$ the midpoints of $BC, CA$ and $AB$ are $D, E$ and $F$, respectively. Prove that the circumcircles of triangles $AEF, BFD$ and $CDE$ intersect all in one point.

2012 Bosnia Herzegovina Team Selection Test, 2

Tags: trigonometry , inequalities

Prove for all positive real numbers $a,b,c$, such that $a^2+b^2+c^2=1$: \[\frac{a^3}{b^2+c}+\frac{b^3}{c^2+a}+\frac{c^3}{a^2+b}\ge \frac{\sqrt{3}}{1+\sqrt{3}}.\]

2001 Bosnia and Herzegovina Team Selection Test, 3

Tags: maximum , combinatorics , subset

Find maximal value of positive integer $n$ such that there exists subset of $S=\{1,2,...,2001\}$ with $n$ elements, such that equation $y=2x$ does not have solutions in set $S \times S$

2022 China Second Round A1, 1

Tags: maximum and minimum , inequalities

$a,b,c,d$ are real numbers so that $a\geq b,c\geq d$,\[|a|+2|b|+3|c|+4|d|=1.\] Let $P=(a-b)(b-c)(c-d)$,find the maximum and minimum value of $P$.

2012 Philippine MO, 4

Tags: induction , algebra

Let $\star$ be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers $x$ and $y$, (i) $(x + 1)\star 0 = (0\star x) + 1$ (ii) $0\star (y + 1) = (y\star 0) + 1$ (iii) $(x + 1)\star (y + 1) = (x\star y) + 1$. If $123\star 456 = 789$, find $246\star 135$.

2007 AIME Problems, 9

Tags: trigonometry , geometry , rectangle , geometric transformation , homothety , number theory

In right triangle $ABC$ with right angle $C$, $CA=30$ and $CB=16$. Its legs $\overline{CA}$ and $\overline{CB}$ are extended beyond $A$ and $B$. Points $O_{1}$ and $O_{2}$ lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center $O_{1}$ is tangent to the hypotenuse and to the extension of leg CA, the circle with center $O_{2}$ is tangent to the hypotenuse and to the extension of leg CB, and the circles are externally tangent to each other. The length of the radius of either circle can be expressed as $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

2009 Today's Calculation Of Integral, 407

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^1 (x \plus{} 3)\sqrt {xe^x}\ dx$.

2020 Balkan MO Shortlist, G5

Tags: geometry , circumcircle

Let $ABC$ be an isosceles triangle with $AB = AC$ and $\angle A = 45^o$. Its circumcircle $(c)$ has center $O, M$ is the midpoint of $BC$ and $D$ is the foot of the perpendicular from $C$ to $AB$. With center $C$ and radius $CD$ we draw a circle which internally intersects $AC$ at the point $F$ and the circle $(c)$ at the points $Z$ and $E$, such that $Z$ lies on the small arc $BC$ and $E$ on the small arc $AC$. Prove that the lines $ZE$, $CO$, $FM$ are concurrent. [i]Brazitikos Silouanos, Greece[/i]