Olympiad problems

2008 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: number theory

Find all positive integers $ a$ and $ b$ such that $ \frac{a^{4}\plus{}a^{3}\plus{}1}{a^{2}b^{2}\plus{}ab^{2}\plus{}1}$ is an integer.

2010 Ukraine Team Selection Test, 4

Tags: algebra , inequalities

For the nonnegative numbers $a, b, c$ prove the inequality: $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge \frac52$$

2015 Online Math Open Problems, 7

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A geometric progression of positive integers has $n$ terms; the first term is $10^{2015}$ and the last term is an odd positive integer. How many possible values of $n$ are there? [i]Proposed by Evan Chen[/i]

2014 Tuymaada Olympiad, 7

Tags: incenter , geometry , analytic geometry , parallelogram

A parallelogram $ABCD$ is given. The excircle of triangle $\triangle{ABC}$ touches the sides $AB$ at $L$ and the extension of $BC$ at $K$. The line $DK$ meets the diagonal $AC$ at point $X$; the line $BX$ meets the median $CC_1$ of trianlge $\triangle{ABC}$ at ${Y}$. Prove that the line $YL$, median $BB_1$ of triangle $\triangle{ABC}$ and its bisector $CC^\prime$ have a common point. [i](A. Golovanov)[/i]

2019 USA TSTST, 6

Tags: integer polynomial , fibonacci , algebra

Suppose $P$ is a polynomial with integer coefficients such that for every positive integer $n$, the sum of the decimal digits of $|P(n)|$ is not a Fibonacci number. Must $P$ be constant? (A [i]Fibonacci number[/i] is an element of the sequence $F_0, F_1, \dots$ defined recursively by $F_0=0, F_1=1,$ and $F_{k+2} = F_{k+1}+F_k$ for $k\ge 0$.) [i]Nikolai Beluhov[/i]

1991 Arnold's Trivium, 80

Tags: calculus , quadratic , integration

Solve the equation \[\int_0^1(x+y)^2u(x)dx=\lambda u(y)+1\]

1996 AMC 8, 11

Tags:

Let $x$ be the number \[0.\underbrace{0000...0000}_{1996\text{ zeros}}1,\] where there are 1996 zeros after the decimal point. Which of the following expressions represents the largest number? $\text{(A)}\ 3+x \qquad \text{(B)}\ 3-x \qquad \text{(C)}\ 3\cdot x \qquad \text{(D)}\ 3/x \qquad \text{(E)}\ x/3$

2018 All-Russian Olympiad, 1

Tags: prime number , prime , number theory , algebra

Suppose $a_1,a_2, \dots$ is an infinite strictly increasing sequence of positive integers and $p_1, p_2, \dots$ is a sequence of distinct primes such that $p_n \mid a_n$ for all $n \ge 1$. It turned out that $a_n-a_k=p_n-p_k$ for all $n,k \ge 1$. Prove that the sequence $(a_n)_n$ consists only of prime numbers.

IV Soros Olympiad 1997 - 98 (Russia), 9.2

Tags: radical , algebra

Solve the equation $$2\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)(x+5)}}}}=x$$

2019 JBMO Shortlist, C1

Tags: combinatorics

Let $S$ be a set of $100$ positive integer numbers having the following property: “Among every four numbers of $S$, there is a number which divides each of the other three or there is a number which is equal to the sum of the other three.” Prove that the set $S$ contains a number which divides all other $99$ numbers of $S$. [i]Proposed by Tajikistan[/i]

2011 Junior Balkan Team Selection Tests - Romania, 1

Tags: decimal representation , digit , number theory

Call a positive integer [i]balanced [/i] if the number of its distinct prime factors is equal to the number of its digits in the decimal representation; for example, the number $385 = 5 \cdot 7 \cdot 11$ is balanced, while $275 = 5^2 \cdot 11$ is not. Prove that there exist only a finite number of balanced numbers.

2012 SEEMOUS, Problem 1

Tags: matrix , number theory , linear algebra

Let $A=(a_{ij})$ be the $n\times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j+j^i$ by $3$ for $i,j=1,2,\ldots,n$. Find the greatest $n$ for which $\det A\ne0$.

II Soros Olympiad 1995 - 96 (Russia), 10.9

Tags: cyclic quadrilateral , newton line , geometry

The opposite sides of a quadrilateral inscribed in a circle intersect at points $K$ and $L$. Let $F$ be the midpoint of $KL$, $E$ and $G$ be the midpoints of the diagonals of the given quadrilateral. It is known that $FE = a$, $FG = b$. Calculate $KL$ in terms of $a$ and $b.$ (It is known that the points $F$, $E$ and $G$ lie on the same straight line. This is true for any quadrilateral, not necessarily inscribed. The indicated straight line is sometimes called the Newton−Gauss line. This fact can be used without proof in proving the problem, as it is known).

2007 Moldova Team Selection Test, 4

Tags: ratio , combinatorial geometry , geometry

Consider five points in the plane, no three collinear. The convex hull of this points has area $S$. Prove that there exist three points of them that form a triangle with area at most $\frac{5-\sqrt 5}{10}S$

2016 NIMO Summer Contest, 4

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Nine people sit in three rows of three chairs each. The probability that two of them, Celery and Drum, sit next to each other in the same row is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by Michael Tang[/i]

2011 Bosnia and Herzegovina Junior BMO TST, 1

Tags: equation , positive integer , number theory

Solve equation $\frac{1}{x}-\frac{1}{y}=\frac{1}{5}-\frac{1}{xy}$, where $x$ and $y$ are positive integers.

2013 Dutch IMO TST, 3

Tags: midpoint , tangent , geometry , circles

Fix a triangle $ABC$. Let $\Gamma_1$ the circle through $B$, tangent to edge in $A$. Let $\Gamma_2$ the circle through C tangent to edge $AB$ in $A$. The second intersection of $\Gamma_1$ and $\Gamma_2$ is denoted by $D$. The line $AD$ has second intersection $E$ with the circumcircle of $\vartriangle ABC$. Show that $D$ is the midpoint of the segment $AE$.

2005 France Pre-TST, 7

Tags:

Prove that a prime of the form $2^{2^n}+1$ cannot be the difference of two fifth powers of two positive integers. Pierre.

2004 National Olympiad First Round, 26

Tags: modular arithmetic

What is the last two digits of base-$3$ representation of $2005^{2003^{2004}+3}$? $ \textbf{(A)}\ 21 \qquad\textbf{(B)}\ 01 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 02 \qquad\textbf{(E)}\ 22 $

2021 Yasinsky Geometry Olympiad, 6

Tags: fixed , tangent circle , geometry

In the circle $\omega$, we draw a chord $BC$, which is not a diameter. Point $A$ moves in a circle $\omega$. $H$ is the orthocenter triangle $ABC$. Prove that for any location of point $A$, a circle constructed on $AH$ as on diameter, touches two fixed circles $\omega_1$ and $\omega_2$. (Dmitry Prokopenko)

1953 Moscow Mathematical Olympiad, 255

Tags: pyramid , 3d geometry , cube , geometry

Divide a cube into three equal pyramids.

2003 China Team Selection Test, 2

Tags: induction , algebra , function

Let $S$ be a finite set. $f$ is a function defined on the subset-group $2^S$ of set $S$. $f$ is called $\textsl{monotonic decreasing}$ if when $X \subseteq Y\subseteq S$, then $f(X) \geq f(Y)$ holds. Prove that: $f(X \cup Y)+f(X \cap Y ) \leq f(X)+ f(Y)$ for $X, Y \subseteq S$ if and only if $g(X)=f(X \cup \{ a \}) - f(X)$ is a $\textsl{monotonic decreasing}$ funnction on the subset-group $2^{S \setminus \{a\}}$ of set $S \setminus \{a\}$ for any $a \in S$.

2020-2021 OMMC, 10

Tags:

Positive integers $a,b,c$ exist such that $a+b+c+1$, $a^2+b^2+c^2 +1$, $a^3+b^3+c^3+1,$ and $a^4+b^4+c^4+7459$ are all multiples of $p$ for some prime $p$. Find the sum of all possible values of $p$ less than $1000$.

2011 ISI B.Math Entrance Exam, 3

Tags: induction , inequalities

For $n\in\mathbb{N}$ prove that \[\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n}\leq\frac{1}{\sqrt{2n+1}}.\]

2003 Indonesia MO, 6

Tags: congruent triangles , geometry , combinatorics , inequalities

The hall in a castle is a regular hexagon where its sides' length is 6 meters. The floor of the hall is to be tiled with equilateral triangular tiles where its sides' length is 50 centimeters. Each tile is divided into three congruent triangles by their altitudes up to its orthocenter (see below). Each of these small triangles are colored such that each tile has different colors and no two tiles have identical colorings. How many colors at least are required? A tile's pattern is: [asy] draw((0,0.000)--(2,0.000)); draw((2,0.000)--(1,1.732)); draw((1,1.732)--(0,0.000)); draw((1,0.577)--(0,0.000)); draw((1,0.577)--(2,0.000)); draw((1,0.577)--(1,1.732)); [/asy]