Olympiad problems

2004 Germany Team Selection Test, 2

Find all functions $f: \Bbb{R}_{0}^{+}\rightarrow \Bbb{R}_{0}^{+}$ with the following properties: (a) We have $f\left( xf\left( y\right) \right) \cdot f\left( y\right) =f\left( x+y\right)$ for all $x$ and $y$. (b) We have $f\left(2\right) = 0$. (c) For every $x$ with $0 < x < 2$, the value $f\left(x\right)$ doesn't equal $0$. [b]NOTE.[/b] We denote by $\Bbb{R}_{0}^{+}$ the set of all non-negative real numbers.

2012 Harvard-MIT Mathematics Tournament, 9

Tags: hmmt , algebra , polynomial , function , trigonometry

How many real triples $(a,b,c)$ are there such that the polynomial $p(x)=x^4+ax^3+bx^2+ax+c$ has exactly three distinct roots, which are equal to $\tan y$, $\tan 2y$, and $\tan 3y$ for some real number $y$?

2009 Ukraine Team Selection Test, 11

Tags: combinatorics , coloring , min , max

Suppose that integers are given $m <n $. Consider a spreadsheet of size $n \times n $, whose cells arbitrarily record all integers from $1 $ to ${{n} ^ {2}} $. Each row of the table is colored in yellow $m$ the largest elements. Similarly, the blue colors the $m$ of the largest elements in each column. Find the smallest number of cells that are colored yellow and blue at a time

2023 Estonia Team Selection Test, 6

Tags: geometry

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

2004 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: inequalities , circumcircle , inradius , geometry

In a convex hexagon $ABCDEF$, triangles $ACE$ and $BDF$ have the same circumradius $R$. If triangle $ACE$ has inradius $r$, prove that \[ \text{Area}(ABCDEF)\le\frac{R}{r}\cdot\text{Area}(ACE).\]

2013 Bogdan Stan, 2

Tags: function , calculus , integration , absolute value

Let be a sequence of continuous functions $ \left( f_n \right)_{n\ge 1} :[0,1]\longrightarrow\mathbb{R} $ satisfying the following properties: $ \text{a) } $ for any natural $ n $ and $ x\in [1/n,1] ,$ it follows $ \left| f_n(x) \right|\leqslant 1/n. $ $ \text{b) } $ for any natural $ n, $ it follows $ \int_0^1 f_n^2(t)dt\leqslant 1. $ Then, $\lim_{n\to 0} \int_0^1\left| f_n(t) \right| dt=0 $ [i]Cristinel Mortici[/i]

2016 Romania Team Selection Test, 1

Tags: combinatorics

Determine the planar finite configurations $C$ consisting of at least $3$ points, satisfying the following conditions; if $x$ and $y$ are distinct points of $C$, there exist $z\in C$ such that $xyz$ are three vertices of equilateral triangles

2010 Postal Coaching, 4

Tags: number theory

How many ordered triples $(a, b, c)$ of positive integers are there such that none of $a, b, c$ exceeds $2010$ and each of $a, b, c$ divides $a + b + c$?

1953 Moscow Mathematical Olympiad, 244

Tags: greatest common divisor , number theory , least common multiple

Prove that $gcd (a + b, lcm(a, b)) = gcd (a, b)$ for any $a, b$.

2016 ASMT, 7

Tags: analytic geometry , geometry

A circle intersects the $y$-axis at two points $(0, a)$ and $(0, b)$ and is tangent to the line $x+100y = 100$ at $(100, 0)$. Compute the sum of all possible values of $ab - a - b$.

1979 IMO Longlists, 24

Tags: search , relatively prime , number theory

Let $a$ and $b$ be coprime integers, greater than or equal to $1$. Prove that all integers $n$ greater than or equal to $(a - 1)(b - 1)$ can be written in the form: \[n = ua + vb, \qquad \text{with} (u, v) \in \mathbb N \times \mathbb N.\]

2023 Belarusian National Olympiad, 11.8

Tags: algebra , geometry

Positive integer $n>2$ is called [i]good[/i] if there exist $n$ distinct points on plane($X_1, \ldots, X_n$), such that for all $1 \leq i \leq n$ vectors $X_iX_1, \ldots, X_iX_n$ can be partitioned into two groups with equal sums. Find all [i]good[/i] numbers

2023 Puerto Rico Team Selection Test, 4

Tags: combinatorics

A frog started from the origin of the coordinate plane and made $3$ jumps. Each time, the frog jumped a distance of $5$ units and landed on a point with integer coordinates. How many different position possibilities end of the frog there?

2015 Iran Team Selection Test, 3

Tags: combinatorics

Find the maximum number of rectangles with sides equal to 1 and 2 and parallel to the coordinate axes such that each two have an area equal to 1 in common.

2021 OMpD, 3

Tags: inequalities , algebra , n-variable inequality , absolute value

Let $a$ and $b$ be positive real numbers, with $a < b$ and let $n$ be a positive integer. Prove that for all real numbers $x_1, x_2, \ldots , x_n \in [a, b]$: $$ |x_1 - x_2| + |x_2 - x_3| + \cdots + |x_{n-1} - x_n| + |x_n - x_1| \leq \frac{2(b - a)}{b + a}(x_1 + x_2 + \cdots + x_n)$$ And determine for what values of $n$ and $x_1, x_2, \ldots , x_n$ the equality holds.

2004 Spain Mathematical Olympiad, Problem 6

Tags: circles , pattern , number theory

We put, forming a circumference of a circle, ${2004}$ bicolored files: white on one side of the file and black on the other. A movement consists in choosing a file with the black side upwards and flipping three files: the one chosen, the one to its right, and the one to its left. Suppose that initially there was only one file with its black side upwards. Is it possible, repeating the movement previously described, to get all of the files to have their white sides upwards? And if we were to have ${2003}$ files, between which exactly one file began with the black side upwards?

2005 China Girls Math Olympiad, 7

Tags: floor function , inequalities

Let $ m$ and $ n$ be positive integers with $ m > n \geq 2.$ Set $ S \equal{} \{1, 2, \ldots, m\},$ and $ T \equal{} \{a_l, a_2, \ldots, a_n\}$ is a subset of S such that every number in $ S$ is not divisible by any two distinct numbers in $ T.$ Prove that \[ \sum^n_{i \equal{} 1} \frac {1}{a_i} < \frac {m \plus{} n}{m}. \]

2013 Princeton University Math Competition, 1

Tags:

A regular pentagon can have the line segments forming its boundary extended to lines, giving an arrangement of lines that intersect at ten points. How many ways are there to choose five points of these ten so that no three of the points are collinear?

Estonia Open Junior - geometry, 1995.1.4

Tags: geometry , right triangle , equal angles

The midpoint of the hypotenuse $AB$ of the right triangle $ABC$ is $K$. The point $M$ on the side $BC$ is taken such that $BM = 2 \cdot MC$. Prove that $\angle BAM = \angle CKM$.

1999 German National Olympiad, 1

Tags: diophantine equation , diophantine , number theory

Find all $x,y$ which satisfy the equality $x^2 +xy+y^2 = 97$, when $x,y$ are a) natural numbers, b) integers

2024 Malaysia IMONST 2, 5

Tags: combinatorics

Janson found $2025$ dogs on a circle. Janson wants to select some (possibly none) of the dogs to take home, such that no two selected dogs have exactly two dogs (whether selected or not) in between them. Let $S_{1}$ be the number of ways for him to do so. Ivan also found $2025$ cats on a circle. Ivan wants to select some (possibly none) of the cats to take home, such that no two selected cats have exactly five cats (whether selected or not) in between them. Let $S_{2}$ be the number of ways for him to do so. a) Prove that $S_{1}=S_{2}$. b) Prove that $S_{1}$ and $S_{2}$ are both perfect cubes.

2008 AMC 10, 10

Tags: trigonometry , power of a point , pythagorean theorem , geometry

Points $ A$ and $ B$ are on a circle of radius $ 5$ and $ AB\equal{}6$. Point $ C$ is the midpoint of the minor arc $ AB$. What is the length of the line segment $ AC$? $ \textbf{(A)}\ \sqrt{10} \qquad \textbf{(B)}\ \frac{7}{2} \qquad \textbf{(C)}\ \sqrt{14} \qquad \textbf{(D)}\ \sqrt{15} \qquad \textbf{(E)}\ 4$

2015 Brazil Team Selection Test, 3

Tags: algebra , function , calculus

Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\] [i]Proposed by Denmark[/i]

1960 IMO Shortlist, 1

Tags: algebra , number theory , divisibility , decimal representation

Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.

2023 Belarusian National Olympiad, 10.3

Tags: algebra , inequalities , inequality

Let $a,b,c$ be positive real numbers, that satisfy $abc=1$. Prove the inequality: $$\frac{ab}{1+c}+\frac{bc}{1+a}+\frac{ca}{1+b} \geq \frac{27}{(a+b+c)(3+a+b+c)}$$