Olympiad problems

2014 Harvard-MIT Mathematics Tournament, 4

[4] Let $D$ be the set of divisors of $100$. Let $Z$ be the set of integers between $1$ and $100$, inclusive. Mark chooses an element $d$ of $D$ and an element $z$ of $Z$ uniformly at random. What is the probability that $d$ divides $z$?

1998 Dutch Mathematical Olympiad, 5

Tags: function

Find all real solutions of the following equation: \[ (x + 1995)(x + 1997)(x + 1999)(x + 2001) + 16 = 0. \]

2010 Singapore Senior Math Olympiad, 1

Tags: geometry

In the $\triangle ABC$ with $AC>BC$ and $\angle B<90^{\circ}$, $D$ is the foot of the perpendicular from $A$ onto $BC$ and $E$ is the foot of perpendicular from $D$ onto $AC$. Let $F$ be the point on the segment $DE$ such that $EF \cdot DC=BD \cdot DE$. Prove that $AF$ is perpendicular to $BF$.

1995 Tournament Of Towns, (454) 3

Tags: perpendicular , geometry

Triangle $ABC$ is inscribed in a circle with center $O$. Let $q$ be the circle passing through $A$, $O$ and $B$. The lines $CA$ and $CB$ intersect $q$ at the points $D$ and $E$ (different from $A$ and $B$). Prove that the lines $CO$ and $DE$ are perpendicular to each other. (S Markelov)

1972 All Soviet Union Mathematical Olympiad, 166

Tags: combinatorial geometry , geometry , square

Each of the $9$ straight lines divides the given square onto two quadrangles with the areas ratio as $2:3$. Prove that there exist three of them intersecting in one point

2011 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: number theory , prime number

If $p>2$ is prime number and $m$ and $n$ are positive integers such that $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}$$ Prove that $p$ divides $m$

2002 Baltic Way, 18

Tags: number theory

Find all integers $n>1$ such that any prime divisor of $n^6-1$ is a divisor of $(n^3-1)(n^2-1)$.

2011 Turkey Team Selection Test, 2

Tags: inequalities

Let $a,b,c$ be positive real numbers satisfying $a^2+b^2+c^2 \geq 3.$ Prove that \[ \frac{(a+1)(b+2)}{(b+1)(b+5)} + \frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2} \]

2010 Philippine MO, 5

Tags: number theory

Determine, with proof, the smallest positive integer $n$ with the following property: For every choice of $n$ integers, there exist at least two whose sum or difference is divisible by $2009$.

2012 Pan African, 1

Tags: induction , sfft , geometry , 3d geometry , invariant , special factorizations , combinatorics

The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them and she writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values of the final number?

2010 Saint Petersburg Mathematical Olympiad, 1

Tags: algebra , polynomial

$f(x)$ is square trinomial. Is it always possible to find polynomial $g(x)$ with fourth degree, such that $f(g(x))=0$ has not roots?

2024 AIME, 11

Tags:

Each vertex of a regular octagon is coloured either red or blue with equal probability. The probability that the octagon can then be rotated in such a way that all of the blue vertices end up at points that were originally red is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

2016 Saudi Arabia GMO TST, 3

Tags: combinatorics

In a school, there are totally $n$ students, with $n \ge 2$. The students take part in $m$ clubs and in each club, there are at least $2$ members (a student may take part in more than $1$ club). Eventually, the Principal notices that: If $2$ clubs share at least $2$ common members then they have different numbers of members. Prove that $$m \le (n - 1)^2$$

2015 VJIMC, 3

Tags: algebra , polynomial

[b]Problem 3[/b] Let $ P(x) = x^{2015} -2x^{2014}+1$ and $ Q(x) = x^{2015} -2x^{2014}-1$. Determine for each of the polynomials $P$ and $Q$ whether it is a divisor of some nonzero polynomial $c_0 + c_{1}x +\ldots + c_{n}x^n$ n whose coefficients $c_i$ are all in the set $ \{ -1, 1\}$.

2010 Contests, 2

Tags: modular arithmetic , number theory

Given a fixed integer $k>0,r=k+0.5$,define $f^1(r)=f(r)=r[r],f^l(r)=f(f^{l-1}(r))(l>1)$ where $[x]$ denotes the smallest integer not less than $x$. prove that there exists integer $m$ such that $f^m(r)$ is an integer.

2022 MMATHS, 11

Tags: complex number , algebra

Denote by $Re(z)$ and $Im(z)$ the real part and imaginary part, respectively, of a complex number $z$; that is, if $z = a + bi$, then $Re(z) = a$ and $Im(z) = b$. Suppose that there exists some real number $k$ such that $Im \left( \frac{1}{w} \right) = Im \left( \frac{k}{w^2} \right) = Im \left( \frac{k}{w^3} \right) $ for some complex number $w$ with $||w||=\frac{\sqrt3}{2}$ , $Re(w) > 0$, and $Im(w) \ne 0$. If $k$ can be expressed as $\frac{\sqrt{a}-b}{c}$ for integers $a$, $b$, $c$ with $a$ squarefree, find $a + b + c$.

1953 AMC 12/AHSME, 29

Tags: geometry

The number of significant digits in the measurement of the side of a square whose computed area is $ 1.1025$ square inches to the nearest ten-thousandth of a square inch is: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 1$

2017 BMT Spring, 2

Tags: geometry , perimeter , equilateral , square , area

Barack is an equilateral triangle and Michelle is a square. If Barack and Michelle each have perimeter $ 12$, find the area of the polygon with larger area.

2007 Austria Beginners' Competition, 3

Tags: inequalities , algebra

For real numbers $x \ge 0$ and $y \ge 0$, write $A= \frac{x+y}{2}$ for the arithmetic mean and $G=\sqrt{xy}$ for the geometric mean of $x$ and $y$. Furthermore, let $W= \frac{\sqrt{x}+\sqrt{y}}{2}$ be the arithmetic mean of $\sqrt{x}$ and $\sqrt{y}$. Prove that $$G\le W^2 \le A.$$ Determine all $x$ and $y$ such that $G= W^2 = A$

1986 Austrian-Polish Competition, 3

Tags: coloring , square , combinatorics

Each point in space is colored either blue or red. Show that there exists a unit square having exactly $0, 1$ or $4$ blue vertices.

2006 Korea National Olympiad, 1

Tags: polynomial , algebra

Given that for reals $a_1,\cdots, a_{2004},$ equation $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0$ has $2006$ positive real solution, find the maximum possible value of $a_1.$

2008 Austria Beginners' Competition, 3

Tags: inequalities , algebra

Prove the inequality $$\frac{a + b}{a^2 -ab + b^2} \le \frac{4}{ |a + b|}$$ for all real numbers $a$ and $b$ with $a + b\ne 0$. When does equality hold?

2024 Chile National Olympiad., 5

Tags: combinatorics

You have a collection of at least two tokens where each one has a number less than or equal to 10 written on it. The sum of the numbers on the tokens is $ S $. Find all possible values of $ S $ that guarantee that the tokens can be separated into two groups such that the sum of each group does not exceed 80.

1969 IMO Shortlist, 70

Tags: geometry , geometric inequality , pentagon , euclidean distance

$(YUG 2)$ A park has the shape of a convex pentagon of area $50000\sqrt{3} m^2$. A man standing at an interior point $O$ of the park notices that he stands at a distance of at most $200 m$ from each vertex of the pentagon. Prove that he stands at a distance of at least $100 m$ from each side of the pentagon.

2017 Purple Comet Problems, 7

Tags: combinatorics

Consider an alphabetized list of all the arrangements of the letters in the word BETWEEN. Then BEEENTW would be in position $1$ in the list, BEEENWT would be in position $2$ in the list, and so forth. Find the position that BETWEEN would be in the list.