Olympiad problems

2017 Balkan MO Shortlist, G2

Let $ABC$ be an acute triangle and $D$ a variable point on side $AC$ . Point $E$ is on $BD$ such that $BE =\frac{BC^2-CD\cdot CA}{BD}$ . As $D$ varies on side $AC$ prove that the circumcircle of $ADE$ passes through a fixed point other than $A$ .

2017 ELMO Shortlist, 3

Tags: number theory , Hi

For each integer $C>1$ decide whether there exist pairwise distinct positive integers $a_1,a_2,a_3,...$ such that for every $k\ge 1$, $a_{k+1}^k$ divides $C^ka_1a_2...a_k$. [i]Proposed by Daniel Liu

2003 IMO Shortlist, 5

Tags: geometry , IMO Shortlist

Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. [i]Proposed by Hojoo Lee[/i]

2017-IMOC, N9

Tags: number theory

Let $a$ be a natural number, $a>3$. Prove there is an infinity of numbers n, for which $a+n|a^{n}+1$

2019 Brazil Team Selection Test, 4

Tags: combinatorics , APMO

Consider a $2018 \times 2019$ board with integers in each unit square. Two unit squares are said to be neighbours if they share a common edge. In each turn, you choose some unit squares. Then for each chosen unit square the average of all its neighbours is calculated. Finally, after these calculations are done, the number in each chosen unit square is replaced by the corresponding average. Is it always possible to make the numbers in all squares become the same after finitely many turns?

2000 Putnam, 6

Tags: Putnam , inequalities , college contests , combinatorics , combinatorial geometry , Probabilistic Method

Let $B$ be a set of more than $\tfrac{2^{n+1}}{n}$ distinct points with coordinates of the form $(\pm 1, \pm 1, \cdots, \pm 1)$ in $n$-dimensional space with $n \ge 3$. Show that there are three distinct points in $B$ which are the vertices of an equilateral triangle.

1997 Estonia National Olympiad, 5

Tags: geometry , tangent circles , circles , Equilateral

There are six small circles in the figure with a radius of $1$ and tangent to a large circle and the sides of the $ABC$ of an equilateral triangle, where touch points are $K, L$ and $M$ respectively with the midpoints of sides $AB, BC$ and $AC$. Find the radius of the large circle and the side of the triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/3/0/f858dcc5840759993ea2722fd9b9b15c18f491.png[/img]

Maryland University HSMC part II, 2000

Tags: algebra , geometry , combinatorics , number theory , Maryland HSMC

[b]p1.[/b] There are $2000$ cans of paint. Show that at least one of the following two statements must be true. There are at least $45$ cans of the same color. There are at least $45$ cans all of different colors. [b]p2.[/b] The measures of the $3$ angles of one triangle are all different from each other but are the same as the measures of the $3$ angles of a second triangle. The lengths of $2$ sides of the first triangle are different from each other but are the same as the lengths of $2$ sides of the second triangle. Must the length of the remaining side of the first triangle be the same as the length of the remaining side of the second triangle? If yes, prove it. If not, provide an example. [b]p3.[/b] Consider the sequence $a_1=1$, $a_2=2$, $a_3=5/2$, ... satisfying $a_{n+1}=a_n+(a_n)^{-1}$ for $n>1$. Show that $a_{10000}>141$. [b]p4.[/b] Prove that no matter how $250$ points are placed in a disk of radius $1$, there is a disk of radius $1/10$ that contains at least $3$ of the points. [b]p5.[/b] Prove that: Given any $11$ integers (not necessarily distinct), one can select $6$ of them so that their sum is divisible by $6$. Given any $71$ integers (not necessarily distinct), one can select $36$ of them so that their sum is divisible by $36$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2018 CMIMC Algebra, 7

Tags: 2018 , algebra , trigonometry

Compute \[\sum_{k=0}^{2017}\dfrac{5+\cos\left(\frac{\pi k}{1009}\right)}{26+10\cos\left(\frac{\pi k}{1009}\right)}.\]

2017 India IMO Training Camp, 2

Tags: combinatorics , IMO Shortlist , Coloring , Ramsey Theory

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

2010 Peru IMO TST, 7

Tags: inequalities

Let $a, b, c$ be positive real numbers such that $a + b + c = 1.$ Prove that $$ \displaystyle{\frac{1}{a + b}+\frac{1}{b + c}+\frac{1}{c + a}+ 3(ab + bc + ca) \geq \frac{11}{2}.}$$

2007 Bosnia and Herzegovina Junior BMO TST, 4

Tags: geometry , circumcircle , incenter , arc midpoint , equal angles

Let $I$ be the incenter of the triangle $ABC$ ($AB < BC$). Let $M$ be the midpoint of $AC$, and let $N$ be the midpoint of the arc $AC$ of the circumcircle of $ABC$ which contains $B$. Prove that $\angle IMA = \angle INB$.

2015 Spain Mathematical Olympiad, 2

Tags: geometry , incenter , circumcircle

In triangle $ABC$, let $A'$ is the symmetrical of $A$ with respect to the circumcenter $O$ of $ABC$. Prove that: [b]a)[/b] The sum of the squares of the tangents segments drawn from $A$ and $A'$ to the incircle of $ABC$ equals $$4R^2-4Rr-2r^2$$ where $R$ and $r$ are the radii of the circumscribed and inscribed circles of $ABC$ respectively. [b]b)[/b] The circle with center $A'$ and radius $A'I$ intersects the circumcircle of $ABC$ in a point $L$ such that $$AL=\sqrt{ AB.AC}$$ where $I$ is the centre of the inscribed circle of $ABC$.

1976 Chisinau City MO, 125

Tags: combinatorics

From twenty different books on mathematics and physics, sets are made containing $5$ books on mathematics and $5$ books on physics each. How many math books should there be for the largest number of possible sets?

2016 AMC 8, 9

Tags: AMC 8 , 2016 AMC 8

What is the sum of the distinct prime integer divisors of $2016$? $\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }16\qquad\textbf{(D) }49\qquad \textbf{(E) }63$

2024 Moldova Team Selection Test, 5

Tags: combinatorial geometry

Consider a natural number $n \ge 3$. A convex polygon with $n$ sides is entirely placed inside a square with side length 1. Prove that we can always find three vertices of this polygon, the triangle formed by which has area not greater than $\frac{8}{n^2}$.

VMEO III 2006 Shortlist, A7

Tags: number theory , floor function , algebra , function

Prove that for all $n\in\mathbb{Z}^+$, we have \[ \sum\limits_{p=1}^n\sum\limits_{q=1}^p\left\lfloor -\frac{1+\sqrt{8q+(2p-1)^2}}{2}\right\rfloor =-\frac{n(n+1)(n+2)}{3} \]

2007 All-Russian Olympiad, 7

Tags: geometry , 3D geometry , tetrahedron , inequalities , geometry proposed

Given a tetrahedron $ T$. Valentin wants to find two its edges $ a,b$ with no common vertices so that $ T$ is covered by balls with diameters $ a,b$. Can he always find such a pair? [i]A. Zaslavsky[/i]

2013 HMNT, 9

Tags: geometry

Let $ABC$ be a triangle and $D$ a point on $BC$ such that $AB =\sqrt2$, $AC =\sqrt3$, $\angle BAD = 30^o$, and $\angle CAD = 45^o$. Find $AD$.

1965 Dutch Mathematical Olympiad, 5

Tags: algebra , functional , functional equation

The function ƒ. which is defined for all real numbers satisfies: $$f(x+y)+f(x-y)=2f(x)+2f(y)$$ Prove that $f(0) = 0$, $f(-x) = f(x)$, $f(2x) = 4 f (x)$, $$f(x + y + z) = f(x + y) + f(y + z) + f(z + x) -f(x) - f(y) -f(z).$$

2014 AMC 10, 9

Tags: geometry , Pythagorean Theorem , AMC

The two legs of a right triangle, which are altitudes, have lengths $2\sqrt3$ and $6$. How long is the third altitude of the triangle? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $

1986 Bulgaria National Olympiad, Problem 1

Tags: number theory

Find the smallest natural number $n$ for which the number $n^2-n+11$ has exactly four prime factors (not necessarily distinct).

1986 Poland - Second Round, 4

Tags: number theory , Perfect Square , diophantine , Diophantine equation

Natural numbers $ x, y, z $ whose greatest common divisor is equal to 1 satisfy the equation $$\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$$ Prove that $ x + y $ is the square of a natural number.

India EGMO 2025 TST, 7

Tags: combinatorics

Rijul and Rohinee are playing a game on an $n\times n$ board alternating turns, with Rijul going first. In each turn, they fill an unfilled cell with a number from $1,2,\cdots, n^2$ such that no number is used twice. Rijul wins if there is any column such that the sum of all its elements is divisible by $n$. Rohinee wins otherwise. For what positive integers $n$ does he have a winning strategy? Proposed by Rohan Goyal

2008 Brazil National Olympiad, 1

Tags: pigeonhole principle , number theory unsolved , number theory

A positive integer is [i]dapper[/i] if at least one of its multiples begins with $ 2008$. For example, $ 7$ is dapper because $ 200858$ is a multiple of $ 7$ and begins with $ 2008$. Observe that $ 200858 \equal{} 28694\times 7$. Prove that every positive integer is dapper.