Olympiad problems

2003 India IMO Training Camp, 3

Tags: function , algebra

Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$, \[f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y).\]

2009 Dutch IMO TST, 4

Tags: functional equation , functional equation in z , algebra

Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.

1967 Polish MO Finals, 3

Tags: combinatorics

There are 100 persons in a hall, everyone knowing at least 67 of the others. Prove that there always exist four of them who know each other

1962 All Russian Mathematical Olympiad, 026

Tags: combinatorics , table

Given positive numbers $a_1, a_2, ..., a_m, b_1, b_2, ..., b_n$. Is known that $$a_1+a_2+...+a_m=b_1+b_2+...+b_n.$$ Prove that you can fill an empty table with $m$ rows and $n$ columns with no more than $(m+n-1)$ positive number in such a way, that for all $i,j$ the sum of the numbers in the $i$-th row will equal to $a_i$, and the sum of the numbers in the $j$-th column -- to $b_j$.

2024 AMC 10, 22

Tags: geometry

Let $\mathcal K$ be the kite formed by joining two right triangles with legs $1$ and $\sqrt3$ along a common hypotenuse. Eight copies of $\mathcal K$ are used to form the polygon shown below. What is the area of triangle $\Delta ABC$? [img]https://cdn.artofproblemsolving.com/attachments/1/3/03abbd4df2932f4a1d16a34c2b9e15b683dedb.png[/img] $\textbf{(A) }2+3\sqrt3\qquad\textbf{(B) }\dfrac92\sqrt3\qquad\textbf{(C) }\dfrac{10+8\sqrt3}{3}\qquad\textbf{(D) }8\qquad\textbf{(E) }5\sqrt3$

2012 JBMO ShortLists, 1

Tags: combinatorics

Along a round table are arranged $11$ cards with the names ( all distinct ) of the $11$ members of the $16^{th}$ JBMO Problem Selection Committee . The cards are arranged in a regular polygon manner . Assume that in the first meeting of the Committee none of its $11$ members sits in front of the card with his name . Is it possible to rotate the table by some angle so that at the end at least two members sit in front of the card with their names ?

Champions Tournament Seniors - geometry, 2017.4

Tags: equal segments , angle bisector , circles , geometry

Let $AD$ be the bisector of triangle $ABC$. Circle $\omega$ passes through the vertex $A$ and touches the side $BC$ at point $D$. This circle intersects the sides $AC$ and $AB$ for the second time at points $M$ and $N$ respectively. Lines $BM$ and $CN$ intersect the circle for the second time $\omega$ at points $P$ and $Q$, respectively. Lines $AP$ and $AQ$ intersect side $BC$ at points $K$ and $L$, respectively. Prove that $KL=\frac12 BC$

2004 India IMO Training Camp, 1

Tags: geometry , circumcircle , trigonometry

Let $ABC$ be an acute-angled triangle and $\Gamma$ be a circle with $AB$ as diameter intersecting $BC$ and $CA$ at $F ( \not= B)$ and $E (\not= A)$ respectively. Tangents are drawn at $E$ and $F$ to $\Gamma$ intersect at $P$. Show that the ratio of the circumcentre of triangle $ABC$ to that if $EFP$ is a rational number.

1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 4

Tags:

We have a square grid of 4 times 4 points. How many triangles are there with vertices on the points? (The three vertices may not lie on a line.) A. 252 B. 256 C. 360 D. 516 E. 560

2010 Tournament Of Towns, 7

Tags: geometry , rectangle , rotation , induction , combinatorics

A square is divided into congruent rectangles with sides of integer lengths. A rectangle is important if it has at least one point in common with a given diagonal of the square. Prove that this diagonal bisects the total area of the important rectangles

2018 CMIMC Number Theory, 4

Tags: number theory

Let $a>1$ be a positive integer. The sequence of natural numbers $\{a_n\}_{n\geq 1}$ is defined such that $a_1 = a$ and for all $n\geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$. Determine the smallest possible value of $a$ such that the numbers $a_1$, $a_2$,$\ldots$, $a_7$ are all distinct.

2011 Putnam, A2

Tags: ratio , geometric series , summation

Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1=b_1=1$ and $b_n=b_{n-1}a_n-2$ for $n=2,3,\dots.$ Assume that the sequence $(b_j)$ is bounded. Prove that \[S=\sum_{n=1}^{\infty}\frac1{a_1\cdots a_n}\] converges, and evaluate $S.$

1991 IberoAmerican, 1

Tags: geometry , 3d geometry , combinatorics

Each vertex of a cube is assigned an 1 or a -1, and each face is assigned the product of the numbers assigned to its vertices. Determine the possible values the sum of these 14 numbers can attain.

2012 District Olympiad, 4

Tags: geometry , 3d geometry , tetrahedron , perpendicular

Consider a tetrahedron $ABCD$ in which $AD \perp BC$ and $AC \perp BD$. We denote by $E$ and $F$ the projections of point $B$ on the lines $AD$ and $AC$, respectively. If $M$ and $N$ are the midpoints of the segments $[AB]$ and $[CD]$, respectively, show that $MN \perp EF$

2012 Online Math Open Problems, 3

Tags:

A lucky number is a number whose digits are only $4$ or $7.$ What is the $17$th smallest lucky number? [i]Author: Ray Li[/i] [hide="Clarifications"] [list=1][*]Lucky numbers are positive. [*]"only 4 or 7" includes combinations of 4 and 7, as well as only 4 and only 7. That is, 4 and 47 are both lucky numbers.[/list][/hide]

2012 Purple Comet Problems, 20

Tags: vector , analytic geometry , graphing lines , slope , algebra , system of equations

Square $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\overline{CD}$, and let $F$ be the point on segment $\overline{AB}$ a distance $17$ from point $A$. Point $G$ is on segment $\overline{EF}$ so that $\overline{EF}$ is perpendicular to segment $\overline{GD}$. The length of segment $\overline{BG}$ can be written as $m\sqrt{n}$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

1996 Baltic Way, 13

Tags: function , algebra

Consider the functions $f$ defined on the set of integers such that \[f(x)=f(x^2+x+1)\] for all integer $x$. Find $(a)$ all even functions, $(b)$ all odd functions of this kind.

2023 Bulgaria EGMO TST, 3

Tags: combinatorics , word , induction

A pair of words consisting only of the letters $a$ and $b$ (with repetitions) is [i]good[/i] if it is $(a,b)$ or of one of the forms $(uv, v)$, $(u, uv)$, where $(u,v)$ is a good pair. Prove that if $(\alpha, \beta)$ is a good pair, then there exists a palindrome $\gamma$ such that $\alpha\beta = a\gamma b$.

1997 Taiwan National Olympiad, 2

Tags: geometry

Given a line segment $AB$ in the plane, find all possible points $C$ such that in the triangle $ABC$, the altitude from $A$ and the median from $B$ have the same length.

2020/2021 Tournament of Towns, P3

Tags: number theory , divisibility

A positive integer number $N{}$ is divisible by 2020. All its digits are different and if any two of them are swapped, the resulting number is not divisible by 2020. How many digits can such a number $N{}$ have? [i]Sergey Tokarev[/i]

2016 Chile TST IMO, 1

Tags:

An equilateral triangle with side length 20 is subdivided using parallels to its sides into $ 20^2 = 400 $ smaller equilateral triangles of side length 1. Some segments of length 1, which are edges of these small triangles, must be colored red in such a way that no small triangle has all three of its edges colored red. Determine the maximum number of segments of length 1 that can be colored red.

2023 Azerbaijan Senior NMO, 3

Tags: algebra , polynomial

Let $m$ be a positive integer. Find polynomials $P(x)$ with real coefficients such that $$(x-m)P(x+2023) = xP(x)$$ is satisfied for all real numbers $x.$

2021 Science ON all problems, 2

Tags: geometry , angle

In triangle $ABC$, we have $\angle ABC=\angle ACB=44^o$. Point $M$ is in its interior such that $\angle MBC=16^o$ and $\angle MCB=30^o$. Prove that $\angle MAC=\angle MBC$. [i] (Andra Elena Mircea)[/i]

2021 Estonia Team Selection Test, 2

Tags: algebra , inequalities

Positive real numbers $a, b, c$ satisfy $abc = 1$. Prove that $$\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \ge \frac32$$

2018 CIIM, Problem 3

Tags:

Let $m$ be an integer and $\mathbb{Z}_m$ the set of integer modulo $m$. An equivalence relation is defined in $\mathbb{Z}_m$ given by, $x \sim y$ if there exists a natural $t$ such that $y \equiv 2^tx \, (\bmod m)$ . Find al values of $m$ such that the number of equivalent classes is even.