Olympiad problems

2012 National Olympiad First Round, 16

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Every cell of $8\times8$ chessboard contains either $1$ or $-1$. It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. At most how many columns are there such that the sum of numbers inside the cells of those columns is less than $-3$? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 2$

2022 IMO Shortlist, N8

Tags: number theory

Prove that $5^n-3^n$ is not divisible by $2^n+65$ for any positive integer $n$.

2020 Korea - Final Round, P3

Tags: algebra , functional equation

Find all $f: \mathbb{Q}_{+} \rightarrow \mathbb{R}$ such that \[ f(x)+f(y)+f(z)=1 \] holds for every positive rationals $x, y, z$ satisfying $x+y+z+1=4xyz$.

2006 China Team Selection Test, 3

Tags: number theory unsolved , number theory

For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that: \[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \] then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number.

1969 Czech and Slovak Olympiad III A, 2

Tags: geometry , Geometric Inequalities , geometrical inequalities , convex , quadrilateral , area

Five different points $O,A,B,C,D$ are given in plane such that \[OA\le OB\le OC\le OD.\] Show that for area $P$ of any convex quadrilateral with vertices $A,B,C,D$ (not necessarily in this order) the inequality \[P\le \frac12(OA+OD)(OB+OC)\] holds and determine when equality occurs.

2024 Rioplatense Mathematical Olympiad, 6

Tags: geometry , rioplatense

Let $ABC$ be an acute triangle with $AB < AC$, and let $H$ be its orthocenter. Let $D$, $E$, $F$ and $M$ be the midpoints of $BC$, $AC$, and $AH$, respectively. Prove that the circumcircles of triangles $AHD$, $BMC$, and $DEF$ pass through a common point.

2009 All-Russian Olympiad Regional Round, 10.4

Tags: geometry , collinear

Circles $\omega_1$ and $\omega_2$ touch externally at the point $O$. Points $A$ and $B$ on the circle $\omega_1$ and points $C$ and $D$ on the circle $\omega_2$ are such that $AC$ and $BD$ are common external tangents to circles. Line $AO$ intersects segment $CD$ at point $M$ and straight line $CO$ intersexts $\omega_1$ again at point $N$. Prove that the points $B$, $M$ and $N$ lie on the same straight line.

1984 IMO Longlists, 45

Tags: algebra unsolved , algebra

Let $X$ be an arbitrary nonempty set contained in the plane and let sets $A_1, A_2,\cdots,A_m$ and $B_1, B_2,\cdots, B_n$ be its images under parallel translations. Let us suppose that $A_1\cup A_2 \cup \cdots\cup A_m \subset B_1 \cup B_2 \cup\cdots\cup B_n$ and that the sets $A_1, A_2,\cdots,A_m$ are disjoint. Prove that $m \le n$.

2010 Stanford Mathematics Tournament, 22

Tags: ratio

We need not restrict our number system radix to be an integer. Consider the phinary numeral system in which the radix is the golden ratio $\phi = \frac{1+\sqrt{5}}{2}$ and the digits $0$ and $1$ are used. Compute $1010100_{\phi}-.010101_{\phi}$

1986 Bundeswettbewerb Mathematik, 1

Tags: combinatorics

The edges of a cube are numbered from $1$ to $12$, then is calculated for each vertex the sum of the numbers of the edges going out from it. a) Prove that these sums cannot all be the same. b) Can eight equal sums result after one of the edge numbers is replaced by the number $13$ ?

2013 Peru IMO TST, 2

Tags: absolute value , algebra , polynomial

Let $a \geq 3$ be a real number, and $P$ a polynomial of degree $n$ and having real coefficients. Prove that at least one of the following numbers is greater than or equal to $1:$ $$|a^0- P(0)|, \ |a^1-P(1)| , \ |a^2-P(2)|, \cdots, |a^{n + 1}-P(n + 1)|.$$

2019 Math Prize for Girls Problems, 4

Tags: Math Prize for Girls

A paper equilateral triangle with area 2019 is folded over a line parallel to one of its sides. What is the greatest possible area of the overlap of folded and unfolded parts of the triangle?

2022 Math Prize for Girls Problems, 7

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The quadrilateral $ABCD$ is an isosceles trapezoid with $AB = CD = 1$, $BC = 2$, and $DA = 1+ \sqrt{3}$. What is the measure of $\angle ACD$ in degrees?

2022 LMT Spring, 6

Tags: algebra

For all $y$, define cubic $f_y (x)$ such that $f_y (0) = y$, $f_y (1) = y +12$, $f_y (2) = 3y^2$, $f_y (3) = 2y +4$. For all $y$, $f_y(4)$ can be expressed in the form $ay^2 +by +c$ where $a,b,c$ are integers. Find $a +b +c$.

1979 Bundeswettbewerb Mathematik, 4

Tags: algebra , polynomial

Prove that the polynomial $P(x) = x^5-x+a$ is irreducible over $\mathbb{Z}$ if $5 \nmid a$.

2009 Moldova National Olympiad, 8.3

Tags: parallel , geometry , circles

The circle $C_1$ of center $O$ and the circle $C_2$ intersect at points $A$ and $B$, so that point $O$ lies on circle $C_2$. The lines $d$ and $e$ are tangent at point $A$ to the circles $C_1$ and $C_2$ respectively. If the line $e$ intersects the circle $C_1$ at point $D$, prove that the lines $BD$ and $d$ are parallel.

LMT Team Rounds 2010-20, B10

Tags: combinatorics

In a certain Zoom meeting, there are $4$ students. How many ways are there to split them into any number of distinguishable breakout rooms, each with at least $ 1$ student?

1992 AMC 12/AHSME, 28

Tags: quadratics , complex numbers , algebra , quadratic formula , AMC

Let $i = \sqrt{-1}$. The product of the real parts of the roots of $z^2 - z = 5 - 5i$ is $ \textbf{(A)}\ -25\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ -5\qquad\textbf{(D)}\ \frac{1}{4}\qquad\textbf{(E)}\ 25 $

2008 ISI B.Stat Entrance Exam, 3

Tags: geometry , 3D geometry , calculus , derivative , function , calculus computations

Study the derivatives of the function \[y=\sqrt{x^3-4x}\] and sketch its graph on the real line.

2005 Denmark MO - Mohr Contest, 4

Tags: number theory

Fourteen students each write an integer number on the board. When they later meet their math teacher Homer Grog, they tell him that no matter what number they erased on the board, then the remaining numbers could be divided into three groups at once sum. They also tell him that the numbers on the board were integer numbers. Is it now possible for Homer Grog to determine what numbers the students wrote on the board?

1999 Putnam, 6

Tags: Putnam , college contests

The sequence $(a_n)_{n\geq 1}$ is defined by $a_1=1,a_2=2,a_3=24,$ and, for $n\geq 4,$ \[a_n=\dfrac{6a_{n-1}^2a_{n-3}-8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}.\] Show that, for all $n$, $a_n$ is an integer multiple of $n$.

2021 Honduras National Mathematical Olympiad, Problem 5

Tags: number theory

A positive integer $m$ is called [i]growing[/i] if its digits, read from left to right, are non-increasing. Prove that for each natural number $n$ there exists a growing number $m$ with $n$ digits such that the sum of its digits is a perfect square.

2025 Macedonian TST, Problem 6

Tags: number theory

Let $n>2$ be an even integer, and let $V$ be an arbitrary set of $8$ distinct integers. Define \[ E(V,n) \;=\; \bigl\{(u,v)\in V\times V : u < v,\ u+v = n^k\text{ for some }k\in\mathbb{N}\bigr\}. \] For each even $n>2$, determine the maximum possible size of the set $E(V,n)$.

2020 Romanian Master of Mathematics Shortlist, G2

Tags: geometry , Euler Line , RMM , RMM 2020 , RMM Shortlist

Let $ABC$ be an acute scalene triangle, and let $A_1, B_1, C_1$ be the feet of the altitudes from $A, B, C$. Let $A_2$ be the intersection of the tangents to the circle $ABC$ at $B, C$ and define $B_2, C_2$ similarly. Let $A_2A_1$ intersect the circle $A_2B_2C_2$ again at $A_3$ and define $B_3, C_3$ similarly. Show that the circles $AA_1A_3, BB_1B_3$, and $CC_1C_3$ all have two common points, $X_1$ and $X_2$ which both lie on the Euler line of the triangle $ABC$. [i]United Kingdom, Joe Benton[/i]

2024 AIME, 11

Tags: AMC , AIME , AIME I

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 $m+n$?