Olympiad problems

2008 Princeton University Math Competition, B2

Tags: combinatorics

Let $P$ be a convex polygon, and let $n \ge 3$ be a positive integer. On each side of $P$, erect a regular $n$-gon that shares that side of $P$, and is outside $P$. If none of the interiors of these regular n-gons overlap, we call P $n$-[i]good[/i]. (a) Find the largest value of $n$ such that every convex polygon is $n$-[i]good[/i]. (b) Find the smallest value of $n$ such that no convex polygon is $n$-[i]good[/i].

2004 Swedish Mathematical Competition, 1

Tags: geometry , circles , area

Two circles in the plane, both of radius $R$, intersect at a right angle. Compute the area of the intersection of the interiors of the two circles.

2001 China Team Selection Test, 1

Tags: algebra , Polynomials , polynomial

Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer

PEN P Problems, 27

Tags: floor function , Additive Number Theory

Determine, with proof, the largest number which is the product of positive integers whose sum is $1976$.

2011 Akdeniz University MO, 2

Tags: number theory , number theory unsolved

Let $a$ and $b$ is roots of the $x^2-6x+1$ equation. [b]a[/b]) Show that, for all $n \in{\mathbb Z^+}$ , $a^n+b^n$ is a integer. [b]b[/b]) Show that, for all $n \in{\mathbb Z^+}$ , $5$ isn't divide $a^n+b^n$

2025 VJIMC, 4

Tags: algebra , polynomial , linear algebra , trace , Minimal Polynomial , matrix

Let $A$ be an $n\times n$ real matrix with minimal polynomial $x^n + x - 1$. Prove that the trace of $(nA^{n-1} + I)^{-1}A^{n-2}$ is zero.

2023 NMTC Junior, P8

Tags: geometry , nmtc , cyclic quadrilateral

$ABCD$ is a cyclic quadrilateral. The midpoints of the diagonals $AC$ and $BD$ are respectively $P$ and $Q$. If $BD$ bisects $\angle AQC$, the prove that $AC$ will bisect $\angle BPD$.

2021 Saudi Arabia IMO TST, 6

Tags: algebra , functional equation , IMO Shortlist , IMO Shortlist 2020 , Iteration

Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ satisfying \[f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)\] for all integers $a$ and $b$

2022-2023 OMMC, 9

Tags:

An ant lies on each corner of a $20 \times 23$ rectangle. Each second, each ant independently and randomly chooses to move one unit vertically or horizontally away from its corner. After $10$ seconds, find the expected area of the convex quadrilateral whose vertices are the positions of the ants.

2021 Harvard-MIT Mathematics Tournament., 10

Tags: Chessboard , combinatorics

Let $n>1$ be a positive integer. Each unit square in an $n\times n$ grid of squares is colored either black or white, such that the following conditions hold: $\bullet$ Any two black squares can be connected by a sequence of black squares where every two consecutive squares in the sequence share an edge; $\bullet$ Any two white squares can be connected by a sequence of white squares where every two consecutive squares in the sequence share an edge; $\bullet$ Any $2\times 2$ subgrid contains at least one square of each color. Determine, with proof, the maximum possible difference between the number of black squares and white squares in this grid (in terms of $n$).

2021 AMC 10 Fall, 20

Tags: 2021 amc 12a , AMC , AMC 12 , AMC 12 A

How many ordered pairs of positive integers $(b,c)$ exist where both $x^2+bx+c=0$ and $x^2+cx+b=0$ do not have distinct, real solutions? $\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 12 \qquad$

2010 Indonesia TST, 2

Tags: geometry , parallelogram , homothety , Inversion , IMO Shortlist , geometry solved , lengths

Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that \[ \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}. \]

1998 Akdeniz University MO, 2

Tags: circle , combinatorics

$100$ points at a circle with radius $1$ $cm$. Show that, we find an another point such that, this point's distance to other $100$ points is greater than $100$ $cm$.

2013 ELMO Shortlist, 2

Tags: induction , logarithms , strong induction , combinatorics unsolved , combinatorics

Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$. [i]Proposed by Calvin Deng[/i]

1985 Dutch Mathematical Olympiad, 2

Tags: modular arithmetic , floor function , number theory proposed , number theory

Among the numbers $ 11n \plus{} 10^{10}$, where $ 1 \le n \le 10^{10}$ is an integer, how many are squares?

2002 AMC 10, 21

Tags: AMC

The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is $ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

2019 International Zhautykov OIympiad, 2

Tags: izho , algebra , number theory , number theory proposed , inequalities

Find the biggest real number $C$, such that for every different positive real numbers $a_1,a_2...a_{2019}$ that satisfy inequality : $\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$

2022 China Team Selection Test, 1

Tags: combinatorics , Coloring

Initially, each unit square of an $n \times n$ grid is colored red, yellow or blue. In each round, perform the following operation for every unit square simultaneously: [list] [*] For a red square, if there is a yellow square that has a common edge with it, then color it yellow. [*] For a yellow square, if there is a blue square that has a common edge with it, then color it blue. [*] For a blue square, if there is a red square that has a common edge with it, then color it red. [/list] It is known that after several rounds, all unit squares of this $n \times n$ grid have the same color. Prove that the grid has became monochromatic no later than the end of the $(2n-2)$-th round.

1955 Moscow Mathematical Olympiad, 318

Tags: maximum , points , combinatorics , combinatorial geometry

What greatest number of triples of points can be selected from $1955$ given points, so that each two triples have one common point?

2011 Sharygin Geometry Olympiad, 7

Tags: geometry , geometric inequality , equal segments

Points $P$ and $Q$ on sides $AB$ and $AC$ of triangle $ABC$ are such that $PB = QC$. Prove that $PQ < BC$.

2000 Slovenia National Olympiad, Problem 1

Tags: Digits

In the expression $4\cdot\text{RAKEC}=\text{CEKAR}$, each letter represents a (decimal) digit. Replace the letters so that the equality is true.

2019 239 Open Mathematical Olympiad, 4

Tags: combinatorics

A $20 \times 20$ treasure map is glued to a torus. A treasure is hidden in a cell of this map. We can ask questions about $1\times 4$ or $4 \times 1$ rectangles so that we find out if there is a treasure in this rectangle or not. The answers to all questions are absolutely true, but they are given only after all rectangles we want to ask are set. What is the least amount of questions needed to be asked so that we can be sure to find the treasure? (If you describe the position of the cells in a torus with numbers $(i, j)$ of row and column, $1 \leq i, j \leq 20$, then two cells are neighbors, if and only if two of the coordinates they have are the same, and the other two differ by $1$ mod $20$.)

2022 Yasinsky Geometry Olympiad, 6

Tags: geometry , incenter , Fixed point , fixed , circumcircle

Let $s$ be an arbitrary straight line passing through the incenter $I$ of the triangle $ABC$ . Line $s$ intersects lines $AB$ and $BC$ at points $D$ and $E$, respectively. Points $P$ and $Q$ are the centers of the circumscribed circles of triangles $DAI$ and $CEI$, respectively, and point $F$ is the second intersection point of these circles. Prove that the circumcircle of the triangle $PQF$ is always passes through a fixed point on the plane regardless of the position of the straight line $s$. (Matvii Kurskyi)

2024 Iran MO (3rd Round), 2

Tags: table , combinatorics , cells , game strategy , Game Theory

Two intelligent people playing a game on the $1403 \times 1403$ table with $1403^2$ cells. The first one in each turn chooses a cell that didn't select before and draws a vertical line segment from the top to the bottom of the cell. The second person in each turn chooses a cell that didn't select before and draws a horizontal line segment from the left to the right of the cell. After $1403^2$ steps the game will be over. The first person gets points equal to the longest verticals line segment and analogously the second person gets point equal to the longest horizonal line segment. At the end the person who gets the more point will win the game. What will be the result of the game?

2023 USA TSTST, 4

Tags: USA TSTST , graph theory , Binomial , Hi

Let $n\ge 3$ be an integer and let $K_n$ be the complete graph on $n$ vertices. Each edge of $K_n$ is colored either red, green, or blue. Let $A$ denote the number of triangles in $K_n$ with all edges of the same color, and let $B$ denote the number of triangles in $K_n$ with all edges of different colors. Prove \[ B\le 2A+\frac{n(n-1)}{3}.\] (The [i]complete graph[/i] on $n$ vertices is the graph on $n$ vertices with $\tbinom n2$ edges, with exactly one edge joining every pair of vertices. A [i]triangle[/i] consists of the set of $\tbinom 32=3$ edges between $3$ of these $n$ vertices.) [i]Proposed by Ankan Bhattacharya[/i]