Olympiad problems

PEN A Problems, 100

Find all positive integers $n$ such that $n$ has exactly $6$ positive divisors $1<d_{1}<d_{2}<d_{3}<d_{4}<n$ and $1+n=5(d_{1}+d_{2}+d_{3}+d_{4})$.

2007 ITest, -1

Tags: analytic geometry , parabola , floor function , probability , number theory , conic , geometry

The Ultimate Question is a 10-part problem in which each question after the first depends on the answer to the previous problem. As in the Short Answer section, the answer to each (of the 10) problems is a nonnegative integer. You should submit an answer for each of the 10 problems you solve (unlike in previous years). In order to receive credit for the correct answer to a problem, you must also correctly answer $\textit{every one}$ $\textit{of the previous parts}$ $\textit{of the Ultimate Question}$.

2021 CMIMC, 2.1

Tags: geometry

Triangle $ABC$ has a right angle at $A$, $AB=20$, and $AC=21$. Circles $\omega_A$, $\omega_B$, and $\omega_C$ are centered at $A$, $B$, and $C$ respectively and pass through the midpoint $M$ of $\overline{BC}$. $\omega_A$ and $\omega_B$ intersect at $X\neq M$, and $\omega_A$ and $\omega_C$ intersect at $Y\neq M$. Find $XY$. [i]Proposed by Connor Gordon[/i]

2019 BMT Spring, Tie 3

Tags: algebra

There are two equilateral triangles with a vertex at $(0, 1)$, with another vertex on the line $y = x + 1$ and with the final vertex on the parabola $y = x^2 + 1$. Find the area of the larger of the two triangles.

2024 IFYM, Sozopol, 6

Tags: combinatorics

Each of 9 girls participates in several (one or more) theater groups, so that there are no two identical groups. Each of them is randomly assigned a positive integer between 1 and 30 inclusive. We call a group \textit{small} if the sum of the numbers of its members does not exceed the sum of any other group. Prove that regardless of which girl participates in which group, the probability that after receiving the numbers there will be a unique small group is at least $ \frac{7}{10} $.

2015 Dutch BxMO/EGMO TST, 5

Tags: algebra , functional equation

Find all functions $f : R \to R$ satisfying $(x^2 + y^2)f(xy) = f(x)f(y)f(x^2 + y^2)$ for all real numbers $x$ and $y$.

1998 Moldova Team Selection Test, 6

Tags: geometry

Two triangles $ABC$ and $BDE$ have vertexes $C$ and $E$ on the same side of the line $AB{}$ and $AB=a<BD$. Denote $\{P\}=CE\cap AB$ and $\gamma=m(\angle CPA)$. Let $r_1$ be the radius of the inscribed cricle of triangle $PAC$ and $r_2$ the radius of the excircle of triangle $PDE$, tangent to the side $DE$. Find $r_1+r_2$.

1962 Poland - Second Round, 3

Tags: 3d geometry , tetrahedron , geometry

Prove that the four segments connecting the vertices of the tetrahedron with the centers of gravity of the opposite faces have a common point.

1987 Iran MO (2nd round), 3

Tags: geometry

Let $L_1, L_2, L_3, L_4$ be four lines in the space such that no three of them are in the same plane. Let $L_1, L_2$ intersect in $A$, $L_2,L_3$ intersect in $B$ and $L_3, L_4$ intersect in $C.$ Find minimum and maximum number of lines in the space that intersect $L_1, L_2, L_3$ and $L_4.$ Justify your answer.

2008 Greece National Olympiad, 2

Tags: number theory

Find all integers $x$ and prime numbers $p$ satisfying $x^8 + 2^{2^x+2} = p$.

2008 Kurschak Competition, 1

Tags: number theory

Denote by $d(n)$ the number of positive divisors of a positive integer $n$. Find the smallest constant $c$ for which $d(n)\le c\sqrt n$ holds for all positive integers $n$.

2023 Sharygin Geometry Olympiad, 8.7

Tags: geometry

The bisector of angle $A$ of triangle $ABC$ meet its circumcircle $\omega$ at point $W$. The circle $s$ with diameter $AH$ ($H$ is the orthocenter of $ABC$) meets $\omega$ for the second time at point $P$. Restore the triangle $ABC$ if the points $A$, $P$, $W$ are given.

2018 Thailand TSTST, 7

Tags: binomial coefficient , algebra , function

Evaluate $\sum_{n=2017}^{2030}\sum_{k=1}^{n}\left\{\frac{\binom{n}{k}}{2017}\right\}$. [i]Note: $\{x\}=x-\lfloor x\rfloor$ for every real numbers $x$.[/i]

2025 Korea - Final Round, P5

Tags: number theory

$S={1,2,...,1000}$ and $T'=\left\{ 1001-t|t \in T\right\}$. A set $P$ satisfies the following three conditions: $1.$ All elements of $P$ are a subset of $S$. $2. A,B \in P \Rightarrow A \cap B \neq \O$ $3. A \in P \Rightarrow A' \in P$ Find the maximum of $|P|$.

1981 IMO, 3

Tags: function , algebra , functional equation

The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.

2015 Stars Of Mathematics, 4

Tags: number theory , combinatorics

Let $n\ge 5$ be a positive integer and let $\{a_1,a_2,...,a_n\}=\{1,2,...,n\}$.Prove that at least $\lfloor \sqrt{n}\rfloor +1$ numbers from $a_1,a_1+a_2,...,a_1+a_2+...+a_n$ leave different residues when divided by $n$.

2011 NIMO Problems, 10

Tags:

The edges and diagonals of convex pentagon $ABCDE$ are all colored either red or blue. How many ways are there to color the segments such that there is exactly one monochromatic triangle with vertices among $A$, $B$, $C$, $D$, $E$; that is, triangles, whose edges are all the same color? [i]Proposed by Eugene Chen [/i]

1984 Kurschak Competition, 3

Tags: number theory

Given are $n$ integers, not necessarily distinct, and two positive integers $p$ and $q$. If the $n$ numbers are not all distinct, choose two equal ones. Add $p$ to one of them and subtract $q$ from the other. If there are still equal ones among the $n$ numbers, repeat this procedure. Prove that after a finite number of steps, all $n$ numbers are distinct.

2017 Balkan MO Shortlist, A2

Tags: algebra

Consider the sequence of rational numbers defined by $x_1=\frac{4}{3}$ and $x_{n+1}=\frac{x_n^2}{x_n^2-x_n+1}$ , $n\geq 1$. Show that the numerator of the lowest term expression of each sum $\sum_{k=1}^{n}x_k$ is a perfect square. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2019 CCA Math Bonanza, I11

Tags:

Let $G$ be the centroid of triangle $ABC$ with $AB=13,BC=14,CA=15$. Calculate the sum of the distances from $G$ to the three sides of the triangle. Note: The [i]centroid[/i] of a triangle is the point that lies on each of the three line segments between a vertex and the midpoint of its opposite side. [i]2019 CCA Math Bonanza Individual Round #11[/i]

2005 Sharygin Geometry Olympiad, 10.2

Tags: similar triangles , cut , triangle , geometry

A triangle can be cut into three similar triangles. Prove that it can be cut into any number of triangles similar to each other.

2016 Israel National Olympiad, 7

Tags: algebra , functional equation , function

Find all functions $f:\mathbb{Z}\rightarrow\mathbb{C}$ such that $f(x(2y+1))=f(x(y+1))+f(x)f(y)$ holds for any two integers $x,y$.

2013 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: combinatorics

A division of a group of people into various groups is called $k$-regular if the number of groups is less or equal to $k$ and two people that know each other are in different groups. Let $A$, $B$, and $C$ groups of people such that there are is no person in $A$ and no person in $B$ that know each other. Suppose that the group $A \cup C$ has an $a$-regular division and the group $B \cup C$ has a $b$-regular division. For each $a$ and $b$, determine the least possible value of $k$ for which it is guaranteed that the group $A \cup B \cup C$ has a $k$-regular division.

2016 Ukraine Team Selection Test, 9

Tags: combinatorics , game

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

1994 China National Olympiad, 4

Tags: polynomial , algebra

Let $f(z)=c_0z^n+c_1z^{n-1}+ c_2z^{n-2}+\cdots +c_{n-1}z+c_n$ be a polynomial with complex coefficients. Prove that there exists a complex number $z_0$ such that $|f(z_0)|\ge |c_0|+|c_n|$, where $|z_0|\le 1$.