This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2024 Taiwan Mathematics Olympiad, 5
Tags: combinatorial geometry
Several triangles are [b]intersecting[/b] if any two of them have non-empty intersections. Show that for any two finite collections of intersecting triangles, there exists a line that intersects all the triangles. [i] Proposed by usjl[/i]

2018 Pan-African Shortlist, A1
Tags: functional equation , algebra , function
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $(f(x + y))^2 = f(x^2) + f(y^2)$ for all $x, y \in \mathbb{Z}$.

2016 Putnam, A3
Tags: mapping
Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that \[f(x)+f\left(1-\frac1x\right)=\arctan x\] for all real $x\ne 0.$ (As usual, $y=\arctan x$ means $-\pi/2<y<\pi/2$ and $\tan y=x.$) Find \[\int_0^1f(x)\,dx.\]

1946 Moscow Mathematical Olympiad, 114
Tags: double factorial , factorial
Prove that for any positive integer $n$ the following identity holds $\frac{(2n)!}{n!}= 2^n \cdot (2n - 1)!!$

1997 Romania Team Selection Test, 2
Tags: modular arithmetic , divisibility , multiplicative nt , multiplicative order , number theory
Suppose that $A$ be the set of all positive integer that can write in form $a^2+2b^2$ (where $a,b\in\mathbb {Z}$ and $b$ is not equal to $0$). Show that if $p$ be a prime number and $p^2\in A$ then $p\in A$. [i]Marcel Tena[/i]

2019 Taiwan TST Round 3, 5
Tags: combinatorics
We have $ n $ kinds of puddings. There are $ a_{i} $ puddings which are $ i $-th type and those $ S = a_{1}+\cdots+a_{n} $ puddings are distinct. Now, for a given arrangement of puddings: $ p_{1}, \dots, p_{n} $. Define $ c_{i} $ to be $$ \#\{1 \le j \le S-1 \ \mid \ p_{j}, p_{j+1} \text{ are the same type.}\} $$ Show that if $ S $ is composite, then the sum of $ \prod_{i=1}^{n}{c_{i}} $ over all possible arrangements is a multiple of $ S $.

2003 Iran MO (3rd Round), 29
Tags: polynomial , induction , algebra
Let $ c\in\mathbb C$ and $ A_c \equal{} \{p\in \mathbb C[z]|p(z^2 \plus{} c) \equal{} p(z)^2 \plus{} c\}$. a) Prove that for each $ c\in C$, $ A_c$ is infinite. b) Prove that if $ p\in A_1$, and $ p(z_0) \equal{} 0$, then $ |z_0| < 1.7$. c) Prove that each element of $ A_c$ is odd or even. Let $ f_c \equal{} z^2 \plus{} c\in \mathbb C[z]$. We see easily that $ B_c: \equal{} \{z,f_c(z),f_c(f_c(z)),\dots\}$ is a subset of $ A_c$. Prove that in the following cases $ A_c \equal{} B_c$. d) $ |c| > 2$. e) $ c\in \mathbb Q\backslash\mathbb Z$. f) $ c$ is a non-algebraic number g) $ c$ is a real number and $ c\not\in [ \minus{} 2,\frac14]$.

2014 All-Russian Olympiad, 1
Tags: number theory
Let $a$ be [i]good[/i] if the number of prime divisors of $a$ is equal to $2$. Do there exist $18$ consecutive good natural numbers?

1966 IMO Longlists, 19
Tags: excircle , geometry , triangle , construction
Construct a triangle given the radii of the excircles.

2010 Canadian Mathematical Olympiad Qualification Repechage, 8
Tags: parallelogram , geometry
Consider three parallelograms $P_1,~P_2,~ P_3$. Parallelogram $P_3$ is inside parallelogram $P_2$, and the vertices of $P_3$ are on the edges of $P_2$. Parallelogram $P_2$ is inside parallelogram $P_1$, and the vertices of $P_2$ are on the edges of $P_1$. The sides of $P_3$ are parallel to the sides of $P_1$. Prove that one side of $P_3$ has length at least half the length of the parallel side of $P_1$.

2023 Stanford Mathematics Tournament, 3
Tags:
What is the least positive integer $x$ for which the expression $x^2 + 3x + 9$ has $3$ distinct prime divisors?

Denmark (Mohr) - geometry, 1995.5
Tags: combinatorial geometry , geometry , circles
In the plane, six circles are given so that none of the circles contain one the center of the other. Show that there is no point that lies in all the circles.

2024 Regional Olympiad of Mexico West, 2
Tags: geometry , collinear
Let $\triangle ABC$ be a triangle and $H$ its orthocenter. We draw the circumference $\mathcal{C}_1$ that passes through $H$ and its tangent to $BC$ at $B$ and the circumference $\mathcal{C}_2$ that passes through $H$ and its tangent to $BC$ at $C$. If $\mathcal{C}_1$ cuts line $AB$ again at $X$ and $\mathcal{C}_2$ cuts line $AC$ again at $Y$. Prove that $X,Y$ and $H$ are collinear.

2011 Princeton University Math Competition, A2
Tags: algebra
Define the sequence of real numbers $\{x_n\}_{n \geq 1}$, where $x_1$ is any real number and \[x_n = 1 - x_1x_2\ldots x_{n-1} \text{ for all } n > 1.\] Show that $x_{2011} > \frac{2011}{2012}$.

2016 AMC 10, 24
Tags:
A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side? $\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\textbf{(D) }300\sqrt{2}\qquad\textbf{(E) } 500$

2022 MIG, 18
Tags:
If the six-digit number $\underline{2}\, \underline{0}\, \underline{2} \, \underline{1} \, \underline{a} \, \underline{b}$ is divisible by $9$, what is the greatest possible value of $a \cdot b$? $\textbf{(A) }18\qquad\textbf{(B) }20\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad\textbf{(E) }42$

2019 JBMO Shortlist, N5
Tags: number theory
Find all positive integers $x, y, z$ such that $45^x-6^y=2019^z$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

2000 All-Russian Olympiad, 5
Tags: induction , algebra
The sequence $a_1 = 1$, $a_2, a_3, \cdots$ is defined as follows: if $a_n - 2$ is a natural number not already occurring on the board, then $a_{n+1} = a_n-2$; otherwise, $a_{n+1} = a_n + 3$. Prove that every nonzero perfect square occurs in the sequence as the previous term increased by $3$.

2012 QEDMO 11th, 7
Tags: combinatorial geometry , combinatorics , rhombus , hexagon
In the following, a rhombus is one with edge length $1$ and interior angles $60^o$ and $120^o$ . Now let $n$ be a natural number and $H$ a regular hexagon with edge length $n$, which is covered with rhombuses without overlapping has been. The rhombuses then appear in three different orientations. Prove that whatever the overlap looks exactly, each of these three orientations can be viewed at the same time.

1995 China Team Selection Test, 3
Tags: vector , combinatorics
21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on?

2024 CCA Math Bonanza, L2.3
Tags:
Call an $8$-digit number [i]cute[/i] if its digits are a permutation of $1,2,\dots,8$. For example, $23615478$ is [i]cute[/i] but $31234587$ is not. Find the number of $8$-digit [i]cute[/i] numbers that are divisible by $11$. [i]Lightning 2.3[/i]

1969 Swedish Mathematical Competition, 1
Tags: number theory , diophantine equation , diophantine
Find all integers m, n such that $m^3 = n^3 + n$.

2014 PUMaC Geometry A, 4
Tags: geometric transformation , reflection , trigonometry , cyclic quadrilateral , perpendicular bisector , trig identities , geometry
Consider the cyclic quadrilateral with side lengths $1$, $4$, $8$, $7$ in that order. What is its circumdiameter? Let the answer be of the form $a\sqrt b+c$, for $b$ squarefree. Find $a+b+c$.

2022 Israel TST, 2
Tags: inequalities
The numbers $a$, $b$, and $c$ are real. Prove that $$(a^5+b^5+c^5+a^3c^2+b^3a^2+c^3b^2)^2\geq 4(a^2+b^2+c^2)(a^5b^3+b^5c^3+c^5a^3)$$

2007 Thailand Mathematical Olympiad, 12
Tags: combinatorics
An alien with four feet wants to wear four identical socks and four identical shoes, where on each foot a sock must be put on before a shoe. How many ways are there for the alien to wear socks and shoes?