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

PEN D Problems, 9
Tags: modular arithmetic , congruence
Show that there exists a composite number $n$ such that $a^n \equiv a \; \pmod{n}$ for all $a \in \mathbb{Z}$.

2025 6th Memorial "Aleksandar Blazhevski-Cane", P1
Tags: number theory , prime number , diophantine equation
Determine all triples of prime numbers $(p, q, r)$ that satisfy \[p2^q + r^2 = 2025.\] Proposed by [i]Ilija Jovcevski[/i]

2024 LMT Fall, 30
Tags: guts
Find \[\sum_{n=1}^{\infty} \frac{\varphi(n)}{(-4)^n-1},\]where $\varphi(n)$ is the number of positive integers $k \le n$ relatively prime to $n$. (Note $\varphi(1)=1$.)

1991 APMO, 3
Tags: inequalities , n-variable inequality
Let $a_1$, $a_2$, $\cdots$, $a_n$, $b_1$, $b_2$, $\cdots$, $b_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = b_1 + b_2 + \cdots + b_n$. Show that \[ \frac{a_1^2}{a_1 + b_1} + \frac{a_2^2}{a_2 + b_2} + \cdots + \frac{a_n^2}{a_n + b_n} \geq \frac{a_1 + a_2 + \cdots + a_n}{2} \]

2005 AIME Problems, 15
Tags: geometry , quadratic , incenter , area of a triangle , heron's formula
Triangle $ABC$ has $BC=20$. The incircle of the triangle evenly trisects the median $AD$. If the area of the triangle is $m \sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n$.

2016 Latvia Baltic Way TST, 13
Tags: geometric inequality , geometry , angle
Suppose that $A, B, C$, and $X$ are any four distinct points in the plane with $$\max \,(BX,CX) \le AX \le BC.$$ Prove that $\angle BAC \le 150^o$.

2020 AIME Problems, 10
Tags: greatest common divisor , number theory
Let $m$ and $n$ be positive integers satisfying the conditions [list] [*] $\gcd(m+n,210) = 1,$ [*] $m^m$ is a multiple of $n^n,$ and [*] $m$ is not a multiple of $n$. [/list] Find the least possible value of $m+n$.

2017 F = ma, 1
Tags: fraction
A motorcycle rides on the vertical walls around the perimeter of a large circular room. The friction coefficient between the motorcycle tires and the walls is $\mu$. How does the minimum $\mu$ needed to prevent the motorcycle from slipping downwards change with the motorcycle's speed, $s$? $\textbf{(A)}\mu \propto s^{0} \qquad \textbf{(B)}\mu \propto s^{-\frac{1}{2}}\qquad \textbf{(C)}\mu \propto s^{-1}\qquad \textbf{(D)}\mu \propto s^{-2}\qquad \textbf{(E)}\text{none of these}$

2018 PUMaC Geometry B, 3
Tags: geometry
Let $\overline{AD}$ be a diameter of a circle. Let point $B$ be on the circle, point $C$ on $\overline{AD}$ such that $A, B, C$ form a right triangle at $C$. The value of the hypotenuse of the triangle is $4$ times the square root of its area. If $\overline{BC}$ has length $30$, what is the length of the radius of the circle?

2023 Thailand Online MO, 10
Tags: combinatorics , combinatorial game
Let $n$ be an even positive integer. Alice and Bob play the following game. Before the start of the game, Alice chooses a set $S$ containing $m$ integers and announces it to Bob. The players then alternate turns, with Bob going first, choosing $i\in\{1,2,\dots, n\}$ that has not been chosen and setting the value of $v_i$ to either $0$ or $1$. At the end of the game, when all of $v_1,v_2,\dots,v_n$ have been set, the expression $$E=v_1\cdot 2^0 + v_2 \cdot 2^1 + \dots + v_n \cdot 2^{n-1}$$ is calculated. Determine the minimum $m$ such that Alice can always ensure that $E\in S$ regardless of how Bob plays.

2015 Bundeswettbewerb Mathematik Germany, 2
Tags: number theory , decimal representation , fraction
In the decimal expansion of a fraction $\frac{m}{n}$ with positive integers $m$ and $n$ you can find a string of numbers $7143$ after the comma. Show $n>1250$. [i]Example:[/i] I mean something like $0.7143$.

2020 Saint Petersburg Mathematical Olympiad, 2.
Tags: number theory
Find all positive integers $n$ such that the sum of the squares of the five smallest divisors of $n$ is a square.

2016 Balkan MO Shortlist, N2
Tags: divisor , maximum , number theory , odd
Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. ($d(n)$ is the number of divisors of the number n including $1$ and $n$ ).

1979 Bundeswettbewerb Mathematik, 4
Tags: algebra , polynomial
Prove that the polynomial $P(x) = x^5-x+a$ is irreducible over $\mathbb{Z}$ if $5 \nmid a$.

2001 Greece JBMO TST, 3
Tags: minimum , algebra , combinatorics
$4$ men stand at the entrance of a dark tunnel. Man $A$ needs $10$ minutes to pass through the tunnel, man $B$ needs $5$ minutes, man $C$ needs $2$ minutes and man $D$ needs $1$ minute. There is only one torch, that may be used from anyone that passes through the tunnel. Additionaly, at most $2$ men can pass through at the same time using the existing torch. Determine the smallest possible time the four men need to reach the exit of the tunnel.

1996 ITAMO, 3
Tags: geometry , 3d geometry , sphere , analytic geometry , symmetry
Given a cube of unit side. Let $A$ and $B$ be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point $A$ and tangent to the three sides with common point $B$.

2017 European Mathematical Cup, 1
Tags: functional equation , natural number
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that the inequality $$f(x)+yf(f(x))\le x(1+f(y))$$ holds for all positive integers $x, y$. Proposed by Adrian Beker.

2017 HMNT, 4
Tags: algebra
[b]M[/b]ary has a sequence $m_2,m_3,m_4,...$ , such that for each $b \ge 2$, $m_b$ is the least positive integer m for which none of the base-$b$ logarithms $log_b(m),log_b(m+1),...,log_b(m+2017)$ are integers. Find the largest number in her sequence.

2014 Portugal MO, 5
Tags: geometry
Let $[ABCD]$ be a convex quadrilateral with area $2014$, and let $P$ be a point on $[AB]$ and $Q$ a point on $[AD]$ such that triangles $[ABQ]$ and $[ADP]$ have area $1$. Let $R$ be the intersection of $[AC]$ and $[PQ]$. Determine $\frac{\overline{RC}}{\overline{RA}}$.

2006 India IMO Training Camp, 1
Tags: linear algebra , matrix , combinatorics
Let $n$ be a positive integer divisible by $4$. Find the number of permutations $\sigma$ of $(1,2,3,\cdots,n)$ which satisfy the condition $\sigma(j)+\sigma^{-1}(j)=n+1$ for all $j \in \{1,2,3,\cdots,n\}$.

2015 Caucasus Mathematical Olympiad, 1
Tags: combinatorics
At the round table, $10$ people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) . It is clear thath I have at least one knight and at least one liar. What is the largest number of those sitting at the table can say: ''Both of my neighbors are knights '' ? (A statement that is at least partially false is considered false.)

2024 BMT, 1
Tags: geometry
Andrew has three identical semicircular mooncake halves, each with radius $3,$ and uses them to construct the following shape, which contains an equilateral triangle in the center. Compute the perimeter around this shape, in bold below. [center] [img] https://cdn.artofproblemsolving.com/attachments/7/2/2314ac2d34cd0706f47bace3eedbb87a91582a.png [/img] [/center]

2004 AIME Problems, 7
Tags:
Let $C$ be the coefficient of $x^2$ in the expansion of the product \[(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x).\] Find $|C|$.

2009 Indonesia TST, 1
Tags: pigeonhole principle , search , number theory
a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime? b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?

2023 HMNT, 10
Tags:
A real number $x$ is chosen uniformly at random from the interval $(0,10).$ Compute the probability that $\sqrt{x}, \sqrt{x+7},$ and $\sqrt{10-x}$ are the side lengths of a non-degenerate triangle.