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

2005 Iran MO (3rd Round), 5

Let $a,b,c\in \mathbb N$ be such that $a,b\neq c$. Prove that there are infinitely many prime numbers $p$ for which there exists $n\in\mathbb N$ that $p|a^n+b^n-c^n$.

2002 Iran MO (3rd Round), 17

Find the smallest natural number $n$ that the following statement holds : Let $A$ be a finite subset of $\mathbb R^{2}$. For each $n$ points in $A$ there are two lines including these $n$ points. All of the points lie on two lines.

2012 ELMO Shortlist, 6

Consider a directed graph $G$ with $n$ vertices, where $1$-cycles and $2$-cycles are permitted. For any set $S$ of vertices, let $N^{+}(S)$ denote the out-neighborhood of $S$ (i.e. set of successors of $S$), and define $(N^{+})^k(S)=N^{+}((N^{+})^{k-1}(S))$ for $k\ge2$. For fixed $n$, let $f(n)$ denote the maximum possible number of distinct sets of vertices in $\{(N^{+})^k(X)\}_{k=1}^{\infty}$, where $X$ is some subset of $V(G)$. Show that there exists $n>2012$ such that $f(n)<1.0001^n$. [i]Linus Hamilton.[/i]

2019 Argentina National Olympiad Level 2, 3

Let $\Gamma$ be a circle of center $S$ and radius $r$ and let be $A$ a point outside the circle. Let $BC$ be a diameter of $\Gamma$ such that $B$ does not belong to the line $AS$ and consider the point $O$ where the perpendicular bisectors of triangle $ABC$ intersect, that is, the circumcenter of $ABC$. Determine all possible locations of point $O$ when $B$ varies in circle $\Gamma$.

2015 Puerto Rico Team Selection Test, 8

Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values ​​of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.

2007 Harvard-MIT Mathematics Tournament, 21

Tags: probability
Bob the bomb-defuser has stumbled upon an active bomb. He opens it up, and finds the red and green wires conveniently located for him to cut. Being a seasoned member of the bomb-squad, Bob quickly determines that it is the green wire that he should cut, and puts his wirecutters on the green wire. But just before he starts to cut, the bomb starts to count down, ticking every second. Each time the bomb ticks, starting at time $t=15$ seconds, Bob panics and has a certain chance to move his wirecutters to the other wire. However, he is a rational man even when panicking, and has a $\frac{1}{2t^2}$ chance of switching wires at time t, regardless of which wire he is about to cut. When the bomb ticks at $t=1$, Bob cuts whatever wire his wirecutters are on, without switching wires. What is the probability that Bob cuts the green wire?

2008 AIME Problems, 5

In trapezoid $ ABCD$ with $ \overline{BC}\parallel\overline{AD}$, let $ BC\equal{}1000$ and $ AD\equal{}2008$. Let $ \angle A\equal{}37^\circ$, $ \angle D\equal{}53^\circ$, and $ m$ and $ n$ be the midpoints of $ \overline{BC}$ and $ \overline{AD}$, respectively. Find the length $ MN$.

2021 Regional Competition For Advanced Students, 1

Let $a$ and $b$ be positive integers and $c$ be a positive real number satisfying $$\frac{a + 1}{b + c}=\frac{b}{a}.$$ Prove that $c \ge 1$ holds. (Karl Czakler)

1985 IMO Shortlist, 12

A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by \[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\] for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$

1984 AMC 12/AHSME, 7

Tags: ratio
When Dave walks to school, he averages 90 steps per minute, each of his steps 75cm long. It takes him 16 minutes to get to school. His brother, Jack, going to the same school by the same route, averages 100 steps per minute, but his steps are only 60 cm long. How long does it take Jack to get to school? $\textbf{(A) }14 \frac{2}{9}\qquad \textbf{(B) }15\qquad \textbf{(C) }18\qquad \textbf{(D) }20\qquad \textbf{(E) }22 \frac{2}{9}$

2022 Greece JBMO TST, 2

Tags: geometry , tangent
Let $ABC$ be an acute triangle with $AB<AC < BC$, inscirbed in circle $\Gamma_1$, with center $O$. Circle $\Gamma_2$, with center point $A$ and radius $AC$ intersects $BC$ at point $D$ and the circle $\Gamma_1$ at point $E$. Line $AD$ intersects circle $\Gamma_1$ at point $F$. The circumscribed circle $\Gamma_3$ of triangle $DEF$, intersects $BC$ at point $G$. Prove that: a) Point $B$ is the center of circle $\Gamma_3$ b) Circumscribed circle of triangle $CEG$ is tangent to $AC$.

2015 China Western Mathematical Olympiad, 2

Two circles $ \left(\Omega_1\right),\left(\Omega_2\right) $ touch internally on the point $ T $. Let $ M,N $ be two points on the circle $ \left(\Omega_1\right) $ which are different from $ T $ and $ A,B,C,D $ be four points on $ \left(\Omega_2\right) $ such that the chords $ AB, CD $ pass through $ M,N $, respectively. Prove that if $ AC,BD,MN $ have a common point $ K $, then $ TK $ is the angle bisector of $ \angle MTN $. * $ \left(\Omega_2\right) $ is bigger than $ \left(\Omega_1\right) $

2009 Dutch IMO TST, 3

Let $a, b$ and $c$ be positive reals such that $a + b + c \ge abc$. Prove that $a^2 + b^2 + c^2 \ge \sqrt3 abc$.

2020 CHMMC Winter (2020-21), 4

Tags: algebra
Let $P(x) = x^3 - 6x^2 - 5x + 4$. Suppose that $y$ and $z$ are real numbers such that \[ zP(y) = P(y - n) + P(y + n) \] for all reals $n$. Evaluate $P(y)$.

1987 Bundeswettbewerb Mathematik, 2

An arrow is assigned to each edge of a polyhedron such that for each vertex, there is an arrow pointing towards that vertex and an arrow pointing away from that vertex. Prove that there exist at least two faces such that the arrows on their boundaries form a cycle.

2001 Cuba MO, 7

Prove that the equation $x^{19} + x^{17} = x^{16 }+ x^7 + a$ for any $a \in R$ has at least two imaginary roots

2025 Abelkonkurransen Finale, 4b

Determine the largest real number \(C\) such that $$\frac{1}{x}+\frac{1}{2y}+\frac{1}{3z}\geqslant C$$ for all real numbers \(x,y,z\neq 0\) satisfying the equation $$\frac{x}{yz}+\frac{4y}{xz}+\frac{9z}{xy}=24$$

2022 Taiwan TST Round 3, G

Tags: geometry
Find all integers $n\geq 3$ for which every convex equilateral $n$-gon of side length $1$ contains an equilateral triangle of side length $1$. (Here, polygons contain their boundaries.)

PEN D Problems, 14

Determine the number of integers $n \ge 2$ for which the congruence \[x^{25}\equiv x \; \pmod{n}\] is true for all integers $x$.

1983 Austrian-Polish Competition, 2

Find all triples of positive integers $(p, q, n)$ with $p$ and $q$ prime, such that $p(p+1)+q(q+1) = n(n+1)$.

2019 Miklós Schweitzer, 5

Tags: inequalities
Let $S \subset \mathbb{R}^d$ be a convex compact body with nonempty interior. Show that there is an $\alpha > 0$ such that if $S = \cap_{i \in I} H_i$, where $I$ is an index set and $(H_i)_{i \in I}$ are halfspaces, then for any $P \in \mathbb{R}^d$, there is an $i \in I$ for which $\mathrm{dist}(P, H_i) \ge \alpha \, \mathrm{dist}(P, S)$.

2007 Indonesia TST, 4

Let $ S$ be a finite family of squares on a plane such that every point on that plane is contained in at most $ k$ squares in $ S$. Prove that $ P$ can be divided into $ 4(k\minus{}1)\plus{}1$ sub-family such that in each sub-family, each pair of squares are disjoint.

2021 Romania National Olympiad, 2

Determine all non-trivial finite rings with am unit element in which the sum of all elements is invertible. [i]Mihai Opincariu[/i]

OIFMAT II 2012, 4

Given a $ \vartriangle ABC $ with $ AB> AC $ and $ \angle BAC = 60^o$. Denote the circumcenter and orthocenter as $ O $ and $ H $ respectively. We also have that $ OH $ intersects $ AB $ in $ P $ and $ AC $ in $ Q $. Prove that $ PO = HQ $.

2017 Online Math Open Problems, 22

Tags:
Let $S=\{(x,y)\mid -1\leq xy\leq 1\}$ be a subset of the real coordinate plane. If the smallest real number that is greater than or equal to the area of any triangle whose interior lies entirely in $S$ is $A$, compute the greatest integer not exceeding $1000A$. [i]Proposed by Yannick Yao[/i]