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.

AND:
OR:
NO:

Found problems: 85335

2022 Dutch BxMO TST, 3
Tags: diophantine equation , diophantine , number theory
Find all pairs $(p, q)$ of prime numbers such that $$p(p^2 -p - 1) = q(2q + 3).$$

2018 Baltic Way, 5
Tags: algebra , polynomial
A polynomial $f(x)$ with real coefficients is called [i]generating[/i], if for each polynomial $\varphi(x)$ with real coefficients there exists a positive integer $k$ and polynomials $g_1(x),\dotsc,g_k(x)$ with real coefficients such that \[\varphi(x)=f(g_1(x))+\dotsc+f(g_k(x)).\] Find all generating polynomials.

2011 Balkan MO Shortlist, C3
Tags: combinatorics , partition
Is it possible to partition the set of positive integer numbers into two classes, none of which contains an infinite arithmetic sequence (with a positive ratio)? What is we impose the extra condition that in each class $\mathcal{C}$ of the partition, the set of difference \begin{align*} \left\{ \min \{ n \in \mathcal{C} \mid n >m \} -m \mid m \in \mathcal{C} \right \} \end{align*} be bounded?

2013 USA Team Selection Test, 4
Tags: function , induction , functional equation , combinatorics , arrows , algebra
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function, and let $f^m$ be $f$ applied $m$ times. Suppose that for every $n \in \mathbb{N}$ there exists a $k \in \mathbb{N}$ such that $f^{2k}(n)=n+k$, and let $k_n$ be the smallest such $k$. Prove that the sequence $k_1,k_2,\ldots $ is unbounded. [i]Proposed by Palmer Mebane, United States[/i]

2011 Today's Calculation Of Integral, 763
Tags: calculus , integration , function , calculus computations , logarithm
Evaluate $\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}dx.$

2025 Malaysian IMO Team Selection Test, 9
Tags: number theory
Given four distinct positive integers $a<b<c<d$ such that $\gcd(a,b,c,d)=1$, find the maximum possible number of integers $1\le n\le 2025$ such that $$a+b+c+d\mid a^n+b^n+c^n+d^n$$ [i]Proposed by Ivan Chan Kai Chin[/i]

2023 Korea - Final Round, 1
Tags: geometry
In a triangle $ABC ~(\overline{AB} < \overline{AC})$, points $D (\neq A, B)$ and $E (\neq A, C)$ lies on side $AB$ and $AC$ respectively. Point $P$ satisfies $\overline{PB}=\overline{PD}, \overline{PC}=\overline{PE}$. $X (\neq A, C)$ is on the arc $AC$ of the circumcircle of triangle $ABC$ not including $B$. Let $Y (\neq A)$ be the intersection of circumcircle of triangle $ADE$ and line $XA$. Prove that $\overline{PX} = \overline{PY}$.

1989 AMC 8, 18
Tags:
Many calculators have a reciprocal key $\boxed{\frac{1}{x}}$ that replaces the current number displayed with its reciprocal. For example, if the display is $\boxed{00004}$ and the $\boxed{\frac{1}{x}}$ key is depressed, then the display becomes $\boxed{000.25}$. If $\boxed{00032}$ is currently displayed, what is the fewest number of times you must depress the $\boxed{\frac{1}{x}}$ key so the display again reads $\boxed{00032}$? $\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

2001 Federal Competition For Advanced Students, Part 2, 2
Tags: algebra
Determine all triples of positive real numbers $(x, y, z)$ such that \[x+y+z=6,\]\[\frac 1x + \frac 1y + \frac 1z = 2 - \frac{4}{xyz}.\]

2008 239 Open Mathematical Olympiad, 2
Tags: geometry
A circumscribed quadrilateral $ABCD$ is given. $E$ and $F$ are the intersection points of opposite sides of the $ABCD$. It turned out that the radii of the inscribed circles of the triangles $AEF$ and $CEF$ are equal. Prove that $AC \bot BD$.

2019 Sharygin Geometry Olympiad, 8
Tags: geometry
What is the least positive integer $k$ such that, in every convex 1001-gon, the sum of any k diagonals is greater than or equal to the sum of the remaining diagonals?

2012 Iran MO (3rd Round), 3
Tags: function , ceiling function , induction , number theory , logarithm
Prove that for each $n \in \mathbb N$ there exist natural numbers $a_1<a_2<...<a_n$ such that $\phi(a_1)>\phi(a_2)>...>\phi(a_n)$. [i]Proposed by Amirhossein Gorzi[/i]

2005 Purple Comet Problems, 18
Tags: geometry , trapezoid , ratio , number theory , relatively prime
The side lengths of a trapezoid are $\sqrt[4]{3}, \sqrt[4]{3}, \sqrt[4]{3}$, and $2 \cdot \sqrt[4]{3}$. Its area is the ratio of two relatively prime positive integers, $m$ and $n$. Find $m + n$.

2022 ITAMO, 6
Tags: geometry
Let $ABC$ be a non-equilateral triangle and let $R$ be the radius of its circumcircle. The incircle of $ABC$ has $I$ as its centre and is tangent to side $CA$ in point $D$ and to side $CB$ in point $E$. Let $A_1$ be the point on line $EI$ such that $A_1I=R$, with $I$ being between $A_1$ and $E$. Let $B_1$ be the point on line $DI$ such that $B_1I=R$, with $I$ being between $B_1$ and $D$. Let $P$ be the intersection of lines $AA_1$ and $BB_1$. (a) Prove that $P$ belongs to the circumcircle of $ABC$. (b) Let us now also suppose that $AB=1$ and $P$ coincides with $C$. Determine the possible values of the perimeter of $ABC$.

2015 ASDAN Math Tournament, 31
Tags:
Compute the sum of the irrational solutions of the equation $$\frac{x^2+16x+54}{x^2+11x+35}=\frac{x^2+13x+35}{x^2+14x+54}.$$

1965 Miklós Schweitzer, 4
Tags: advanced fields , algebra , function , domain , combinatorial geometry
The plane is divided into domains by $ n$ straight lines in general position, where $ n \geq 3$. Determine the maximum and minimum possible number of angular domains among them. (We say that $ n$ lines are in general position if no two are parallel and no three are concurrent.)

2005 Purple Comet Problems, 14
Tags:
Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?

2007 Bundeswettbewerb Mathematik, 4
Tags: combinatorics
A regular hexagon, as shown in the attachment, is dissected into 54 congruent equilateral triangles by parallels to its sides. Within the figure we yield exactly 37 points which are vertices of at least one of those triangles. Those points are enumerated in an arbitrary way. A triangle is called [i]clocky[/i] if running in a clockwise direction from the vertex with the smallest assigned number, we pass a medium number and finally reach the vertex with the highest number. Prove that at least 19 out of 54 triangles are clocky.

2011 Sharygin Geometry Olympiad, 20
Tags: midpoint , diagonal , geometry , cyclic quadrilateral , tangential quadrilateral
Quadrilateral $ABCD$ is circumscribed around a circle with center $I$. Points $M$ and $N$ are the midpoints of diagonals $AC$ and $BD$. Prove that $ABCD$ is cyclic quadrilateral if and only if $IM : AC = IN : BD$. [i]Nikolai Beluhov and Aleksey Zaslavsky[/i]

1951 Moscow Mathematical Olympiad, 188
Tags: algebra , inequalities
Prove that $x^{12} - x^9 + x^4 - x + 1 > 0$ for all $x$.

2023 AMC 10, 1
Tags: speed
Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes towards $B$ at 18 miles per hour. Leaving at the same time, Beth bikes toward $A$ at 12 miles per hour. How many miles from City $A$ will they be when they meet? $\textbf{(A) }20\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$

2001 India Regional Mathematical Olympiad, 2
Tags: gauss
Find all primes $p$ and $q$ such that $p^2 + 7pq + q^2$ is a perfect square.

2018 PUMaC Individual Finals A, 2
Tags: function
Find all functions $f:\mathbb{R^{+}}\to\mathbb{R^+}$ such that for all $x,y\in\mathbb{R^+}$ it holds that $$f\left(xy\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{x+y}\right)\right)=f\left(xy\left(\frac{1}{x}+\frac{1}{y}\right)\right)+f(x)f\left(\frac{y}{x+y}\right).$$

2025 Alborz Mathematical Olympiad, P3
Tags: 3d geometry , combination , partition , combinatorics
Is it possible to partition three-dimensional space into tetrahedra (not necessarily regular) such that there exists a plane that intersects the edges of each tetrahedron at exactly 4 or 0 points? Proposed by Arvin Taheri

2021 SEEMOUS, Problem 1
Tags:
Let $f: [0, 1] \to \mathbb{R}$ be a continuous strictly increasing function such that \[ \lim_{x \to 0^+} \frac{f(x)}{x}=1. \] (a) Prove that the sequence $(x_n)_{n \ge 1}$ defined by \[ x_n=f \left(\frac{1}{1} \right)+f \left(\frac{1}{2} \right)+\cdots+f \left(\frac{1}{n} \right)-\int_1^n f \left(\frac{1}{x} \right) \mathrm dx \] is convergent. (b) Find the limit of the sequence $(y_n)_{n \ge 1}$ defined by \[ y_n=f \left(\frac{1}{n+1} \right)+f \left(\frac{1}{n+2} \right)+\cdots+f \left(\frac{1}{2021n} \right). \]