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 India IMOTC, 3
Tags: number theory
Let $P(x) \in \mathbb{Q}[x]$ be a polynomial with rational coefficients and degree $d\ge 2$. Prove there is no infinite sequence $a_0, a_1, \ldots$ of rational numbers such that $P(a_i)=a_{i-1}+i$ for all $i\ge 1$. [i]Proposed by Pranjal Srivastava and Rohan Goyal[/i]

Russian TST 2016, P3
Tags: number theory , rational number
Prove that any rational number can be represented as a product of four rational numbers whose sum is zero.

1999 Hungary-Israel Binational, 1
Tags: polynomial , algebra
$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$, for all $ n \in N$. Let $ r_n$ be the average of $ g_n(x)$'s roots. If $ r_{19}\equal{}99$, find $ r_{99}$.

2023 Harvard-MIT Mathematics Tournament, 17
Tags: guts
An equilateral triangle lies in the Cartesian plane such that the $x$-coordinates of its vertices are pairwise distinct and all satisfy the equation $x^3-9x^2 + 10x + 5 = 0.$ Compute the side length of the triangle.

2010 Kazakhstan National Olympiad, 1
Tags: ellipse , rectangle , geometry , conic
Triangle $ABC$ is given. Consider ellipse $ \Omega _1$, passes through $C$ with focuses in $A$ and $B$. Similarly define ellipses $ \Omega _2 , \Omega _3$ with focuses $B,C$ and $C,A$ respectively. Prove, that if all ellipses have common point $D$ then $A,B,C,D$ lies on the circle. Ellipse with focuses $X,Y$, passes through $Z$- locus of point $T$, such that $XT+YT=XZ+YZ$

2025 India STEMS Category A, 3
Tags: geometry , a-hm point , reflection
Let $ABC$ be an acute scalene triangle with orthocenter $H$. Let $M$ be the midpoint of $BC$. $N$ is the point on line $AM$ such that $(BMN)$ is tangent to $AB$. Finally, let $H'$ be the reflection of $H$ in $B$. Prove that $\angle ANH'=90^{\circ}$. [i]Proposed by Malay Mahajan and Siddharth Choppara[/i]

2003 Alexandru Myller, 2
Tags: linear algebra , matrix
Let be two $ 3\times 3 $ real matrices that have the property that $$ AX=\begin{pmatrix}0\\0\\0\end{pmatrix}\implies BX=\begin{pmatrix}0\\0\\0\end{pmatrix} , $$ for any three-dimensional vectors $ X. $ Prove that there exists a $ 3\times 3 $ real matrix $ C $ such that $ B=CA. $

2023 BMT, 4
Tags: number theory
Given positive integers $a \ge 2$ and $k$, let $m_a(k)$ denote the remainder when $k$ is divided by $a$. Compute the number of positive integers, $n$, less than 500 such that $m_2(m_5(m_{11}(n))) = 1$.

2024 Miklos Schweitzer, 6
Tags:
During heat diffusion, we say that the evolution of temperature at a point $x \in \mathbb{R}^n$ is astonishing if it changes monotonicity infinitely many times. Can it happen that the temperature evolves astonishingly at every point $x \in \mathbb{R}^n$? More precisely, does there exist a nonnegative $u \in C^2((0, +\infty) \times \mathbb{R}^n)$ solving the heat equation $\partial_t u = \Delta u$, such that $u(t,x) \to 0$ for every $x$ as $t \to \infty$, and for every $x \in \mathbb{R}^n$, the function $t \mapsto u(t,x)$ changes monotonicity infinitely many times on $(0, \infty)$?

1988 Romania Team Selection Test, 1
Tags: 3d geometry , sphere , geometric transformation , rotation , geometry
Consider a sphere and a plane $\pi$. For a variable point $M \in \pi$, exterior to the sphere, one considers the circular cone with vertex in $M$ and tangent to the sphere. Find the locus of the centers of all circles which appear as tangent points between the sphere and the cone. [i]Octavian Stanasila[/i]

2001 All-Russian Olympiad, 3
Tags: geometry , circumcircle , ratio , trigonometry , geometric transformation , homothety , power of a point
Let the circle $ {\omega}_{1}$ be internally tangent to another circle $ {\omega}_{2}$ at $ N$.Take a point $ K$ on $ {\omega}_{1}$ and draw a tangent $ AB$ which intersects $ {\omega}_{2}$ at $ A$ and $ B$. Let $M$ be the midpoint of the arc $ AB$ which is on the opposite side of $ N$. Prove that, the circumradius of the $ \triangle KBM$ doesnt depend on the choice of $ K$.

2001 Bulgaria National Olympiad, 2
Tags: parallelogram , circumcircle , power of a point , radical axis , geometry
Suppose that $ABCD$ is a parallelogram such that $DAB>90$. Let the point $H$ to be on $AD$ such that $BH$ is perpendicular to $AD$. Let the point $M$ to be the midpoint of $AB$. Let the point $K$ to be the intersecting point of the line $DM$ with the circumcircle of $ADB$. Prove that $HKCD$ is concyclic.

2023/2024 Tournament of Towns, 2
Tags: combinatorics
2. There are three hands on a clock. Each of them rotates in a normal direction at some non-zero speed, which can be wrong. In the morning the long and the short hands coincided. Just in three hours after that moment the long and the mid-length hands coincided. After next four hours the short and the mid-length hands coincided. Will it necessarily occur that all three hands will coincide? Alexandr Yuran

2023 AMC 10, 12
Tags:
How many three-digit positive integers $N$ satisfy the following properties? - The number $N$ is divisible by $7$. - The number formed by reversing the digits of $N$ is divisible by $5$. $\textbf{(A) }13\qquad\textbf{(B) }14\qquad\textbf{(C) }15\qquad\textbf{(D) }16\qquad\textbf{(E) }17$

2005 Miklós Schweitzer, 4
Tags: abstract algebra , free group
Let F be a countable free group and let $F = H_1> H_2> H_3> \cdots$ be a descending chain of finite index subgroups of group F. Suppose that $\cap H_i$ does not contain any nontrivial normal subgroups of F. Prove that there exist $g_i\in F$ for which the conjugated subgroups $H_i^{g_i}$ also form a chain, and $\cap H_i^{g_i}=\{1\}$. [hide=Note]Nielsen-Schreier Theorem might be useful.[/hide]

2010 Contests, 3
Tags: circumcircle , geometric transformation , reflection , geometry
Let $ABC$ be a triangle,$O$ its circumcenter and $R$ the radius of its circumcircle.Denote by $O_{1}$ the symmetric of $O$ with respect to $BC$,$O_{2}$ the symmetric of $O$ with respect to $AC$ and by $O_{3}$ the symmetric of $O$ with respect to $AB$. (a)Prove that the circles $C_{1}(O_{1},R)$, $C_{2}(O_{2},R)$, $C_{3}(O_{3},R)$ have a common point. (b)Denote by $T$ this point.Let $l$ be an arbitary line passing through $T$ which intersects $C_{1}$ at $L$, $C_{2}$ at $M$ and $C_{3}$ at $K$.From $K,L,M$ drop perpendiculars to $AB,BC,AC$ respectively.Prove that these perpendiculars pass through a point.

2016 EGMO TST Turkey, 6
Tags: number theory , squarefree
Prove that for every square-free integer $n>1$, there exists a prime number $p$ and an integer $m$ satisfying \[ p \mid n \quad \text{and} \quad n \mid p^2+p\cdot m^p. \]

2011 Saudi Arabia Pre-TST, 4.1
Tags: ratio , fixed , geometry , semicircle
On a semicircle of diameter $AB$ and center $C$, consider vari­able points $M$ and $N$ such that $MC \perp NC$. The circumcircle of triangle $MNC$ intersects $AB$ for the second time at $P$. Prove that $\frac{|PM-PN|}{PC}$ constant and find its value.

2024 AMC 12/AHSME, 2
Tags: factorial
What is $10! - 7! \cdot 6!$? $ \textbf{(A) }-120 \qquad \textbf{(B) }0 \qquad \textbf{(C) }120 \qquad \textbf{(D) }600 \qquad \textbf{(E) }720 \qquad $

2019 Saint Petersburg Mathematical Olympiad, 3
Tags: inequalities
Let $a, b$ and $c$ be non-zero natural numbers such that $c \geq b$ . Show that $$a^b\left(a+b\right)^c>c^b a^c.$$

1995 Yugoslav Team Selection Test, Problem 2
Tags: number theory
A natural number $n$ has exactly $1995$ units in its binary representation. Show that $n!$ is divisible by $2^{n-1995}$.

2000 All-Russian Olympiad, 4
Tags: inequalities , algebra , combinatorics , sequence
Let $a_1, a_2, \cdots, a_n$ be a sequence of nonnegative integers. For $k=1,2,\cdots,n$ denote \[ m_k = \max_{1 \le l \le k} \frac{a_{k-l+1} + a_{k-l+2} + \cdots + a_k}{l}. \] Prove that for every $\alpha > 0$ the number of values of $k$ for which $m_k > \alpha$ is less than $\frac{a_1+a_2+ \cdots +a_n}{\alpha}.$

2012 Romania Team Selection Test, 4
Tags: algebra , function , domain , inequalities
Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.

1964 Miklós Schweitzer, 3
Tags: superior algebra , abstract algebra , geometric series
Prove that the intersection of all maximal left ideals of a ring is a (two-sided) ideal.

1972 IMO Shortlist, 7
Tags: geometry , 3d geometry , tetrahedron
Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.