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

2016 India IMO Training Camp, 2

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^3+f(y)\right)=x^2f(x)+y,$$for all $x,y\in\mathbb{R}.$ (Here $\mathbb{R}$ denotes the set of all real numbers.)

2004 Harvard-MIT Mathematics Tournament, 1

Tags: inequalities
How many ordered pairs of integers $(a,b)$ satisfy all of the following inequalities? \begin{eqnarray*} a^2 + b^2 &<& 16 \\ a^2 + b^2 &<& 8a \\ a^2 + b^2 &<& 8b \end{eqnarray*}

PEN K Problems, 1

Prove that there is a function $f$ from the set of all natural numbers into itself such that $f(f(n))=n^2$ for all $n \in \mathbb{N}$.

2009 Benelux, 1

Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ that satisfy the following two conditions: [list]$\bullet\ f(n)$ is a perfect square for all $n\in\mathbb{Z}_{>0}$ $\bullet\ f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}_{>0}$.[/list]

2014 Indonesia MO Shortlist, A3

Prove for each positive real number $x, y, z$, $$\frac{x^2y}{x+2y}+\frac{y^2z}{y+2z}+\frac{z^2x}{z+2x}<\frac{(x+y+z)^2}{8}$$

2019 AMC 12/AHSME, 25

Tags: geometry
Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\circ$, $60^\circ$, and $60.001^\circ$. For each positive integer $n$ define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\triangle A_nB_nC_n$ is obtuse? $\phantom{}$ $\textbf{(A) } 10 \qquad \textbf{(B) }11 \qquad \textbf{(C) } 13\qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15$

Kvant 2019, M2581

In a social network with a fixed finite setback of users, each user had a fixed set of [i]followers[/i] among the other users. Each user has an initial positive integer rating (not necessarily the same for all users). Every midnight, the rating of every user increases by the sum of the ratings that his followers had just before midnight. Let $m$ be a positive integer. A hacker, who is not a user of the social network, wants all the users to have ratings divisible by $m$. Every day, he can either choose a user and increase his rating by 1, or do nothing. Prove that the hacker can achieve his goal after some number of days. [i]Vladislav Novikov[/i]

2011 USA Team Selection Test, 5

Let $c_n$ be a sequence which is defined recursively as follows: $c_0 = 1$, $c_{2n+1} = c_n$ for $n \geq 0$, and $c_{2n} = c_n + c_{n-2^e}$ for $n > 0$ where $e$ is the maximal nonnegative integer such that $2^e$ divides $n$. Prove that \[\sum_{i=0}^{2^n-1} c_i = \frac{1}{n+2} {2n+2 \choose n+1}.\]

1962 Kurschak Competition, 1

Show that the number of ordered pairs $(a, b)$ of positive integers with lowest common multiple $n$ is the same as the number of positive divisors of $n^2$.

2018 AIME Problems, 14

Tags:
The incircle of $\omega$ of $\triangle ABC$ is tangent to $\overline{BC}$ at $X$. Let $Y \neq X$ be the other intersection of $\overline{AX}$ with $\omega$. Points $P$ and $Q$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, so that $\overline{PQ}$ is tangent to $\omega$ at $Y$. Assume that $AP=3, PB = 4, AC=8$, and $AQ = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

1985 IMO Longlists, 78

The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by \[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\] Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$

2023 Turkey EGMO TST, 4

Let $n$ be a positive integer and $P,Q$ be polynomials with real coefficients with $P(x)=x^nQ(\frac{1}{x})$ and $P(x) \geq Q(x)$ for all real numbers $x$. Prove that $P(x)=Q(x)$ for all real number $x$.

2017 BAMO, B

Tags:
Two three-dimensional objects are said to have the same coloring if you can orient one object (by moving or turning it) so that it is indistinguishable from the other. For example, suppose we have two unit cubes sitting on a table, and the faces of one cube are all black except for the top face which is red, and the the faces of the other cube are all black except for the bottom face, which is colored red. Then these two cubes have the same coloring. In how many different ways can you color the edges of a regular tetrahedron, coloring two edges red, two edges black, and two edges green? (A regular tetrahedron has four faces that are each equilateral triangles. The figure below depicts one coloring of a tetrahedron, using thick, thin, and dashed lines to indicate three colors.)

2025 Francophone Mathematical Olympiad, 2

Let $n \geqslant 2$ be an integer. We consider a square grid of size $2n \times 2n$ divided into $4n^2$ unit squares. The grid is called [i]balanced[/i] if: [list] [*]Each cell contains a number equal to $-1$, $0$ or $1$. [*]The absolute value of the sum of the numbers in the grid does not exceed $4n$. [/list] Determine, as a function of $n$, the smallest integer $k \geqslant 1$ such that any balanced grid always contains an $n \times n$ square whose absolute sum of the $n^2$ cells is less than or equal to $k$.

2020 MMATHS, 1

A positive integer $n$ is called an untouchable number if there is no positive integer $m$ for which the sum of the factors of $m$ (including $m$ itself) is $n + m$. Find the sum of all of the untouchable numbers between $1$ and $10$ (inclusive)

2004 ITAMO, 3

(a) Is $2005^{2004}$ the sum of two perfect squares? (b) Is $2004^{2005}$ the sum of two perfect squares?

2022 BMT, Tie 2

A positive integer is called extra-even if all of its digits are even. Compute the number of positive integers $n$ less than or equal to $2022$ such that both $n$ and $2n$ are both extra-even.

2022 Bangladesh Mathematical Olympiad, 6

About $5$ years ago, Joydip was researching on the number $2017$. He understood that $2017$ is a prime number. Then he took two integers $a,b$ such that $0<a,b <2017$ and $a+b\neq 2017.$ He created two sequences $A_1,A_2,\dots ,A_{2016}$ and $B_1,B_2,\dots, B_{2016}$ where $A_k$ is the remainder upon dividing $ak$ by $2017$, and $B_k$ is the remainder upon dividing $bk$ by $2017.$ Among the numbers $A_1+B_1,A_2+B_2,\dots A_{2016}+B_{2016}$ count of those that are greater than $2017$ is $N$. Prove that $N=1008.$

2001 All-Russian Olympiad Regional Round, 8.5

Tags: algebra
Let $a, b, c, d, e$ and $f$ be some numbers, and $ a \cdot c \cdot e \ne 0$.It is known that the values of the expressions $|ax+b|+|cx+d| $and $|ex+f|$ equal at all values of $x$. Prove that $ad = bc$.

1998 Iran MO (3rd Round), 3

Let $n(r)$ be the maximum possible number of points with integer coordinates on a circle with radius $r$ in Cartesian plane. Prove that $n(r) < 6\sqrt[3]{3 \pi r^2}.$

2001 District Olympiad, 4

Tags: logarithm , algebra
Solve the equation: \[2^{\lg x}+8=(x-8)^{\frac{1}{\lg 2}}\] Note: $\lg x=\log_{10}x$. [i]Daniel Jinga [/i]

2019 Azerbaijan BMO TST, 2

Let $ABC$ be a triangle inscribed in circle $\Gamma$ with center $O$. Let $H$ be the orthocenter of triangle $ABC$ and let $K$ be the midpoint of $OH$. Tangent of $\Gamma$ at $B$ intersects the perpendicular bisector of $AC$ at $L$. Tangent of $\Gamma$ at $C$ intersects the perpendicular bisector of $AB$ at $M$. Prove that $AK$ and $LM$ are perpendicular. by Michael Sarantis, Greece

2012 Baltic Way, 3

(a) Show that the equation \[\lfloor x \rfloor (x^2 + 1) = x^3,\] where $\lfloor x \rfloor$ denotes the largest integer not larger than $x$, has exactly one real solution in each interval between consecutive positive integers. (b) Show that none of the positive real solutions of this equation is rational.

1991 AMC 12/AHSME, 19

Triangle $ABC$ has a right angle at $C$, $AC = 3$ and $BC = 4$. Triangle $ABD$ has a right angle at $A$ and $AD = 12$. Points $C$ and $D$ are on opposite sides of $\overline{AB}$. The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$. If $\frac{DE}{DB} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then $m + n = $ [asy] size(170); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair C=origin, A=(0,3), B=(4,0), D=(7.2,12.6), E=(7.2,0); draw(A--C--B--A--D--B--E--D); label("$A$",A,W); label("$B$",B,S); label("$C$",C,SW); label("$D$",D,NE); label("$E$",E,SE); [/asy] $ \textbf{(A)}\ 25\qquad\textbf{(B)}\ 128\qquad\textbf{(C)}\ 153\qquad\textbf{(D)}\ 243\qquad\textbf{(E)}\ 256 $

2013 AMC 10, 5

Tags: algebra
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $\$105$, Dorothy paid $\$125$, and Sammy paid $\$175$. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$? $ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 $