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

2006 Harvard-MIT Mathematics Tournament, 3
Tags: calculus , analytic geometry
At time $0$, an ant is at $(1,0)$ and a spider is at $(-1,0)$. The ant starts walking counterclockwise around the unit circle, and the spider starts creeping to the right along the $x$-axis. It so happens that the ant's horizontal speed is always half the spider's. What will the shortest distance ever between the ant and the spider be?

2024 Baltic Way, 12
Tags: geometry
Let $ABC$ be an acute triangle with circumcircle $\omega$ such that $AB<AC$. Let $M$ be the midpoint of the arc $BC$ of~$\omega$ containing the point~$A$, and let $X\neq M$ be the other point on $\omega$ such that $AX=AM$. Points $E$ and $F$ are chosen on sides $AC$ and $AB$ of the triangle $ABC$ such that $EX=EC$ and $FX=FB$. Prove that $AE=AF$.

2014 Singapore Senior Math Olympiad, 17
Tags:
Let $n$ be a positive integer such that $12n^2+12n+11$ is a $4$-digit number with all $4$ digits equal. Determine the value of $n$.

2000 Manhattan Mathematical Olympiad, 1
Tags: polynomial , algebra
Prove there exists no polynomial $f(x)$, with integer coefficients, such that $f(7) = 11$ and $f(11) = 13$.

2025 SEEMOUS, P4
Tags: real analysis , series , sequence
Let $(a_n)_{n\geq 1}$ be a monotone decreasing sequence of real numbers that converges to $0$. Prove that $\sum_{n=1}^{\infty}\frac{a_n}{n}$ is convergent if and only if the sequence $(a_n\ln n)_{n\geq 1}$ is bounded and $\sum_{n=1}^{\infty} (a_n-a_{n+1})\ln n$ is convergent.

2012 Mathcenter Contest + Longlist, 2
Tags: number theory , prime
Let $p=2^n+1$ and $3^{(p-1)/2}+1\equiv 0 \pmod p$. Show that $p$ is a prime. [i](Zhuge Liang) [/i]

1980 Kurschak Competition, 2
Tags: number theory
Let $n > 1$ be an odd integer. Prove that a necessary and sufficient condition for the existence of positive integers $x$ and $y$ satisfying $$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}$$ is that $n$ has a prime divisor of the form $4k - 1$.

1951 AMC 12/AHSME, 39
Tags:
A stone is dropped into a well and the report of the stone striking the bottom is heard $ 7.7$ seconds after it is dropped. Assume that the stone falls $ 16t^2$ feet in $ t$ seconds and that the velocity of sound is $ 1120$ feet per second. The depth of the well is: $ \textbf{(A)}\ 784 \text{ ft.} \qquad\textbf{(B)}\ 342 \text{ ft.} \qquad\textbf{(C)}\ 1568 \text{ ft.} \qquad\textbf{(D)}\ 156.8 \text{ ft.} \qquad\textbf{(E)}\ \text{none of these}$

1986 Traian Lălescu, 1.1
Tags: equation , system of equations , algebra
Solve: $$ \left\{ \begin{matrix} x+y=\sqrt{4z -1} \\ y+z=\sqrt{4x -1} \\ z+x=\sqrt{4y -1}\end{matrix}\right. . $$

2006 Stanford Mathematics Tournament, 6
Tags: function , ratio , geometric series
The expression $16^n+4^n+1$ is equiavalent to the expression $(2^{p(n)}-1)/(2^{q(n)}-1)$ for all positive integers $n>1$ where $p(n)$ and $q(n)$ are functions and $\tfrac{p(n)}{q(n)}$ is constant. Find $p(2006)-q(2006)$.

2019 India IMO Training Camp, P3
Tags: algebra
Let $n\ge 2$ be an integer. Solve in reals: \[|a_1-a_2|=2|a_2-a_3|=3|a_3-a_4|=\cdots=n|a_n-a_1|.\]

2021-2022 OMMC, 10
Tags:
A real number $x$ satisfies $2 + \log_{25} x + \log_8 5 = 0$. Find \[\log_2 x - (\log_8 5)^3 - (\log_{25} x)^3.\] [i]Proposed by Evan Chang[/i]

2003 India IMO Training Camp, 10
Tags: modular arithmetic , number theory
Let $n$ be a positive integer greater than $1$, and let $p$ be a prime such that $n$ divides $p-1$ and $p$ divides $n^3-1$. Prove that $4p-3$ is a square.

1999 Harvard-MIT Mathematics Tournament, 9
Tags: algebra
Evaluate $$\sum^{17}_{n=2} \frac{n^2+n+1}{n^4+2n^3-n^2-2n}.$$

2023 Taiwan TST Round 1, 1
Tags: algebra
Let $\mathbb{Q}_{>1}$ be the set of rational numbers greater than $1$. Let $f:\mathbb{Q}_{>1}\to \mathbb{Z}$ be a function that satisfies \[f(q)=\begin{cases} q-3&\textup{ if }q\textup{ is an integer,}\\ \lceil q\rceil-3+f\left(\frac{1}{\lceil q\rceil-q}\right)&\textup{ otherwise.} \end{cases}\] Show that for any $a,b\in\mathbb{Q}_{>1}$ with $\frac{1}{a}+\frac{1}{b}=1$, we have $f(a)+f(b)=-2$. [i]Proposed by usjl[/i]

2017 Hanoi Open Mathematics Competitions, 13
Tags: inequalities , distance , geometric inequalities , geometry
Let $ABC$ be a triangle. For some $d>0$ let $P$ stand for a point inside the triangle such that $|AB| - |P B| \ge d$, and $|AC | - |P C | \ge d$. Is the following inequality true $|AM | - |P M | \ge d$, for any position of $M \in BC $?

2020 Bangladesh Mathematical Olympiad National, Problem 7
Tags: coordinate geometry , algebra , geometry
$f$ is a function on the set of complex numbers such that $f(z)=1/(z*)$, where $z*$ is the complex conjugate of $z$. $S$ is the set of complex numbers $z$ such that the real part of $f(z)$ lies between $1/2020$ and $1/2018$. If $S$ is treated as a subset of the complex plane, the area of $S$ can be expressed as $m× \pi$ where $m$ is an integer. What is the value of $m$?

2000 IMO Shortlist, 2
Tags: number theory , divisor
For a positive integer $n$, let $d(n)$ be the number of all positive divisors of $n$. Find all positive integers $n$ such that $d(n)^3=4n$.

IMSC 2023, 5
Tags: combinatorics
In the plane, $2022$ points are chosen such that no three points lie on the same line. Each of the points is coloured red or blue such that each triangle formed by three distinct red points contains at least one blue point. What is the largest possible number of red points? [i]Proposed by Art Waeterschoot, Belgium[/i]

2022 Portugal MO, 1
Tags: combinatorics
Raul's class has $15$ students, all with different heights. The Mathematics teacher wants to place them in a queue so that, at the beginning of the queue, they are ordered in ascending order of heights, from then on, they are ordered in descending order and Raul, who He is the tallest in the class, he cannot be at the extremes. In how many different ways is it possible to form this queue?

1988 IberoAmerican, 2
Tags: number theory
Let $a,b,c,d,p$ and $q$ be positive integers satisfying $ad-bc=1$ and $\frac{a}{b}>\frac{p}{q}>\frac{c}{d}$. Prove that: $(a)$ $q\ge b+d$ $(b)$ If $q=b+d$, then $p=a+c$.

2013 Bogdan Stan, 4
Tags: seqeunce , cesaro-stolz , real analysis , sequence
Let be a sequence $ \left( x_n \right)_{n\ge 1} $ having the property that $$ \lim_{n\to\infty } \left( 14(n+2)x_{n+2} -15(n+1)x_{n+1} +nx_n \right) =13. $$ Show that $ \left( x_n \right)_{n\ge 1} $ is convergent and calculate its limit. [i]Cosmin Nițu[/i]

2013 BMT Spring, 4
Tags: algebra
Find the sum of all real numbers $x$ such that $x^2 = 5x + 6\sqrt{x} - 3$.

1991 Arnold's Trivium, 57
Tags:
Find the dimension of the solution space of the problem $\partial u/\partial \overline{z} = \delta(z - i)$ for $\text{Im } z \ge 0$, $\text{Im } u(z) = 0$ for $\text{Im } z = 0$, $u\to 0$ as $z\to\infty$.

2010 Iran MO (2nd Round), 4
Tags: polynomial , inequalities , algebra
Let $P(x)=ax^3+bx^2+cx+d$ be a polynomial with real coefficients such that \[\min\{d,b+d\}> \max\{|{c}|,|{a+c}|\}\] Prove that $P(x)$ do not have a real root in $[-1,1]$.