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

1994 Putnam, 5
Tags: limit
Let $(r_n)_{n\ge 0}$ be a sequence of positive real numbers such that $\lim_{n\to \infty} r_n = 0$. Let $S$ be the set of numbers representable as a sum \[ r_{i_1} + r_{i_2} +\cdots + r_{i_{1994}} ,\] with $i_1 < i_2 < \cdots< i_{1994}.$ Show that every nonempty interval $(a, b)$ contains a nonempty subinterval $(c, d)$ that does not intersect $S$.

1974 IMO Longlists, 25
Tags: trigonometry , limit , algebra
Let $f : \mathbb R \to \mathbb R$ be of the form $f(x) = x + \epsilon \sin x,$ where $0 < |\epsilon| \leq 1.$ Define for any $x \in \mathbb R,$ \[x_n=\underbrace{f \ o \ \ldots \ o \ f}_{n \text{ times}} (x).\] Show that for every $x \in \mathbb R$ there exists an integer $k$ such that $\lim_{n\to \infty } x_n = k\pi.$

2007 Princeton University Math Competition, 7
Tags: geometry
$A, B, C$, and $D$ are all on a circle, and $ABCD$ is a convex quadrilateral. If $AB = 13$, $BC = 13$, $CD = 37$, and $AD = 47$, what is the area of $ABCD$?

2024 Israel TST, P3
Tags: combinatorics
Let $0<c<1$ and $n$ a positive integer. Alice and Bob are playing a game. Bob writes $n$ integers on the board, not all equal. On a player's turn, they erase two numbers from the board and write their arithmetic mean instead. Alice starts and performs at most $cn$ moves. After her, Bob makes moves until there are only two numbers left on the board. Alice wins if these two numbers are different, and otherwise, Bob wins. For which values of $c$ does Alice win for all large enough $n$?

2022 Israel TST, 2
Tags: function , algebra , polynomial
Let $f: \mathbb{Z}^2\to \mathbb{R}$ be a function. It is known that for any integer $C$ the four functions of $x$ \[f(x,C), f(C,x), f(x,x+C), f(x, C-x)\] are polynomials of degree at most $100$. Prove that $f$ is equal to a polynomial in two variables and find its maximal possible degree. [i]Remark: The degree of a bivariate polynomial $P(x,y)$ is defined as the maximal value of $i+j$ over all monomials $x^iy^j$ appearing in $P$ with a non-zero coefficient.[/i]

2016 AMC 8, 21
Tags: probability
A box contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn? $\textbf{(A) }\frac{3}{10}\qquad\textbf{(B) }\frac{2}{5}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{3}{5}\qquad \textbf{(E) }\frac{7}{10}$

2006 Moldova Team Selection Test, 2
Tags: analytic geometry , trigonometry , geometry
Consider a right-angled triangle $ABC$ with the hypothenuse $AB=1$. The bisector of $\angle{ACB}$ cuts the medians $BE$ and $AF$ at $P$ and $M$, respectively. If ${AF}\cap{BE}=\{P\}$, determine the maximum value of the area of $\triangle{MNP}$.

1983 IMO Longlists, 2
Tags: combinatorics
Seventeen cities are served by four airlines. It is noted that there is direct service (without stops) between any two cities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings.

2019 Stanford Mathematics Tournament, 7
Tags: geometry
Let $G$ be the centroid of triangle $ABC$ with $AB = 9$, $BC = 10,$ and $AC = 17$. Denote $D$ as the midpoint of $BC$. A line through $G$ parallel to $BC$ intersects $AB$ at $M$ and $AC$ at $N$. If $BG$ intersects $CM$ at $E$ and $CG$ intersects $BN$ at $F$, compute the area of triangle $DEF$.

Novosibirsk Oral Geo Oly VIII, 2020.2
Tags: geometry , perimeter , chessboard , combinatorics
Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?

2003 Romania National Olympiad, 2
Tags: function , real analysis , integration , definite integral , titu andreescu
Let be an odd natural number $ n\ge 3. $ Find all continuous functions $ f:[0,1]\longrightarrow\mathbb{R} $ that satisfy the following equalities. $$ \int_0^1 \left( f\left(\sqrt[k]{x}\right) \right)^{n-k} dx=k/n,\quad\forall k\in\{ 1,2,\ldots ,n-1\} $$ [i]Titu Andreescu[/i]

2005 Tournament of Towns, 5
Tags: geometry , 3d geometry
Prove that if a regular icosahedron and a regular dodecahedron have a common circumsphere, then they have a common insphere. [i](7 points)[/i]

2002 Tournament Of Towns, 3
Tags: modular arithmetic , algebra , polynomial , quadratic , number theory
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.

2012 France Team Selection Test, 1
Tags: function , algebra
Let $k>1$ be an integer. A function $f:\mathbb{N^*}\to\mathbb{N^*}$ is called $k$-[i]tastrophic[/i] when for every integer $n>0$, we have $f_k(n)=n^k$ where $f_k$ is the $k$-th iteration of $f$: \[f_k(n)=\underbrace{f\circ f\circ\cdots \circ f}_{k\text{ times}}(n)\] For which $k$ does there exist a $k$-tastrophic function?

Russian TST 2016, P2
Tags: algebra , inequalities
Prove that \[1+\frac{2^1}{1-2^1}+\frac{2^2}{(1-2^1)(1-2^2)}+\cdots+\frac{2^{2016}}{(1-2^1)\cdots(1-2^{2016})}>0.\]

Kyiv City MO 1984-93 - geometry, 1993.8.3
Tags: geometry , angle bisector , equal segments
In the triangle $ABC$, $\angle .ACB = 60^o$, and the bisectors $AA_1$ and $BB_1$ intersect at the point $M$. Prove that $MB_1 = MA_1$.

2021 Science ON grade VII, 1
Tags: number theory , set
Supoose $A$ is a set of integers which contains all integers that can be written as $2^a-2^b$, $a,b\in \mathbb{Z}_{\ge 1}$ and also has the property that $a+b\in A$ whenever $a,b\in A$. Prove that if $A$ contains at least an odd number, then $A=\mathbb{Z}$. [i] (Andrei Bâra)[/i]

2014 Online Math Open Problems, 6
Tags: modular arithmetic , least common multiple
Let $L_n$ be the least common multiple of the integers $1,2,\dots,n$. For example, $L_{10} = 2{,}520$ and $L_{30} = 2{,}329{,}089{,}562{,}800$. Find the remainder when $L_{31}$ is divided by $100{,}000$. [i]Proposed by Evan Chen[/i]

2019 IOM, 3
Tags: circumcircle , incenter , geometry
In a non-equilateral triangle $ABC$ point $I$ is the incenter and point $O$ is the circumcenter. A line $s$ through $I$ is perpendicular to $IO$. Line $\ell$ symmetric to like $BC$ with respect to $s$ meets the segments $AB$ and $AC$ at points $K$ and $L$, respectively ($K$ and $L$ are different from $A$). Prove that the circumcenter of triangle $AKL$ lies on the line $IO$. [i]Dušan Djukić[/i]

2018 PUMaC Geometry B, 8
Tags: geometry , parallelogram , angle bisector
Let $ABCD$ be a parallelogram such that $AB = 35$ and $BC = 28$. Suppose that $BD \perp BC$. Let $\ell_1$ be the reflection of $AC$ across the angle bisector of $\angle BAD$, and let $\ell_2$ be the line through $B$ perpendicular to $CD$. $\ell_1$ and $\ell_2$ intersect at a point $P$. If $PD$ can be expressed in simplest form as $\frac{m}{n}$, find $m + n$.

1999 Harvard-MIT Mathematics Tournament, 1
Tags: calculus , function , integration
Find all twice differentiable functions $f(x)$ such that $f^{\prime \prime}(x)=0$, $f(0)=19$, and $f(1)=99$.

1997 All-Russian Olympiad Regional Round, 8.4
Tags: combinatorics
The company employs 50,000 people. For each of them, the sum of the number of his immediate superiors and his immediate subordinates is equal to 7. On Monday, each employee of the enterprise issues an order and gives a copy of this order to each of his direct subordinates (if there are any). Further, every day an employee takes all the basics he received on the previous day and either distributes them copies to all your direct subordinates, or, if any, he is not there, he carries out orders himself. It turned out that on Friday no papers were transferred to the institution. Prove that the enterprise has at least 97 bosses over whom there are no bosses.

2013 Harvard-MIT Mathematics Tournament, 3
Tags: hmmt , modular arithmetic
Find the rightmost non-zero digit of the expansion of $(20)(13!)$.

2015 BMT Spring, 7
Tags: geometry
Define $A = (1, 0, 0)$, $B = (0, 1, 0)$, and $P$ as the set of all points $(x, y, z)$ such that $x+y+z = 0$. Let $P$ be the point on $P$ such that $d = AP + P B$ is minimized. Find $d^2$.

2020 Germany Team Selection Test, 1
Tags: combinatorics , induction , local argument
You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.