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

1999 Akdeniz University MO, 1
Tags: number theory
Prove that, we find infinite numbers such that, this number writeable $1999k+1$ for $k \in {\mathbb N}$ and all digits are equal.

2020 Baltic Way, 16
Tags: number theory
Richard and Kaarel are taking turns to choose numbers from the set $\{1,\dots,p-1\}$ where $p > 3$ is a prime. Richard is the first one to choose. A number which has been chosen by one of the players cannot be chosen again by either of the players. Every number chosen by Richard is multiplied with the next number chosen by Kaarel. Kaarel wins the game if at any moment after his turn the sum of all of the products calculated so far is divisible by $p$. Richard wins if this does not happen, i.e. the players run out of numbers before any of the sums is divisible by $p$. Can either of the players guarantee their victory regardless of their opponent's moves and if so, which one?

1980 Spain Mathematical Olympiad, 6
Tags: perfect square , consecutive , number theory
Prove that if the product of four consecutive natural numbers is added one unit, the result is a perfect square.

2024 239 Open Mathematical Olympiad, 2
Tags: combinatorics
A rich knight has a chest and a lot of coins, so every day he puts into the chest some quantity of coins - among the numbers $1, 2, \ldots, 100$. If there exist two days on which he added equal quantities of coins (say, $k$ coins) and he has added in total at most $100k$ coins on the days between these two days, he stops putting coins into the chest. Prove that this will necessarily happen eventually.

2009 Princeton University Math Competition, 5
Tags: inequalities
Suppose that for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$.

2014 France Team Selection Test, 6
Tags: inequalities
Let $n$ be a positive integer and $x_1,x_2,\ldots,x_n$ be positive reals. Show that there are numbers $a_1,a_2,\ldots, a_n \in \{-1,1\}$ such that the following holds: \[a_1x_1^2+a_2x_2^2+\cdots+a_nx_n^2 \ge (a_1x_1+a_2x_2 +\cdots+a_nx_n)^2\]

1996 AMC 8, 7
Tags:
Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has $4$ goldfish at the same time that Gretel has $128$ goldfish, then in how many months from that time will they have the same number of goldfish? $\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8$

2022 Grosman Mathematical Olympiad, P3
Tags: combinatorics , combinatorial geometry
An ant crawled a total distance of $1$ in the plane and returned to its original position (so that its path is a closed loop of length $1$; the width is considered to be $0$). Prove that there is a circle of radius $\frac{1}{4}$ containing the path. Illustration of an example path:

2019 Bundeswettbewerb Mathematik, 2
Tags: digit , divisible , divisibility , number theory
The lettes $A,C,F,H,L$ and $S$ represent six not necessarily distinct decimal digits so that $S \ne 0$ and $F \ne 0$. We form the two six-digit numbers $SCHLAF$ and $FLACHS$. Show that the difference of these two numbers is divisible by $271$ if and only if $C=L$ and $H=A$. [i]Remark:[/i] The words "Schlaf" and "Flachs" are German for "sleep" and "flax".

1994 Taiwan National Olympiad, 4
Tags: arithmetic sequence , number theory
Prove that there are infinitely many positive integers $n$ with the following property: For any $n$ integers $a_{1},a_{2},...,a_{n}$ which form in arithmetic progression, both the mean and the standard deviation of the set $\{a_{1},a_{2},...,a_{n}\}$ are integers. [i]Remark[/i]. The mean and standard deviation of the set $\{x_{1},x_{2},...,x_{n}\}$ are defined by $\overline{x}=\frac{x_{1}+x_{2}+...+x_{n}}{n}$ and $\sqrt{\frac{\sum (x_{i}-\overline{x})^{2}}{n}}$, respectively.

2010 Bosnia Herzegovina Team Selection Test, 5
Tags: inequalities
Let $a$,$b$ and $c$ be sides of a triangle such that $a+b+c\le2$. Prove that $-3<{\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}-\frac{a^3}{c}-\frac{b^3}{a}-\frac{c^3}{b}}<3$

MBMT Team Rounds, 2020.9
Tags:
Consider a regular pentagon $ABCDE$, and let the intersection of diagonals $\overline{CA}$ and $\overline{EB}$ be $F$. Find $\angle AFB$. [i]Proposed by Justin Chen[/i]

2022 AMC 10, 2
Tags: rates
Mike cycled $15$ laps in $57$ minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first $27$ minutes? $\textbf{(A) } 5 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13$

1976 Swedish Mathematical Competition, 5
Tags: analysis , inequalities , function , algebra
$f(x)$ is defined for $x \geq 0$ and has a continuous derivative. It satisfies $f(0)=1$, $f'(0)=0$ and $(1+f(x))f''(x)=1+x$. Show that $f$ is increasing and that $f(1) \leq 4/3$.

1990 All Soviet Union Mathematical Olympiad, 524
Tags: geometry , regular polygon , angle bisector , angle
$A, B, C$ are adjacent vertices of a regular $2n$-gon and $D$ is the vertex opposite to $B$ (so that $BD$ passes through the center of the $2n$-gon). $X$ is a point on the side $AB$ and $Y$ is a point on the side $BC$ so that $XDY = \frac{\pi}{2n}$. Show that $DY$ bisects $\angle XYC$.

2023 Math Prize for Girls Problems, 17
Tags:
Let $C$ be a unit cube. Let $D$ be a translate of $C$ such that one corner of $D$ is located at the center of $C$ and one corner of $C$ is located at the center of $D$. Let $D^\prime$ be the image of $D$ under a $60^\circ$ clockwise rotation about the line that passes through both cube centers when looking from the center of $D$ to the center of $C$. What is the volume of the intersection of $C$ with $D^\prime$?

2021 Ukraine National Mathematical Olympiad, 5
Tags: diophantine equation , diophantine , number theory
Are there natural numbers $(m,n,k)$ that satisfy the equation $m^m+ n^n=k^k$ ?

1998 AIME Problems, 6
Tags: geometry , parallelogram , similar triangles
Let $ABCD$ be a parallelogram. Extend $\overline{DA}$ through $A$ to a point $P,$ and let $\overline{PC}$ meet $\overline{AB}$ at $Q$ and $\overline{DB}$ at $R.$ Given that $PQ=735$ and $QR=112,$ find $RC.$

2023 AMC 12/AHSME, 25
Tags: trigonometry , sequence
There is a unique sequence of integers $a_1, a_2, \cdots a_{2023}$ such that $$ \tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x} $$ whenever $\tan 2023x$ is defined. What is $a_{2023}?$ $\textbf{(A) } -2023 \qquad\textbf{(B) } -2022 \qquad\textbf{(C) } -1 \qquad\textbf{(D) } 1 \qquad\textbf{(E) } 2023$

2009 International Zhautykov Olympiad, 3
Tags: inequalities , trigonometry , trig identities , law of cosines , geometry
For a convex hexagon $ ABCDEF$ with an area $ S$, prove that: \[ AC\cdot(BD\plus{}BF\minus{}DF)\plus{}CE\cdot(BD\plus{}DF\minus{}BF)\plus{}AE\cdot(BF\plus{}DF\minus{}BD)\geq 2\sqrt{3}S \]

1948 Moscow Mathematical Olympiad, 141
Tags: number theory , sum , reciprocal
The sum of the reciprocals of three positive integers is equal to $1$. What are all the possible such triples?

2014 Federal Competition For Advanced Students, P2, 2
Tags: functional equation , algebra
Let $S$ be the set of all real numbers greater than or equal to $1$. Determine all functions$ f: S \to S$, so that for all real numbers $x ,y \in S$ with $x^2 -y^2 \in S$ the condition $f (x^2 -y^2) = f (xy)$ is fulfilled.

2004 Miklós Schweitzer, 2
Tags: graph theory
Write $t(G)$ for the number of complete quadrilaterals in the graph $G$ and $e_G(S)$ for the number of edges spanned by a subset $S$ of vertices of $G$. Let $G_1, G_2$ be two (simple) graphs on a common underlying set $V$ of vertices, $|V|-n$, and assume that $|e_{G_1}(S)-e_{G_2}(S)|<\frac{n^2}{1000}$ holds for any subset $S\subseteq V$. Prove that $|t(G_1)-t(G_2)|\le \frac{n^4}{1000}$.

2014 AMC 12/AHSME, 13
Tags: binomial coefficient , counting , distinguishability , complementary counting
A fancy bed and breakfast inn has $5$ rooms, each with a distinctive color-coded decor. One day $5$ friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than $2$ friends per room. In how many ways can the innkeeper assign the guests to the rooms? $\textbf{(A) }2100\qquad \textbf{(B) }2220\qquad \textbf{(C) }3000\qquad \textbf{(D) }3120\qquad \textbf{(E) }3125\qquad$

2006 Junior Balkan Team Selection Tests - Romania, 3
Tags: trigonometry , inequalities , algebra
Find all real numbers $ a$ and $ b$ such that \[ 2(a^2 \plus{} 1)(b^2 \plus{} 1) \equal{} (a \plus{} 1)(b \plus{} 1)(ab \plus{} 1). \] [i]Valentin Vornicu[/i]