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 Korea Junior Math Olympiad, 4
Tags: transformation , analytic geometry , geometric transformation , algebra
In the coordinate plane, de fine $M = \{(a, b),a,b \in Z\}$. A transformation $S$, which is de fined on $M$, sends $(a,b)$ to $(a + b, b)$. Transformation $T$, also de fined on $M$, sends $(a, b)$ to $(-b, a)$. Prove that for all $(a, b) \in M$, we can use $S,T$ denitely to map it to $(g,0)$.

2003 Baltic Way, 17
Tags: ceiling function , number theory
All the positive divisors of a positive integer $n$ are stored into an increasing array. Mary is writing a programme which decides for an arbitrarily chosen divisor $d > 1$ whether it is a prime. Let $n$ have $k$ divisors not greater than $d$. Mary claims that it suffices to check divisibility of $d$ by the first $\left\lceil\frac{k}{2}\right\rceil$ divisors of $n$: $d$ is prime if and only if none of them but $1$ divides $d$. Is Mary right?

2024 Bundeswettbewerb Mathematik, 3
Tags: ratio , area of a triangle , geometry , inequalities
Let $ABC$ be a triangle. For a point $P$ in its interior, we draw the threee lines through $P$ parallel to the sides of the triangle. This partitions $ABC$ in three triangles and three quadrilaterals. Let $V_A$ be the area of the quadrilateral which has $A$ as one vertex. Let $D_A$ be the area of the triangle which has a part of $BC$ as one of its sides. Define $V_B, D_B$ and $V_C, D_C$ similarly. Determine all possible values of $\frac{D_A}{V_A}+\frac{D_B}{V_B}+\frac{D_C}{V_C}$, as $P$ varies in the interior of the triangle.

1981 All Soviet Union Mathematical Olympiad, 324
Tags: distance , combinatorial geometry , geometry , rectangle
Six points are marked inside the $3\times 4$ rectangle. Prove that there is a pair of marked points with the distance between them not greater than $\sqrt5$.

2008 Indonesia TST, 3
Tags: combinatorics
$10$ people attended a party. For every $3$ people, there exist at least $2$ people who don’t know each other. Prove that there exist $4$ people who don’t know each other.

2007 Harvard-MIT Mathematics Tournament, 1
Tags: calculus , limit
Compute: \[\lim_{x\to 0}\text{ }\dfrac{x^2}{1-\cos(x)}\]

TNO 2024 Junior, 5
Tags: number theory
The nine digits from 1 to 9 are to be placed around a circle so that the average of any three consecutive digits is a multiple of 3. Is this possible? Justify your answer.

2002 Olympic Revenge, 5
Tags: combinatorics
In a "Hanger Party", the guests are initially dressed. In certain moments, the host chooses a guest, and the chosen guest and all his friends will wear its respective clothes if they are naked, and undress it if they are dressed. It is possible that, in some moment, the guests are naked, independent of their mutual friendships? (Suppose friendship is reciprocal.)

1967 All Soviet Union Mathematical Olympiad, 088
Tags: number theory
Prove that there exists a number divisible by $5^{1000}$ not containing a single zero in its decimal notation.

1989 Irish Math Olympiad, 4
Tags: number theory
Note that $12^2=144$ ends in two $4$s and $38^2=1444$ end in three $4$s. Determine the length of the longest string of equal nonzero digits in which the square of an integer can end.

2023 AMC 10, 6
Tags:
An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is $21$. What is the value of the cube? $\textbf{(A)}~42\qquad\textbf{(B)}~63\qquad\textbf{(C)}~84\qquad\textbf{(D)}~126\qquad\textbf{(E)}~252$

2022 Moldova Team Selection Test, 3
Tags: combinatorics , number theory
Let $n$ be a positive integer. On a board there are written all integers from $1$ to $n$. Alina does $n$ moves consecutively: for every integer $m$ $(1 \leq m \leq n)$ the move $m$ consists in changing the sign of every number divisible by $m$. At the end Alina sums the numbers. Find this sum.

2018 Online Math Open Problems, 5
Tags:
A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$-sided die, with faces labeled $0,1,2,\ldots, 2018$, and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on $k$, and the mouse has at least $k$ slices of cheese remaining, then the mouse eats $k$ slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone? [i]Proposed by Brandon Wang[/i]

1995 Poland - First Round, 7
Tags: inequalities
Nonnegative numbers $a, b, c, p, q, r$ satisfy the conditions: $a + b + c = p + q + r = 1; ~~~~~~ p, q, r \leq \frac{1}{2}$. Prove that $8abc \leq pa + qb + rc$ and determine when equality holds.

2003 Turkey Junior National Olympiad, 3
Tags:
How many subsets of $\{1,2,3,4,5,6,7,8,9,10,11\}$ contain no two consequtive numbers?

1985 Traian Lălescu, 1.1
Tags: analytic geometry , symmetry , geometry
We are given two concurrent lines $ d_1 $ and $ d_2. $ Find, analytically, the acute angle formed by them such that for any point $ A $ the equation $ A=A_4 $ holds, where $ A_1 $ is the symmetric of $ A $ with respect to $ d_1, $ $ A_2 $ is the symmetric of $ A_1 $ with respect to $ d_2, $ $ A_3 $ is the symmetric of $ A_2 $ with respect to $ d_1, $ and $ A_4 $ is the symmetric of $ A_3 $ with respect to $ d_2. $

2022 Stanford Mathematics Tournament, 4
Tags:
Frank mistakenly believes that the number $1011$ is prime and for some integer $x$ writes down $(x+1)^{1011}\equiv x^{1011}+1\pmod{1011}$. However, it turns out that for Frank's choice of $x$, this statement is actually true. If $x$ is positive and less than $1011$, what is the sum of the possible values of $x$?

1982 Austrian-Polish Competition, 1
Tags: gcd , number theory , greatest common divisor
Find all pairs $(n, m)$ of positive integers such that $gcd ((n + 1)^m - n, (n + 1)^{m+3} - n) > 1$.

2007 Tournament Of Towns, 2
Tags: geometric transformation , combinatorics , geometry
A convex figure $F$ is such that any equilateral triangle with side $1$ has a parallel translation that takes all its vertices to the boundary of $F$. Is $F$ necessarily a circle?

2016 Saudi Arabia GMO TST, 4
Tags: game , combinatorics
There are totally $16$ teams participating in a football tournament, each team playing with every other exactly $1$ time. In each match, the winner gains $3$ points, the loser gains $0$ point and each teams gain $1$ point for the tie match. Suppose that at the end of the tournament, each team gains the same number of points. Prove that there are at least $4$ teams that have the same number of winning matches, the same number of losing matches and the same number of tie matches.

2011 Saudi Arabia Pre-TST, 4.2
Tags: pentagon , cyclic , geometry
Pentagon $ABCDE$ is inscribed in a circle. Distances from point $E$ to lines $AB$ , $BC$ and $CD$ are equal to $a, b$ and $c$, respectively. Find the distance from point $E$ to line $AD$.

2015 Princeton University Math Competition, B3
Tags: algebra
Andrew and Blair are bored in class and decide to play a game. They pick a pair $(a, b)$ with $1 \le a, b \le 100$. Andrew says the next number in the geometric series that begins with $a,b$ and Blair says the next number in the arithmetic series that begins with $a,b$. For how many pairs $(a, b)$ is Andrew's number minus Blair's number a positive perfect square?

VMEO IV 2015, 10.3
Tags: number theory , divide , divisible
Given a positive integer $k$. Find the condition of positive integer $m$ over $k$ such that there exists only one positive integer $n$ satisfying $$n^m | 5^{n^k} + 1,$$

2012 Hanoi Open Mathematics Competitions, 7
Tags: number theory , perfect square
Prove that the number $a =\overline{{1...1}{5...5}6}$ is a perfect square (where $1$s are $2012$ in total and $5$s are $2011$ in total)

1988 AIME Problems, 4
Tags:
Suppose that $|x_i| < 1$ for $i = 1, 2, \dots, n$. Suppose further that \[ |x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|. \] What is the smallest possible value of $n$?