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

1968 Miklós Schweitzer, 11
Tags: probability , probability and stats
Let $ A_1,...,A_n$ be arbitrary events in a probability field. Denote by $ C_k$ the event that at least $ k$ of $ A_1,...,A_n$ occur. Prove that \[ \prod_{k=1}^n P(C_k) \leq \prod_{k=1}^n P(A_k).\] [i]A. Renyi[/i]

1996 USAMO, 4
Tags: induction , function , modular arithmetic , strong induction , combinatorics
An $n$-term sequence $(x_1, x_2, \ldots, x_n)$ in which each term is either 0 or 1 is called a [i]binary sequence of length [/i]$n$. Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.

2010 Contests, 4
Tags: inequalities
If $a,b,c\in (0,1)$ satisfy $a+b+c=2$ , prove that $\frac{abc}{(1-a)(1-b)(1-c)}\ge 8$

1987 IMO Longlists, 49
Tags: analytic geometry , ceiling function , combinatorics
In the coordinate system in the plane we consider a convex polygon $W$ and lines given by equations $x = k, y = m$, where $k$ and $m$ are integers. The lines determine a tiling of the plane with unit squares. We say that the boundary of $W$ intersects a square if the boundary contains an interior point of the square. Prove that the boundary of $W$ intersects at most $4 \lceil d \rceil $ unit squares, where $d$ is the maximal distance of points belonging to $W$ (i.e., the diameter of $W$) and $\lceil d \rceil$ is the least integer not less than $d.$

MathLinks Contest 1st, 1
Tags: number theory
Let $a, m$ be two positive integers, $a \ne 10^k$, for all non-negative integers $k$ and $d_1, d_2, ... , d_m$ random decimal$^1$ digits with $d_1 > 0$. Prove that there exists some positive integer $n$ for which the representation in the decimal base of the number $a^n$ begins with the digits $d_1, d_2, ... , d_m$ in this order. $^1$ lesser or equal with $9$

2005 Tournament of Towns, 5
Tags: geometry , 3d geometry , invariance
A cube lies on the plane. After being rolled a few times (over its edges), it is brought back to its initial location with the same face up. Could the top face have been rotated by 90 degrees? [i](5 points)[/i]

2008 Stars Of Mathematics, 2
Tags: combinatorics
The $ 2^N$ vertices of the $ N$-dimensional hypercube $ \{0,1\}^N$ are labelled with integers from $ 0$ to $ 2^N \minus{} 1$, by, for $ x \equal{} (x_1,x_2,\ldots ,x_N)\in \{0,1\}^N$, \[v(x) \equal{} \sum_{k \equal{} 1}^{N}x_k2^{k \minus{} 1}.\] For which values $ n$, $ 2\leq n \leq 2^n$ can the vertices with labels in the set $ \{v|0\leq v \leq n \minus{} 1\}$ be connected through a Hamiltonian circuit, using edges of the hypercube only? [i]E. Bazavan & C. Talau[/i]

2019 Spain Mathematical Olympiad, 5
Tags: minimum value , inequalities
We consider all pairs (x, y) of real numbers such that $0\leq x \leq y \leq 1$.Let $M (x,y)$ the maximum value of the set $$A=\{xy, 1-x-y+xy, x+y-2xy\}.$$ Find the minimum value that $M(x,y)$ can take for all these pairs $(x,y)$.

2003 Korea - Final Round, 1
Tags: inequalities , combinatorics
Some computers of a computer room have a following network. Each computers are connected by three cable to three computers. Two arbitrary computers can exchange data directly or indirectly (through other computers). Now let's remove $K$ computers so that there are two computers, which can not exchange data, or there is one computer left. Let $k$ be the minimum value of $K$. Let's remove $L$ cable from original network so that there are two computers, which can not exchange data. Let $l$ be the minimum value of $L$. Show that $k=l$.

2018 Philippine MO, 3
Tags: linear algebra , matrix , combinatorics
Let $n$ be a positive integer. An $n \times n$ matrix (a rectangular array of numbers with $n$ rows and $n$ columns) is said to be a platinum matrix if: [list=i] [*] the $n^2$ entries are integers from $1$ to $n$; [*] each row, each column, and the main diagonal (from the upper left corner to the lower right corner) contains each integer from $1$ to $n$ exactly once; and [*] there exists a collection of $n$ entries containing each of the numbers from $1$ to $n$, such that no two entries lie on the same row or column, and none of which lie on the main diagonal of the matrix. [/list] Determine all values of $n$ for which there exists an $n \times n$ platinum matrix.

2013 Romania Team Selection Test, 1
Tags: inequalities , algebra , polynomial , absolute value , triangle inequality , number theory
Let $a$ and $b$ be two square-free, distinct natural numbers. Show that there exist $c>0$ such that \[ \left | \{n\sqrt{a}\}-\{n\sqrt{b}\} \right |>\frac{c}{n^3}\] for every positive integer $n$.

2005 MOP Homework, 1
Tags: geometry , rectangle , pigeonhole principle , rotation , combinatorics
Two rooks on a chessboard are said to be attacking each other if they are placed in the same row or column of the board. (a) There are eight rooks on a chessboard, none of them attacks any other. Prove that there is an even number of rooks on black fields. (b) How many ways can eight mutually non-attacking rooks be placed on the 9 £ 9 chessboard so that all eight rooks are on squares of the same color.

1994 Portugal MO, 1
Tags: number theory
Determine the smallest natural number that has exactly $1994$ divisors.

2017 Junior Regional Olympiad - FBH, 2
Tags: milk , equation
In three cisterns of milk lies $780$ litres of milk. When we pour off from first cistern quarter of milk, from second cistern fifth of milk and from third cistern $\frac{3}{7}$ of milk, in all cisterns remain same amount of milk. How many milk is in cisterns?

2022 Moldova EGMO TST, 3
Tags: combinatorics
Find the smallest nonnegative integer $n$ such that in every set of $n$ numbers there are always two distinct numbers such that their sum or difference is divisible by $2022$.

2003 Greece National Olympiad, 2
Tags: algebra
Find all real solutions of the system \[\begin{cases}x^2 + y^2 - z(x + y) = 2, \\ y^2 + z^2 - x(y + z) = 4, \\ z^2 + x^2 - y(z + x) = 8.\end{cases}\]

2008 Junior Balkan Team Selection Tests - Romania, 1
Tags: induction , modular arithmetic , number theory , logarithm
Let $ p$ be a prime number, $ p\not \equal{} 3$, and integers $ a,b$ such that $p\mid a+b$ and $ p^2\mid a^3 \plus{} b^3$. Prove that $ p^2\mid a \plus{} b$ or $ p^3\mid a^3 \plus{} b^3$.

2016 Harvard-MIT Mathematics Tournament, 10
Tags:
The incircle of a triangle $ABC$ is tangent to $BC$ at $D$. Let $H$ and $\Gamma$ denote the orthocenter and circumcircle of $\triangle ABC$. The $B$-mixtilinear incircle, centered at $O_B$, is tangent to lines $BA$ and $BC$ and internally tangent to $\Gamma$. The $C$-mixtilinear incircle, centered at $O_C$, is defined similarly. Suppose that $\overline{DH} \perp \overline{O_BO_C}$, $AB = \sqrt3$ and $AC = 2$. Find $BC$.

2007 Bosnia and Herzegovina Junior BMO TST, 4
Tags: geometry , circumcircle , incenter , arc midpoint , equal angles
Let $I$ be the incenter of the triangle $ABC$ ($AB < BC$). Let $M$ be the midpoint of $AC$, and let $N$ be the midpoint of the arc $AC$ of the circumcircle of $ABC$ which contains $B$. Prove that $\angle IMA = \angle INB$.

2019 AMC 10, 5
Tags:
What is the greatest number of consecutive integers whose sum is $45 ?$ $\textbf{(A) } 9 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 90 \qquad\textbf{(E) } 120$

2021 BMT, Tie 1
Tags: geometry
Regular hexagon $NOSAME$ with side length $1$ and square $UDON$ are drawn in the plane such that $UDON$ lies outside of $NOSAME$. Compute $[SAND] + [SEND]$, the sum of the areas of quadrilaterals $SAND$ and $SEND$.

2009 Germany Team Selection Test, 2
Tags: combinatorics
In Skinien there 2009 towns where each of them is connected with exactly 1004 other town by a highway. Prove that starting in an arbitrary town one can make a round trip along the highways such that each town is passed exactly once and finally one returns to its starting point.

1991 Denmark MO - Mohr Contest, 5
Tags: point , geometry , combinatorial geometry , combinatorics , circles
Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.

2015 Stars Of Mathematics, 1
Tags: inequalities
Let $a,b,c\ge 0$ be three real numbers such that $$ab+bc+ca+2abc=1.$$ Prove that $\sqrt{a}+\sqrt{b}+\sqrt{c}\ge 2$ and determine equality cases.

2014 Thailand Mathematical Olympiad, 4
Tags: polynomial , algebra
Find $P(x)\in Z[x]$ st : $P(n)|2557^{n}+213.2014$ with any $n\in N^{*}$