Found problems: 85335
2021 Spain Mathematical Olympiad, 2
Given a positive integer $n$, we define $\lambda (n)$ as the number of positive integer solutions of $x^2-y^2=n$. We say that $n$ is [i]olympic[/i] if $\lambda (n) = 2021$. Which is the smallest olympic positive integer? Which is the smallest olympic positive odd integer?
2000 Harvard-MIT Mathematics Tournament, 11
Let $M$ be the maximum possible value of $x_1x_2+x_2x_3+\cdots +x_5x_1$ where $x_1, x_2, \cdots x_5$ is a permutation of $(1,2,3,4,5)$ and let $N$ be the number of permutations for which this maximum is attained. Evaluate $M+N$.
2020 IMO Shortlist, N6
For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that
$$ \frac {\varphi ( d(n))}{d(\varphi(n))}\le C$$
for all $n\ge 1$
[i]Cyprus[/i]
1991 AIME Problems, 13
A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $1/2$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?
1981 Vietnam National Olympiad, 2
Consider the polynomials
\[f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;\]
\[g(p) = p^3 + 2p + m.\]
Find all integral values of $m$ for which $f$ is divisible by $g$.
2003 Turkey Team Selection Test, 1
Let $M = \{(a,b,c,d)|a,b,c,d \in \{1,2,3,4\} \text{ and } abcd > 1\}$. For each $n\in \{1,2,\dots, 254\}$, the sequence $(a_1, b_1, c_1, d_1)$, $(a_2, b_2, c_2, d_2)$, $\dots$, $(a_{255}, b_{255},c_{255},d_{255})$ contains each element of $M$ exactly once and the equality \[|a_{n+1} - a_n|+|b_{n+1} - b_n|+|c_{n+1} - c_n|+|d_{n+1} - d_n| = 1\] holds. If $c_1 = d_1 = 1$, find all possible values of the pair $(a_1,b_1)$.
2022 IMO Shortlist, N1
A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number.
(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)
2023 Iranian Geometry Olympiad, 1
We are given an acute triangle $ABC$. The angle bisector of $\angle BAC$ cuts $BC$ at $P$. Points $D$ and $E$ lie on segments $AB$ and $AC$, respectively, so that $BC \parallel DE$. Points $K$ and $L$ lie on segments $PD$ and $PE$, respectively, so that points $A$, $D$, $E$, $K$, $L$ are concyclic. Prove that points $B$, $C$, $K$, $L$ are also concyclic.
[i]Proposed by Patrik Bak, Slovakia [/i]
VI Soros Olympiad 1999 - 2000 (Russia), 11.1
$16$ different natural numbers are written on the board, none of which exceeds $30$. Prove that there must be two coprime numbers among the written numbers.
2014 Cezar Ivănescu, 3
[b]a)[/b] Prove that, for any point in the interior of a triangle, there are two points on the sides of this triangle such that the resultant of the vectors from the interior point those two points is the vector $ 0. $
[b]b)[/b] Prove that, for any point in the interior of a triangle, there are three points on the sides of this triangle such that the resultant of the vectors from the interior point those three points is the vector $ 0. $
2009 Germany Team Selection Test, 3
Initially, on a board there a positive integer. If board contains the number $x,$ then we may additionally write the numbers $2x+1$ and $\frac{x}{x+2}.$ At some point 2008 is written on the board. Prove, that this number was there from the beginning.
1989 Poland - Second Round, 3
Given is a trihedral angle $ OABC $ with a vertex $ O $ and a point $ P $ in its interior. Let $ V $ be the volume of a parallelepiped with two vertices at points $ O $ and $ P $, whose three edges are contained in the rays $ \overrightarrow{OA} $, $ \overrightarrow{OB} $, $ \overrightarrow{OC} $. Calculate the minimum volume of a tetrahedron whose three faces are contained in the faces of the trihedral angle $OABC$ and the fourth face contains the point $P$.
1955 AMC 12/AHSME, 13
The fraction $ \frac{a^{\minus{}4}\minus{}b^{\minus{}4}}{a^{\minus{}2}\minus{}b^{\minus{}2}}$ is equal to:
$ \textbf{(A)}\ a^{\minus{}6}\minus{}b^{\minus{}6} \qquad
\textbf{(B)}\ a^{\minus{}2}\minus{}b^{\minus{}2} \qquad
\textbf{(C)}\ a^{\minus{}2}\plus{}b^{\minus{}2} \\
\textbf{(D)}\ a^2\plus{}b^2 \qquad
\textbf{(E)}\ a^2\minus{}b^2$
2016 China Northern MO, 4
Can we put intengers $1,2,\cdots,12$ on a circle, number them $a_1,a_2,\cdots,a_{12}$ in order. For any $1\leq i<j\leq12$, $|a_i-a_j|\neq|i-j|$?
2010 International Zhautykov Olympiad, 3
A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figures of the third kind is even.
[img]http://up.iranblog.com/Files7/dda310bab8b6455f90ce.jpg[/img]
2012 Bosnia And Herzegovina - Regional Olympiad, 1
Find all possible values of $$\frac{1}{a}\left(\frac{1}{b}+\frac{1}{c}+\frac{1}{b+c}\right)+\frac{1}{b}\left(\frac{1}{c}+\frac{1}{a}+\frac{1}{c+a}\right)+\frac{1}{c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{a+b}\right)-\frac{1}{a+b+c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}$$ where $a$, $b$ and $c$ are positive real numbers such that $ab+bc+ca=abc$
1992 IMO Longlists, 23
An [i]Egyptian number[/i] is a positive integer that can be expressed as a sum of positive integers, not necessarily distinct, such that the sum of their reciprocals is $1$. For example, $32 = 2 + 3 + 9 + 18$ is Egyptian because $\frac 12 +\frac 13 +\frac 19 +\frac{1}{18}=1$ . Prove that all integers greater than $23$ are [i]Egyptian[/i].
2021 Chile National Olympiad, 1
Consider the sequence of numbers defined by $a_1 = 7$, $a_2 = 7^7$ , $ ...$ , $a_n = 7^{a_{n-1}}$ for $n \ge 2$. Determine the last digit of the decimal representation of $a_{2021}$.
2022 Brazil Team Selection Test, 3
Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right.
Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors.
[i]Proposed by Gurgen Asatryan, Armenia[/i]
2022 Sharygin Geometry Olympiad, 1
Let $O$ and $H$ be the circumcenter and the orthocenter respectively of triangle $ABC$. Itis known that $BH$ is the bisector of angle $ABO$. The line passing through $O$ and parallel to $AB$ meets $AC$ at $K$. Prove that $AH = AK$
2010 Germany Team Selection Test, 1
The quadrilateral $ABCD$ is a rhombus with acute angle at $A.$ Points $M$ and $N$ are on segments $\overline{AC}$ and $\overline{BC}$ such that $|DM| = |MN|.$ Let $P$ be the intersection of $AC$ and $DN$ and let $R$ be the intersection of $AB$ and $DM.$ Prove that $|RP| = |PD|.$
2006 Macedonia National Olympiad, 3
Let $a,b,c$ be real numbers distinct from $0$ and $1$, with $a+b+c=1$. Prove that
\[8\left(\frac{1}{2}-ab-bc-ca\right)\left(\frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} \right)\ge 9 \]
2007 Iran Team Selection Test, 2
Suppose $n$ lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in $[0,\frac{\pi}2]$. Find the largest value of the sum of the $\binom n2$ angles between line.
[i]By Aliakbar Daemi[/i]
2015 China Western Mathematical Olympiad, 8
Let $k$ be a positive integer, and $n=\left(2^k\right)!$ .Prove that $\sigma(n)$ has at least a prime divisor larger than $2^k$, where $\sigma(n)$ is the sum of all positive divisors of $n$.
1993 Tournament Of Towns, (397) 5
Four frogs sit on the vertices of a square, one on each vertex. They jump in arbitrary order but not simultaneously. Each frog jumps to the point symmetrical to its take-off position with respect to the centre of gravity of the three other frogs. Can one of them jump (at a given time) exactly on to one of the others (considering their locations as points)?
(A Andzans)