Found problems: 85335
1990 Turkey Team Selection Test, 2
For real numbers $x_i$, the statement \[ x_1 + x_2 + x_3 = 0 \Rightarrow x_1x_2 + x_2x_3 + x_3x_1 \leq 0\] is always true. (Prove!)
For which $n\geq 4$ integers, the statement \[x_1 + x_2 + \dots + x_n = 0 \Rightarrow x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1 \leq 0\] is always true. Justify your answer.
2010 National Olympiad First Round, 27
Let $P$ be a polynomial with each root is real and each coefficient is either $1$ or $-1$. The degree of $P$ can be at most ?
$ \textbf{(A)}\ 5
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ \text{None}
$
1966 IMO Shortlist, 35
Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.
2022 Auckland Mathematical Olympiad, 2
The number $12$ is written on the whiteboard. Each minute, the number on the board is either multiplied or divided by one of the numbers $2$ or $3$ (a division is possible only if the result is an integer) . Prove that the number that will be written on the board in exactly one hour will not be equal to $54$.
2011 AMC 12/AHSME, 12
A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river $9$ hours after leaving dock $A.$ How many hours did it take the power boat to go from $A $ to $B$?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 3.5 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 4.5 \qquad
\textbf{(E)}\ 5
$
2009 Romania Team Selection Test, 2
Let $m<n$ be two positive integers, let $I$ and $J$ be two index sets such that $|I|=|J|=n$ and $|I\cap J|=m$, and let $u_k$, $k\in I\cup J$ be a collection of vectors in the Euclidean plane such that \[|\sum_{i\in I}u_i|=1=|\sum_{j\in J}u_j|.\] Prove that \[\sum_{k\in I\cup J}|u_k|^2\geq \frac{2}{m+n}\] and find the cases of equality.
1970 IMO Longlists, 22
In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$
2020 Austrian Junior Regional Competition, 3
Given is an isosceles trapezoid $ABCD$ with $AB \parallel CD$ and $AB> CD$. The projection from $D$ on $ AB$ is $E$. The midpoint of the diagonal $BD$ is $M$. Prove that $EM$ is parallel to $AC$.
(Karl Czakler)
2004 Tournament Of Towns, 5
Two 10-digit integers are called neighbours if they differ in exactly one digit (for example, integers $1234567890$ and $1234507890$ are neighbours). Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbours.
2010 Dutch IMO TST, 2
Find all functions $f : R \to R$ which satisfy $f(x) = max_{y\in R} (2xy - f(y))$ for all $x \in R$.
1995 Tuymaada Olympiad, 4
It is known that the merchant’s $n$ clients live in locations laid along the ring road. Of these, $k$ customers have debts to the merchant for $a_1,a_2,...,a_k$ rubles, and the merchant owes the remaining $n-k$ clients, whose debts are $b_1,b_2,...,b_{n-k}$ rubles, moreover, $a_1+a_2+...+a_k=b_1+b_2+...+b_{n-k}$. Prove that a merchant who has no money can pay all his debts and have paid all the customer debts, by starting a customer walk along the road from one of points and not missing any of their customers.
1898 Eotvos Mathematical Competition, 1
Determine all positive integers $n$ for which $2^n + 1$ is divisible by $3$.
2013 Switzerland - Final Round, 7
Let $O$ be the center of the circle of the triangle $ABC$ with $AB \ne AC$. Furthermore, let $S$ and $T$ be points on the rays $AB$ and $AC$, such that $\angle ASO = \angle ACO$ and $\angle ATO = \angle ABO$. Show that $ST$ bisects the segment $BC$.
2021 New Zealand MO, 3
Let $\{x_1, x_2, x_3, ..., x_n\}$ be a set of $n$ distinct positive integers, such that the sum of any $3$ of them is a prime number. What is the maximum value of $n$?
VMEO III 2006, 12.4
For every positive integer $n$, the symbol $a_n/b_n$ is the simplest form of the fraction $1+1/2+...+1/n$.
Prove that for every pair of positive integers $(M, N)$ we can always find a positive integer $m$ where $(a_n, N) = 1$ for all $n = m, m + 1, ...,m + M$.
1997 India National Olympiad, 3
If $a,b,c$ are three real numbers and \[ a + \dfrac{1}{b} = b + \dfrac{1}{c} = c + \dfrac{1}{a} = t \] for some real number $t$, prove that $abc + t = 0 .$
2006 MOP Homework, 5
Find all pairs of positive integers (m, n) for which it is possible to
paint each unit square of an m*n chessboard either black or white
in such way that, for any unit square of the board, the number
of unit squares which are painted the same color as that square
and which have at least one common vertex with it (including the
square itself) is even.
2014 Contests, 1
Show that \[\cos(56^{\circ}) \cdot \cos(2 \cdot 56^{\circ}) \cdot \cos(2^2\cdot 56^{\circ})\cdot . . . \cdot \cos(2^{23}\cdot 56^{\circ}) = \frac{1}{2^{24}} .\]
2008 iTest Tournament of Champions, 2
Let $A$ be the number of $12$-digit words that can be formed by from the alphabet $\{0,1,2,3,4,5,6\}$ if each pair of neighboring digits must differ by exactly $1$. Find the remainder when $A$ is divided by $2008$.
2019 Adygea Teachers' Geometry Olympiad, 1
Inside the quadrangle, a point is taken and connected with the midpoint of all sides. Areas of the three out of four formed quadrangles are $S_1, S_2, S_3$. Find the area of the fourth quadrangle.
2015 Iran Team Selection Test, 1
Point $A$ is outside of a given circle $\omega$. Let the tangents from $A$ to $\omega$ meet $\omega$ at $S, T$ points $X, Y$ are midpoints of $AT, AS$ let the tangent from $X$ to $\omega$ meet $\omega$ at $R\neq T$. points $P, Q$ are midpoints of $XT, XR$ let $XY\cap PQ=K, SX\cap TK=L$ prove that quadrilateral $KRLQ$ is cyclic.
2018 Putnam, A4
Let $m$ and $n$ be positive integers with $\gcd(m, n) = 1$, and let
\[a_k = \left\lfloor \frac{mk}{n} \right\rfloor - \left\lfloor \frac{m(k-1)}{n} \right\rfloor\]
for $k = 1, 2, \dots, n$. Suppose that $g$ and $h$ are elements in a group $G$ and that
\[gh^{a_1} gh^{a_2} \cdots gh^{a_n} = e,\]
where $e$ is the identity element. Show that $gh = hg$. (As usual, $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)
1983 Tournament Of Towns, (045) 2
Find all natural numbers $k$ which can be represented as the sum of two relatively prime numbers not equal to $1$.
2011 IMAR Test, 1
Let $A_0A_1A_2$ be a triangle and let $P$ be a point in the plane, not situated on the circle $A_0A_1A_2$. The line $PA_k$ meets again the circle $A_0A_1A_2$ at point $B_k, k = 0, 1, 2$. A line $\ell$ through the point $P$ meets the line $A_{k+1}A_{k+2}$ at point $C_k, k = 0, 1, 2$. Show that the lines $B_kC_k, k = 0, 1, 2$, are concurrent and determine the locus of their concurrency point as the line $\ell$ turns about the point $P$.
2013 South East Mathematical Olympiad, 6
$n>1$ is an integer. The first $n$ primes are $p_1=2,p_2=3,\dotsc, p_n$. Set $A=p_1^{p_1}p_2^{p_2}...p_n^{p_n}$.
Find all positive integers $x$, such that $\dfrac Ax$ is even, and $\dfrac Ax$ has exactly $x$ divisors