Found problems: 85335
1989 All Soviet Union Mathematical Olympiad, 494
Show that the $120$ five digit numbers which are permutations of $12345$ can be divided into two sets with each set having the same sum of squares.
2016 Tuymaada Olympiad, 3
Altitudes $AA_1$, $BB_1$, $CC_1$ of an acute triangle $ABC$ meet at $H$.
$A_0$, $B_0$, $C_0$ are the midpoints of $BC$, $CA$, $AB$ respectively. Points $A_2$, $B_2$, $C_2$ on the segments $AH$, $BH$, $HC_1$ respectively are such that $\angle A_0B_2A_2 = \angle B_0C_2B_2 = \angle C_0A_2C_2 =90^\circ$.
Prove that the lines $AC_2$, $BA_2$, $CB_2$ are concurrent.
1998 All-Russian Olympiad, 7
A tetrahedron $ABCD$ has all edges of length less than $100$, and contains two nonintersecting spheres of diameter $1$. Prove that it contains a sphere of diameter $1.01$.
2011 AMC 10, 21
Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine?
$ \textbf{(A)}\ \frac{7}{11}\qquad\textbf{(B)}\ \frac{9}{13}\qquad\textbf{(C)}\ \frac{11}{15}\qquad\textbf{(D)}\ \frac{15}{19}\qquad\textbf{(E)}\ \frac{15}{16} $
2018 Online Math Open Problems, 17
Let $S$ be the set of all subsets of $\left\{2,3,\ldots,2016\right\}$ with size $1007$, and for a nonempty set $T$ of numbers, let $f(T)$ be the product of the elements in $T$. Determine the remainder when \[ \sum_{T\in S}\left(f(T)-f(T)^{-1}\right)^2\] is divided by $2017$. Note: For $b$ relatively prime to $2017$, we say that $b^{-1}$ is the unique positive integer less than $2017$ for which $2017$ divides $bb^{-1} -1$.
[i]Proposed by Tristan Shin[/i]
2019 Saudi Arabia JBMO TST, 3
Is there positive integer $n$, such that
$n+2$ divides $S=1^{2019}+2^{2019}+...+n^{2019}$
2005 District Olympiad, 1
Prove that for all $a\in\{0,1,2,\ldots,9\}$ the following sum is divisible by 10:
\[ S_a = \overline{a}^{2005} + \overline{1a}^{2005} + \overline{2a}^{2005} + \cdots + \overline{9a}^{2005}. \]
LMT Guts Rounds, 18
Congruent unit circles intersect in such a way that the center of each circle lies on the circumference of the other. Let $R$ be the region in which two circles overlap. Determine the perimeter of $R.$
2002 National Olympiad First Round, 35
For each integer $i=0,1,2, \dots$, there are eight balls each weighing $2^i$ grams. We may place balls as much as we desire into given $n$ boxes. If the total weight of balls in each box is same, what is the largest possible value of $n$?
$
\textbf{a)}\ 8
\qquad\textbf{b)}\ 10
\qquad\textbf{c)}\ 12
\qquad\textbf{d)}\ 15
\qquad\textbf{e)}\ 16
$
2012 IMAC Arhimede, 6
Let $a,b,c$ be positive real numbers that satisfy the condition $a + b + c = 1$. Prove the inequality
$$\frac{a^{-3}+b}{1-a}+\frac{b^{-3}+c}{1-b}+\frac{c^{-3}+a}{1-c}\ge 123$$
2010 Stanford Mathematics Tournament, 8
Find all solutions of $\frac{a}{x}=\frac{x-a}{a}$ for $x$.
2004 AMC 12/AHSME, 25
For each integer $ n\geq 4$, let $ a_n$ denote the base-$ n$ number $ 0.\overline{133}_n$. The product $ a_4a_5 \dotsm a_{99}$ can be expressed as $ \frac {m}{n!}$, where $ m$ and $ n$ are positive integers and $ n$ is as small as possible. What is the value of $ m$?
$ \textbf{(A)}\ 98 \qquad \textbf{(B)}\ 101 \qquad \textbf{(C)}\ 132\qquad \textbf{(D)}\ 798\qquad \textbf{(E)}\ 962$
1993 India Regional Mathematical Olympiad, 7
In the group of ten persons, each person is asked to write the sum of the ages of all the other nine persons. Of all ten sums form the nine-element set $\{ 82, 83,84,85,87,89,90,91,92 \}$, find the individual ages of the persons, assuming them to be whole numbers.
2001 Putnam, 5
Let $a$ and $b$ be real numbers in the interval $\left(0,\tfrac{1}{2}\right)$, and let $g$ be a continuous real-valued function such that $g(g(x))=ag(x)+bx$ for all real $x$. Prove that $g(x)=cx$ for some constant $c$.
2015 HMMT Geometry, 4
Let $ABCD$ be a cyclic quadrilateral with $AB=3$, $BC=2$, $CD=2$, $DA=4$. Let lines perpendicular to $\overline{BC}$ from $B$ and $C$ meet $\overline{AD}$ at $B'$ and $C'$, respectively. Let lines perpendicular to $\overline{BC}$ from $A$ and $D$ meet $\overline{AD}$ at $A'$ and $D'$, respectively. Compute the ratio $\frac{[BCC'B']}{[DAA'D']}$, where $[\overline{\omega}]$ denotes the area of figure $\overline{\omega}$.
2012 Kazakhstan National Olympiad, 3
Let $ a,b,c,d>0$ for which the following conditions::
$a)$ $(a-c)(b-d)=-4$
$b)$ $\frac{a+c}{2}\geq\frac{a^{2}+b^{2}+c^{2}+d^{2}}{a+b+c+d}$
Find the minimum of expression $a+c$
1996 North Macedonia National Olympiad, 2
Let $P$ be the set of all polygons in the plane and let $M : P \to R$ be a mapping that satisfies:
(i) $M(P) \ge 0$ for each polygon $P$,
(ii) $M(P) = x^2$ if $P$ is an equilateral triangle of side $x$,
(iii) If a polygon $P$ is partitioned into polygons $S$ and $T$, then $M(P) = M(S)+ M(T)$,
(iv) If polygons $P$ and $T$ are congruent, then $M(P) = M(T )$.
Determine $M(P)$ if $P$ is a rectangle with edges $x$ and $y$.
2014 Ukraine Team Selection Test, 4
The $A$-excircle of the triangle $ABC$ touches the side $BC$ at point $K$. The circumcircles of triangles $AKB$ and $AKC$ intersect for the second time with the bisector of angle $A$ at points $X$ and $Y$ respectively. Let $M$ be the midpoint of $BC$. Prove that the circumcenter of triangle $XYM$ lies on $BC$.
2009 AMC 10, 25
For $ k>0$, let $ I_k\equal{}10\ldots 064$, where there are $ k$ zeros between the $ 1$ and the $ 6$. Let $ N(k)$ be the number of factors of $ 2$ in the prime factorization of $ I_k$. What is the maximum value of $ N(k)$?
$ \textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10$
2005 Tournament of Towns, 5
In triangle $ABC$ bisectors $AA_1, BB_1$ and $CC_1$ are drawn. Given $\angle A : \angle B : \angle C = 4 : 2 : 1$, prove that $A_1B_1 = A_1C_1$.
[i](7 points)[/i]
2018 Iran MO (1st Round), 17
Two positive integers $m$ and $n$ are both less than $500$ and $\text{lcm}(m,n) = (m-n)^2$. What is the maximum possible value of $m+n$?
2010 Math Prize for Girls Olympiad, 3
Let $p$ and $q$ be integers such that $q$ is nonzero. Prove that
\[
\Bigl\lvert \frac{p}{q} - \sqrt{7} \Bigr\rvert \ge
\frac{24 - 9\sqrt{7}}{q^2} \, .
\]
2022 VN Math Olympiad For High School Students, Problem 8
Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$.
Prove that: $k(m)$ is even for all $m>2.$
2009 Vietnam National Olympiad, 4
Let $ a$, $ b$, $ c$ be three real numbers. For each positive integer number $ n$, $ a^n \plus{} b^n \plus{} c^n$ is an integer number. Prove that there exist three integers $ p$, $ q$, $ r$ such that $ a$, $ b$, $ c$ are the roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} r \equal{} 0$.
1996 Miklós Schweitzer, 8
Prove that a simply connected, closed manifold (i.e., compact, no boundary) cannot contain a closed, smooth submanifold of codimension 1, with odd Euler characteristic.