Found problems: 85335
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.
2021 EGMO, 1
The number 2021 is fantabulous. For any positive integer $m$, if any element of the set $\{m, 2m+1, 3m\}$ is fantabulous, then all the elements are fantabulous. Does it follow that the number $2021^{2021}$ is fantabulous?
2019 Online Math Open Problems, 18
Define a function $f$ as follows. For any positive integer $i$, let $f(i)$ be the smallest positive integer $j$ such that there exist pairwise distinct positive integers $a,b,c,$ and $d$ such that $\gcd(a,b)$, $\gcd(a,c)$, $\gcd(a,d)$, $\gcd(b,c)$, $\gcd(b,d)$, and $\gcd(c,d)$ are pairwise distinct and equal to $i, i+1, i+2, i+3, i+4,$ and $j$ in some order, if any such $j$ exists; let $f(i)=0$ if no such $j$ exists. Compute $f(1)+f(2)+\dots +f(2019)$.
[i]Proposed by Edward Wan[/i]
2024 Austrian MO National Competition, 3
Initially, the numbers $1, 2, \dots, 2024$ are written on a blackboard. Trixi and Nana play a game, taking alternate turns. Trixi plays first.
The player whose turn it is chooses two numbers $a$ and $b$, erases both, and writes their (possibly negative) difference $a-b$ on the blackboard. This is repeated until only one number remains on the blackboard after $2023$ moves. Trixi wins if this number is divisible by $3$, otherwise Nana wins.
Which of the two has a winning strategy?
[i](Birgit Vera Schmidt)[/i]
2007 Middle European Mathematical Olympiad, 2
A set of balls contains $ n$ balls which are labeled with numbers $ 1,2,3,\ldots,n.$ We are given $ k > 1$ such sets. We want to colour the balls with two colours, black and white in such a way, that
(a) the balls labeled with the same number are of the same colour,
(b) any subset of $ k\plus{}1$ balls with (not necessarily different) labels $ a_{1},a_{2},\ldots,a_{k\plus{}1}$ satisfying the condition $ a_{1}\plus{}a_{2}\plus{}\ldots\plus{}a_{k}\equal{} a_{k\plus{}1}$, contains at least one ball of each colour.
Find, depending on $ k$ the greatest possible number $ n$ which admits such a colouring.
2006 Bundeswettbewerb Mathematik, 4
A piece of paper with the shape of a square lies on the desk. It gets dissected step by step into smaller pieces: in every step, one piece is taken from the desk and cut into two pieces by a straight cut; these pieces are put back on the desk then.
Find the smallest number of cuts needed to get $100$ $20$-gons.
1978 Poland - Second Round, 1
Prove that for positive real numbers $x$ and $y$ smaller than or equal to $1/2$,
\[\frac{(x+y)^2}{xy} \geq \frac{(2-xy)^2}{(1-x)(1-y)}.\]
1957 Moscow Mathematical Olympiad, 370
* Three equal circles are tangent to each other externally and to the fourth circle internally. Tangent lines are drawn to the circles from an arbitrary point on the fourth circle. Prove that the sum of the lengths of two tangent lines equals the length of the third tangent.
1961 All Russian Mathematical Olympiad, 003
Prove that among $39$ sequential natural numbers there always is a number with the sum of its digits divisible by $11$.
2012 Tuymaada Olympiad, 2
Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$.
[i]Proposed by A. Golovanov, M. Ivanov, K. Kokhas[/i]
2014 District Olympiad, 2
Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a differentiable function, with continuous derivative, and let
\[ s_{n}=\sum_{k=1}^{n}f\left( \frac{k}{n}\right) \]
Prove that the sequence $(s_{n+1}-s_{n})_{n\in{\mathbb{N}}^{\ast}}$ converges to $\int_{0}^{1}f(x)\mathrm{d}x$.
2008 Harvard-MIT Mathematics Tournament, 1
Four students from Harvard, one of them named Jack, and five students from MIT, one of them named Jill, are going to see a Boston Celtics game. However, they found out that only $ 5$ tickets remain, so $ 4$ of them must go back. Suppose that at least one student from each school must go see the game, and at least one of Jack and Jill must go see the game, how many ways are there of choosing which $ 5$ people can see the game?