Found problems: 85335
2006 Baltic Way, 20
A $12$-digit positive integer consisting only of digits $1,5$ and $9$ is divisible by $37$. Prove that the sum of its digits is not equal to $76$.
2014 Saudi Arabia GMO TST, 3
Let $ABCDE$ be a cyclic pentagon such that the diagonals $AC$ and $AD$ intersect $BE$ at $P$ and $Q$, respectively, with $BP \cdot QE = PQ^2$. Prove that $BC \cdot DE = CD \cdot PQ$.
2016 Abels Math Contest (Norwegian MO) Final, 2b
Find all non-negative integers $x, y$ and $z$ such that $x^3 + 2y^3 + 4z^3 = 9!$
1998 National Olympiad First Round, 22
$ \left(x_{1} x_{2} \ldots x_{1998} \right)$ shows a number with 1998 digits in decimal system. How many numbers $ \left(x_{1} x_{2} \ldots x_{1998} \right)$ are there such that $ \left(x_{1} x_{2} \ldots x_{1998} \right) \equal{} 7\cdot 10^{1996} \left(x_{1} \plus{} x_{2} \plus{} \ldots \plus{} x_{1998} \right)$ ?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$
1987 IMO Longlists, 25
Numbers $d(n,m)$, with $m, n$ integers, $0 \leq m \leq n$, are defined by $d(n, 0) = d(n, n) = 0$ for all $n \geq 0$ and
\[md(n,m) = md(n-1,m)+(2n-m)d(n-1,m-1) \text{ for all } 0 < m < n.\]
Prove that all the $d(n,m)$ are integers.
2002 Iran Team Selection Test, 6
Assume $x_{1},x_{2},\dots,x_{n}\in\mathbb R^{+}$, $\sum_{i=1}^{n}x_{i}^{2}=n$, $\sum_{i=1}^{n}x_{i}\geq s>0$ and $0\leq\lambda\leq1$. Prove that at least $\left\lceil\frac{s^{2}(1-\lambda)^{2}}n\right\rceil$ of these numbers are larger than $\frac{\lambda s}{n}$.
1987 IMO, 2
In an acute-angled triangle $ABC$ the interior bisector of angle $A$ meets $BC$ at $L$ and meets the circumcircle of $ABC$ again at $N$. From $L$ perpendiculars are drawn to $AB$ and $AC$, with feet $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.
2013 ITAMO, 2
In triangle $ABC$, suppose we have $a> b$, where $a=BC$ and $b=AC$. Let $M$ be the midpoint of $AB$, and $\alpha, \beta$ are inscircles of the triangles $ACM$ and $BCM$ respectively. Let then $A'$ and $B'$ be the points of tangency of $\alpha$ and $\beta$ on $CM$. Prove that $A'B'=\frac{a - b}{2}$.
2016 Turkey Team Selection Test, 6
In a triangle $ABC$ with $AB=AC$, let $D$ be the midpoint of $[BC]$. A line passing through $D$ intersects $AB$ at $K$, $AC$ at $L$. A point $E$ on $[BC]$ different from $D$, and a point $P$ on $AE$ is taken such that $\angle KPL=90^\circ-\frac{1}{2}\angle KAL$ and $E$ lies between $A$ and $P$. The circumcircle of triangle $PDE$ intersects $PK$ at point $X$, $PL$ at point $Y$ for the second time. Lines $DX$ and $AB$ intersect at $M$, and lines $DY$ and $AC$ intersect at $N$. Prove that the points $P,M,A,N$ are concyclic.
2024 Harvard-MIT Mathematics Tournament, 12
Compute the number of quadruples $(a,b,c,d)$ of positive integers satisfying $$12a+21b+28c+84d=2024.$$
2010 ELMO Shortlist, 7
Find the smallest real number $M$ with the following property: Given nine nonnegative real numbers with sum $1$, it is possible to arrange them in the cells of a $3 \times 3$ square so that the product of each row or column is at most $M$.
[i]Evan O' Dorney.[/i]
Kyiv City MO Juniors Round2 2010+ geometry, 2022.8.4
Points $D, E, F$ are selected on sides $BC, CA, AB$ correspondingly of triangle $ABC$ with $\angle C = 90^\circ$ such that $\angle DAB = \angle CBE$ and $\angle BEC = \angle AEF$. Show that $DB = DF$.
[i](Proposed by Mykhailo Shtandenko)[/i]
1997 Rioplatense Mathematical Olympiad, Level 3, 6
Let $N$ be the set of positive integers.
Determine if there is a function $f: N\to N$ such that $f(f(n))=2n$, for all $n$ belongs to $N$.
2001 Stanford Mathematics Tournament, 1
$ABCD$ is a square with sides of unit length. Points $E$ and $F$ are taken on sides $AB$ and $AD$ respectively so that $AE = AF$ and the quadrilateral $CDFE$ has maximum area. What is this maximum area?
2021 Macedonian Team Selection Test, Problem 6
Let $ABC$ be an acute triangle such that $AB<AC$ with orthocenter $H$. The altitudes $BH$ and $CH$ intersect $AC$ and $AB$ at $B_{1}$ and $C_{1}$. Denote by $M$ the midpoint of $BC$. Let $l$ be the line parallel to $BC$ passing through $A$. The circle around $ CMC_{1}$ meets the line $l$ at points $X$ and $Y$, such that $X$ is on the same side of the line $AH$ as $B$ and $Y$ is on the same side of $AH$ as $C$. The lines $MX$ and $MY$ intersect $CC_{1}$ at $U$ and $V$ respectively. Show that the circumcircles of $ MUV$ and $ B_{1}C_{1}H$ are tangent.
[i] Authored by Nikola Velov[/i]
1963 IMO, 1
Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.
2006 Pre-Preparation Course Examination, 2
Show that there exists a continuos function $f: [0,1]\rightarrow [0,1]$ such that it has no periodic orbit of order $3$ but it has a periodic orbit of order $5$.
2014 Iran Team Selection Test, 2
find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.
2014 AIME Problems, 5
Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$.
2018 Iran MO (1st Round), 10
Consider a triangle $ABC$ in which $AB=AC=15$ and $BC=18$. Points $D$ and $E$ are chosen on $CA$ and $CB$, respectively, such that $CD=5$ and $CE=3$. The point $F$ is chosen on the half-line $\overrightarrow{DE}$ so that $EF=8$. If $M$ is the midpoint of $AB$ and $N$ is the intersection of $FM$ and $BC$, what is the length of $CN$?
1977 AMC 12/AHSME, 8
For every triple $(a,b,c)$ of non-zero real numbers, form the number \[ \frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}. \] The set of all numbers formed is
$\textbf{(A)}\ {0} \qquad
\textbf{(B)}\ \{-4,0,4\} \qquad
\textbf{(C)}\ \{-4,-2,0,2,4\} \qquad
\textbf{(D)}\ \{-4,-2,2,4\} \qquad
\textbf{(E)}\ \text{none of these}$
2007 Serbia National Math Olympiad, 2
In a scalene triangle $ABC , AD, BE , CF$ are the angle bisectors $(D \in BC , E \in AC , F \in AB)$. Points $K_{a}, K_{b}, K_{c}$ on the incircle of triangle $ABC$ are such that $DK_{a}, EK_{b}, FK_{c}$ are tangent to the incircle and $K_{a}\not\in BC , K_{b}\not\in AC , K_{c}\not\in AB$. Let $A_{1}, B_{1}, C_{1}$ be the midpoints of sides $BC , CA, AB$ , respectively. Prove that the lines $A_{1}K_{a}, B_{1}K_{b}, C_{1}K_{c}$ intersect on the incircle of triangle $ABC$.
2011 Iran Team Selection Test, 2
Find all natural numbers $n$ greater than $2$ such that there exist $n$ natural numbers $a_{1},a_{2},\ldots,a_{n}$ such that they are not all equal, and the sequence $a_{1}a_{2},a_{2}a_{3},\ldots,a_{n}a_{1}$ forms an arithmetic progression with nonzero common difference.
1999 USAMTS Problems, 1
The digits of the three-digit integers $a, b,$ and $c$ are the nine nonzero digits $1,2,3,\cdots 9$ each of them appearing exactly once. Given that the ratio $a:b:c$ is $1:3:5$, determine $a, b,$ and $c$.
1989 Romania Team Selection Test, 3
Find all pair $(m,n)$ of integer ($m >1,n \geq 3$) with the following property:If an $n$-gon can be partitioned into $m$ isoceles triangles,then the $n$-gon has two congruent sides.