Found problems: 85335
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.
2010 VJIMC, Problem 2
Prove or disprove that if a real sequence $(a_n)$ satisfies $a_{n+1}-a_n\to0$ and $a_{2n}-2a_n\to0$ as $n\to\infty$, then $a_n\to0$.
1977 Putnam, B5
Suppose that $a_1,a_2,\dots a_n$ are real $(n>1)$ and $$A+ \sum_{i=1}^{n} a^2_i< \frac{1}{n-1} (\sum_{i=1}^{n} a_i)^2.$$ Prove that $A<2a_ia_j$ for $1\leq i<j\leq n.$
2017 Azerbaijan BMO TST, 1
Let $a, b,c$ be positive real numbers.
Prove that $ \sqrt{a^3b+a^3c}+\sqrt{b^3c+b^3a}+\sqrt{c^3a+c^3b}\ge \frac43 (ab+bc+ca)$
2015 Brazil National Olympiad, 6
Let $\triangle ABC$ be a scalene triangle and $X$, $Y$ and $Z$ be points on the lines $BC$, $AC$ and $AB$, respectively, such that $\measuredangle AXB = \measuredangle BYC = \measuredangle CZA$. The circumcircles of $BXZ$ and $CXY$ intersect at $P$. Prove that $P$ is on the circumference which diameter has ends in the ortocenter $H$ and in the baricenter $G$ of $\triangle ABC$.
2016 ISI Entrance Examination, 8
Suppose that $(a_n)_{n\geq 1}$ is a sequence of real numbers satisfying $a_{n+1} = \frac{3a_n}{2+a_n}$.
(i) Suppose $0 < a_1 <1$, then prove that the sequence $a_n$ is increasing and hence show that $\lim_{n \to \infty} a_n =1$.
(ii) Suppose $ a_1 >1$, then prove that the sequence $a_n$ is decreasing and hence show that $\lim_{n \to \infty} a_n =1$.
2007 Singapore Junior Math Olympiad, 4
The difference between the product and the sum of two different integers is equal to the sum of their GCD (greatest common divisor) and LCM (least common multiple). Findall these pairs of numbers. Justify your answer.
2015 CIIM, Problem 5
There are $n$ people seated on a circular table that have seats numerated from 1 to $n$ clockwise. Let $k$ be a fix integer with $2 \leq k \leq n$. The people can change their seats. There are two types of moves permitted:
1. Each person moves to the next seat clockwise.
2. Only the ones in seats 1 and $k$ exchange their seats.
Determine, in function of $n$ and $k$, the number of possible configurations of people in the table that can be attain by using a sequence of permitted moves.
2005 Regional Competition For Advanced Students, 3
For which values of $ k$ and $ d$ has the system $ x^3\plus{}y^3\equal{}2$ and $ y\equal{}kx\plus{}d$ no real solutions $ (x,y)$?
2000 Junior Balkan Team Selection Tests - Moldova, 1
Show that the expression $(a + b + 1) (a + b - 1) (a - b + 1) (- a + b + 1)$, where $a =\sqrt{1 + x^2}$, $b =\sqrt{1 + y^2}$ and $x + y = 1$ is constant ¸and be calculated that constant value.
2016 Japan MO Preliminary, 10
Boy A and $2016$ flags are on a circumference whose length is $1$ of a circle. He wants to get all flags by moving on the circumference. He can get all flags by moving distance $l$ regardless of the positions of boy A and flags. Find the possible minimum value as $l$ like this.
Note that boy A doesn’t have to return to the starting point to leave gotten flags.