Found problems: 85335
2012 Mathcenter Contest + Longlist, 1 sl8
For matrices $A=[a_{ij}]_{m \times m}$ and $B=[b_{ij}]_{m \times m}$ where $A,B \in \mathbb{Z} ^{m \times m}$ let $A \equiv B \pmod{n}$ only if $a_{ij} \equiv b_{ij} \pmod{n}$ for every $i,j \in \{ 1,2,...,m \}$, that's $A-B=nZ$ for some $Z \in \mathbb{Z}^{m \times m}$. (The symbol $A \in \mathbb{Z} ^{m \times m}$ means that every element in $A$ is an integer.)
Prove that for $A \in \mathbb{Z} ^{m \times m}$ there is $B \in \mathbb{Z} ^{m \times m}$ , where $AB \equiv I \pmod{n }$ only if $(\det (A),n)=1$ and find the value of $B$ in the form of $A$ where $I$ represents the dimensional identity matrix $m \times m$.
[i](PP-nine)[/i]
1998 Italy TST, 2
In a triangle $ABC$, points $H,M,L$ are the feet of the altitude from $C$, the median from $A$, and the angle bisector from $B$, respectively. Show that if triangle $HML$ is equilateral, then so is triangle $ABC$.
2018 CCA Math Bonanza, T10
The irrational number $\alpha>1$ satisfies $\alpha^2-3\alpha-1=0$. Given that there is a fraction $\frac{m}{n}$ such that $n<500$ and $\left|\alpha-\frac{m}{n}\right|<3\cdot10^{-6}$, find $m$.
[i]2018 CCA Math Bonanza Team Round #10[/i]
ICMC 7, 1
Let $F_n{}$ denote the $n{}$-th Fibonacci number. Prove that $3^{2023}$ divides \[3^2\cdot F_4+3^3\cdot F_6+3^4\cdot F_8+\dots+3^{2023}F_{4046}.\][i]Proposed by Dylan Toh[/i]
2010 Brazil Team Selection Test, 4
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
2011 Morocco National Olympiad, 2
Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle.
$(a)$ Prove that
\[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\]
$(b)$ When do we have equality?
2001 Korea Junior Math Olympiad, 4
Some $n \geq 3$ cities are connected with railways, so that you can travel from one city to every other, not necessarily directly. However, the railways are structured in such a way that there is only one way to get from one city to another, assuming you don't pass through the same city again. Let $A$ be the set of these cities and railways. Show that there exists a Subset of $A$, let's say $C$, such that
(1) $C$ has at least $[(n+1)/2]$ cities as its element.
(2) No two elements of $C$ are directly connected with railways.
2006 Pan African, 2
Let $a, b, c$ be three non-zero integers. It is known that the sums $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are integers. Find these sums.
2007 Kazakhstan National Olympiad, 3
Let $p$ be a prime such that $2^{p-1}\equiv 1 \pmod{p^2}$. Show that $(p-1)(p!+2^n)$ has at least three distinct prime divisors for each $n\in \mathbb{N}$ .
1986 IMO Longlists, 48
Let $P$ be a convex $1986$-gon in the plane. Let $A,D$ be interior points of two distinct sides of P and let $B,C$ be two distinct interior points of the line segment $AD$. Starting with an arbitrary point $Q_1$ on the boundary of $P$, define recursively a sequence of points $Q_n$ as follows:
given $Q_n$ extend the directed line segment $Q_nB$ to meet the boundary of $P$ in a point $R_n$ and then extend $R_nC$ to meet the boundary of $P$ again in a point, which is defined to be $Q_{n+1}$. Prove that for all $n$ large enough the points $Q_n$ are on one of the sides of $P$ containing $A$ or $D$.
2013 VTRMC, Problem 2
Let $ABC$ be a right-angled triangle with $\angle ABC=90^\circ$, and let $D$ be on $AB$ such that $AD=2DB$. What is the maximum possible value of $\angle ACD$?
2023 Sharygin Geometry Olympiad, 10.7
There are $43$ points in the space: $3$ yellow and $40$ red. Any four of them are not coplanar. May the number of triangles with red vertices hooked with the triangle with yellow vertices be equal to $2023$? Yellow triangle is hooked with the red one if the boundary of the red triangle meet the part of the plane bounded by the yellow triangle at the unique point. The triangles obtained by the transpositions of vertices are identical.
2001 Abels Math Contest (Norwegian MO), 3b
The diagonals $AC$ and $BD$ in the convex quadrilateral $ABCD$ intersect in $S$. Let $F_1$ and $F_2$ be the areas of $\vartriangle ABS$ and $\vartriangle CSD$. and let $F$ be the area of the quadrilateral $ABCD$. Show that $\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}$
2002 Taiwan National Olympiad, 2
A lattice point $X$ in the plane is said to be [i]visible[/i] from the origin $O$ if the line segment $OX$ does not contain any other lattice points. Show that for any positive integer $n$, there is square $ABCD$ of area $n^{2}$ such that none of the lattice points inside the square is visible from the origin.
2012 Greece Team Selection Test, 3
Let $a,b,c$ be positive real numbers satisfying $a+b+c=3$.Prove that $\sum_{sym} \frac{a^{2}}{(b+c)^{3}}\geq \frac{3}{8}$
2023 HMNT, 2
Suppose rectangle $FOLK$ and square $LORE$ are on the plane such that $RL = 12$ and $RK = 11$. Compute the product of all possible areas of triangle $RKL$.
2006 India IMO Training Camp, 3
Let $ABC$ be an equilateral triangle, and let $D,E$ and $F$ be points on $BC,BA$ and $AB$ respectively. Let $\angle BAD= \alpha, \angle CBE=\beta$ and $\angle ACF =\gamma$. Prove that if $\alpha+\beta+\gamma \geq 120^\circ$, then the union of the triangular regions $BAD,CBE,ACF$ covers the triangle $ABC$.
1993 India National Olympiad, 1
The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ intersect at $P$. Let $O$ be the circumcenter of triangle $APB$ and $H$ be the orthocenter of triangle $CPD$. Show that the points $H,P,O$ are collinear.
2013 Turkey Team Selection Test, 3
For all real numbers $x,y,z$ such that $-2\leq x,y,z \leq 2$ and $x^2+y^2+z^2+xyz = 4$, determine the least real number $K$ satisfying \[\dfrac{z(xz+yz+y)}{xy+y^2+z^2+1} \leq K.\]
1986 Traian Lălescu, 2.4
Show that there is an unique group $ G $ (up to isomorphism) of order $ 1986 $ which has the property that there is at most one subgroup of it having order $ n, $ for every natural number $ n. $
2008 IMC, 3
Let $p$ be a polynomial with integer coefficients and let $a_1<a_2<\cdots <a_k$ be integers. Given that $p(a_i)\ne 0\forall\; i=1,2,\cdots, k$.
[list]
(a) Prove $\exists\; a\in \mathbb{Z}$ such that
\[ p(a_i)\mid p(a)\;\;\forall i=1,2,\dots ,k \]
(b) Does there exist $a\in \mathbb{Z}$ such that
\[ \prod_{i=1}^{k}p(a_i)\mid p(a) \][/list]
2006 ISI B.Stat Entrance Exam, 8
Show that there exists a positive real number $x\neq 2$ such that $\log_2x=\frac{x}{2}$. Hence obtain the set of real numbers $c$ such that
\[\frac{\log_2x}{x}=c\]
has only one real solution.
2003 Iran MO (3rd Round), 25
Let $ A,B,C,Q$ be fixed points on plane. $ M,N,P$ are intersection points of $ AQ,BQ,CQ$ with $ BC,CA,AB$. $ D',E',F'$ are tangency points of incircle of $ ABC$ with $ BC,CA,AB$. Tangents drawn from $ M,N,P$ (not triangle sides) to incircle of $ ABC$ make triangle $ DEF$. Prove that $ DD',EE',FF'$ intersect at $ Q$.
2023 Saint Petersburg Mathematical Olympiad, 7
Let $\ell_1, \ell_2$ be two non-parallel lines and $d_1, d_2$ be positive reals. The set of points $X$, such that $dist(X, \ell_i)$ is a multiple of $d_i$ is called a $\textit{grid}$. Let $A$ be finite set of points, not all collinear. A triangle with vertices in $A$ is called $\textit{empty}$ if no points from $A$ lie inside or on the sides of the triangle. Given that all empty triangles have the same area, show that $A$ is the intersection of a grid $L$ and a convex polygon $F$.
2020 Jozsef Wildt International Math Competition, W34
Let $a,b,c>0.$ Prove that$$\frac{a^3+b^2c+bc^2}{bc}+\frac{b^3+c^2a+ca^2}{ca}+\frac{c^3+a^2b+ab^2}{ab}\geq 3(a+b+c)$$
$$\frac{bc}{a^3+b^2c+bc^2}+\frac{ca}{b^3+c^2a+ca^2}+\frac{ab}{c^3+a^2b+ab^2}\leq \frac{1}{3}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})$$