Found problems: 85335
1999 Spain Mathematical Olympiad, 1
The lines $t$ and $ t'$, tangent to the parabola $y = x^2$ at points $A$ and $B$ respectively, intersect at point $C$. The median of triangle $ABC$ from $C$ has length $m$. Find the area of $\triangle ABC$ in terms of $m$.
2000 Mexico National Olympiad, 1
Circles $A,B,C,D$ are given on the plane such that circles $A$ and $B$ are externally tangent at $P, B$ and $C$ at $Q, C$ and $D$ at $R$, and $D$ and $A$ at $S$. Circles $A$ and $C$ do not meet, and so do not $B$ and $D$.
(a) Prove that the points $P,Q,R,S$ lie on a circle.
(b) Suppose that $A$ and $C$ have radius $2, B$ and $D$ have radius $3$, and the distance between the centers of $A$ and $C$ is $6$. Compute the area of the quadrilateral $PQRS$.
2017 Taiwan TST Round 2, 2
Let $ABC$ be a triangle such that $BC>AB$, $L$ be the internal angle bisector of $\angle ABC$. Let $P,Q$ be the feet from $A,C$ to $L$, respectively. Suppose $M,N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively. Let $O$ be the circumcenter of triangle $PQM$, and the circumcircle intersects $AC$ at point $H$. Prove that $O,M,N,H$ are concyclic.
2011 Argentina Team Selection Test, 6
Each square of $1\times 1$, of a $n\times n$ grid is colored using red or blue, in such way that between all the $2\times 2$ subgrids, there are all the possible colorations of a $2\times 2$ grid using red or blue, (colorations that can be obtained by using rotation or symmetry, are said to be different, so there are 16 possibilities). Find:
a) The minimum value of $n$.
b) For that value, find the least possible number of red squares.
PEN H Problems, 16
Find all pairs $(a,b)$ of different positive integers that satisfy the equation $W(a)=W(b)$, where $W(x)=x^{4}-3x^{3}+5x^{2}-9x$.
2015 Romania National Olympiad, 3
Let be two nonnegative real numbers $ a,b $ with $ b>a, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ of real numbers such that the sequence $ \left( \frac{x_1+x_2+\cdots +x_n}{n^a} \right)_{n\ge 1} $ is bounded.
Show that the sequence $ \left( x_1+\frac{x_2}{2^b} +\frac{x_3}{3^b} +\cdots +\frac{x_n}{n^b} \right)_{n\ge 1} $ is convergent.
2005 Kyiv Mathematical Festival, 1
Prove that there exists a positive integer $ n$ such that for every $ x\ge0$ the inequality $ (x\minus{}1)(x^{2005}\minus{}2005x^{n\plus{}1}\plus{}2005x^n\minus{}1)\ge0$ holds.
1987 IMO Longlists, 70
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.[i](IMO Problem 2)[/i]
[i]Proposed by Soviet Union.[/i]
1979 Miklós Schweitzer, 4
For what values of $ n$ does the group $ \textsl{SO}(n)$ of all orthogonal transformations of determinant $ 1$ of the $ n$-dimensional Euclidean space possess a closed regular subgroup?($ \textsl{G}<\textsl{SO}(n)$ is called $ \textit{regular}$ if for any elements $ x,y$ of the unit sphere there exists a unique $ \varphi \in \textsl{G}$ such that $ \varphi(x)\equal{}y$.)
[i]Z. Szabo[/i]
1986 IMO Longlists, 23
Let $I$ and $J$ be the centers of the incircle and the excircle in the angle $BAC$ of the triangle $ABC$. For any point $M$ in the plane of the triangle, not on the line $BC$, denote by $I_M$ and $J_M$ the centers of the incircle and the excircle (touching $BC$) of the triangle $BCM$. Find the locus of points $M$ for which $II_MJJ_M$ is a rectangle.
1969 AMC 12/AHSME, 35
Let $L(m)$ be the $x$-coordinate of the left end point of the intersection of the graphs of $y=x^2-6$ and $y=m$, where $-6<m<6$. Let $r=[L(-m)-L(m)]/m$. Then, as $m$ is made arbitrarily close to zero, the value of $r$ is:
$\textbf{(A) }\text{arbitrarily close to zero}\qquad
\textbf{(B) }\text{arbitrarily close to }\tfrac1{\sqrt6}\qquad$
$\textbf{(C) }\text{arbitrarily close to }\tfrac2{\sqrt6}\qquad\,\,\,
\textbf{(D) }\text{arbitrarily large}\qquad$
$\textbf{(E) }\text{undetermined}$
2018 Korea Junior Math Olympiad, 6
Let there be a figure with $9$ disks and $11$ edges, as shown below.
We will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write $0$ in disk $A$, and $1$ in disk $I$. Find the minimum sum of all real numbers written in $11$ edges.
LMT Team Rounds 2021+, 3
Farmer Boso has a busy farm with lots of animals. He tends to $5b$ cows, $5a +7$ chickens, and $b^{a-5}$ insects. Note that each insect has $6$ legs. The number of cows is equal to the number of insects. The total number of legs present amongst his animals can be expressed as $\overline{LLL }+1$, where $L$ stands for a digit. Find $L$.
1961 AMC 12/AHSME, 5
Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$. Then $S$ equals:
${{ \textbf{(A)}\ (x-2)^4 \qquad\textbf{(B)}\ (x-1)^4 \qquad\textbf{(C)}\ x^4 \qquad\textbf{(D)}\ (x+1)^4 }\qquad\textbf{(E)}\ x^4+1} $
2000 CentroAmerican, 2
Determine all positive integers $ n$ such that it is possible to tile a $ 15 \times n$ board with pieces shaped like this:
[asy]size(100); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(1,2)); draw((2,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); draw((5,0)--(6,0)); draw((4,1)--(7,1)); draw((4,2)--(7,2)); draw((5,3)--(6,3)); draw((4,1)--(4,2)); draw((5,0)--(5,3)); draw((6,0)--(6,3)); draw((7,1)--(7,2));[/asy]
1964 Spain Mathematical Olympiad, 2
The RTP tax is a function $f(x)$, where $x$ is the total of the annual profits (in pesetas). Knowing that:
a) $f(x)$ is a continuous function
b) The derivative $\frac{df(x)}{dx}$ on the interval $0 \leq 6000$ is constant and equals zero; in the interval $6000< x < P$ is constant and equals $1$; and when $x>P$ is constant and equal 0.14.
c) $f(0)=0$ and $f(140000)=14000$.
Determine the value of the amount $P$ (in pesetas) and represent graphically the function $y=f(x)$.
2023 Taiwan TST Round 2, A
Find all functions $f : \mathbb{R} \to \mathbb{R}$, such that
$$f\left(xy+f(y)\right)f(x)=x^2f(y)+f(xy)$$
for all $x,y \in \mathbb{R}$
[i]Proposed by chengbilly[/i]
1997 Belarusian National Olympiad, 1
We call the sum of any $k$ of $n$ given numbers (with distinct indices) a $k$-sum. Given $n$, find all $k$ such that, whenever more than half of $k$-sums of numbers $a_{1},a_{2},...,a_{n}$ are positive, the sum $a_{1}+a_{2}+...+a_{n}$ is positive as well.
1968 All Soviet Union Mathematical Olympiad, 104
Three spheres are constructed so that the edges $[AB], [BC], [AD]$ of the tetrahedron $ABCD$ are their respective diameters. Prove that the spheres cover all the tetrahedron.
2019 ASDAN Math Tournament, 4
Suppose $Z, Y$ , and $W$ are points on a circle such that lengths $ZY = Y W$. Extend $ZY$ and let $X$ be a point on $ZY$ where $ZY = Y X$. If $XW$ is a tangent of the circle, what is $\angle W XY$ ?
2011 Belarus Team Selection Test, 1
Let $A$ be the sum of all $10$ distinct products of the sides of a convex pentagon, $S$ be the area of the pentagon.
a) Prove thas $S \le \frac15 A$.
b) Does there exist a constant $c<\frac15$ such that $S \le cA$ ?
I.Voronovich
2013 ELMO Shortlist, 5
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that
\[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \]
and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$.
[i]Proposed by Victor Wang[/i]
2014 Greece Team Selection Test, 3
Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.
1996 APMO, 1
Let $ABCD$ be a quadrilateral $AB = BC = CD = DA$. Let $MN$ and $PQ$ be two segments perpendicular to the diagonal $BD$ and such that the distance between them is $d > \frac{BD}{2}$, with $M \in AD$, $N \in DC$, $P \in AB$, and $Q \in BC$. Show that the perimeter of hexagon $AMNCQP$ does not depend on the position of $MN$ and $PQ$ so long as the distance between them remains constant.
2009 Argentina Iberoamerican TST, 1
In the vertexes of a regular $ 31$-gon there are written the numbers from $ 1$ to $ 31$, ordered increasingly, clockwise oriented.
We are allowed to perform an operation which consists in taking any three vertexes, namely the ones who have written $ a$,$ b$, and $ c$ and change them into $ c$, $ a\minus{}\frac{1}{10}$ and $ b\plus{}\frac{1}{10}$ respectively ( $ a$ becomes $ c$, $ b$ becomes $ a\minus{}\frac{1}{10}$ and $ c$ turns into $ b\plus{}\frac{1}{10}$
Prove that after applying several operations we can reach the state in which the numbers in the vertexes are the numbers from $ 1$ to $ 31$, ordered increasingly,anti-clockwise oriented.