Found problems: 85335
2008 Germany Team Selection Test, 2
Tracey baked a square cake whose surface is dissected in a $ 10 \times 10$ grid. In some of the fields she wants to put a strawberry such that for each four fields that compose a rectangle whose edges run in parallel to the edges of the cake boundary there is at least one strawberry. What is the minimum number of required strawberries?
KoMaL A Problems 2018/2019, A. 745
A clock hand is attached to every face of a convex polyhedron. Each hand always points towards a neighboring face and every minute, exactly one of the hands turns clockwise to point at the next face. Suppose that the hands on neighboring faces never point towards one another. Show that one of the hands makes only finitely many turns.
2004 Purple Comet Problems, 7
A rectangle has area $1100$. If the length is increased by ten percent and the width is
decreased by ten percent, what is the area of the new rectangle?
2007 Oral Moscow Geometry Olympiad, 6
A point $P$ is fixed inside the circle. $C$ is an arbitrary point of the circle, $AB$ is a chord passing through point $B$ and perpendicular to the segment $BC$. Points $X$ and $Y$ are projections of point $B$ onto lines $AC$ and $BC$. Prove that all line segments $XY$ are tangent to the same circle.
(A. Zaslavsky)
1993 Romania Team Selection Test, 3
Suppose that each of the diagonals $AD,BE,CF$ divides the hexagon $ABCDEF$ into two parts of the same area and perimeter. Does the hexagon necessarily have a center of symmetry?
1986 AMC 12/AHSME, 8
The population of the United States in 1980 was $226,504,825$. The area of the country is $3,615,122$ square miles. The are $(5280)^{2}$ square feet in one square mile. Which number below best approximates the average number of square feet per person?
$ \textbf{(A)}\ 5,000\qquad\textbf{(B)}\ 10,000\qquad\textbf{(C)}\ 50,000\qquad\textbf{(D)}\ 100,000\qquad\textbf{(E)}\ 500,000 $
2015 Taiwan TST Round 2, 1
Let $f(x)=\sum_{i=0}^{n}a_ix^i$ and $g(x)=\sum_{i=0}^{n}b_ix^i$, where $a_n$,$b_n$ can be zero.
Called $f(x)\ge g(x)$ if exist $r$ such that $\forall i>r,a_i=b_i,a_r>b_r$ or $f(x)=g(x)$.
Prove that: if the leading coefficients of $f$ and $g$ are positive, then $f(f(x))+g(g(x))\ge f(g(x))+g(f(x))$
2023 VN Math Olympiad For High School Students, Problem 11
Given a triangle $ABC$ inscribed in $(O)$ with $2$ symmedians $AD, CF(D,F$ are on the sides $BC, AB,$ respectively$).$ The ray $DF$ intersects $(O)$ at $P.$ The line passing through $P$ and perpendicular to $OA$ intersects $AB,AC$ at $Q,R,$ respectively$.$
Compute the ratio $\dfrac{PR}{PQ}.$
2007 Baltic Way, 12
Let $M$ be a point on the arc $AB$ of the circumcircle of the triangle $ABC$ which does not contain $C$. Suppose that the projections of $M$ onto the lines $AB$ and $BC$ lie on the sides themselves, not on their extensions. Denote these projections by $X$ and $Y$, respectively. Let $K$ and $N$ be the midpoints of $AC$ and $XY$, respectively. Prove that $\angle MNK=90^{\circ}$ .
2023 Tuymaada Olympiad, 7
$3n$ people forming $n$ families of a mother, a father and a child, stand in a circle. Every two neighbours can exchange places except the case when a parent exchanges places with his/her child (this is forbidden). For what $n$ is it possible to obtain every arrangement of those people by such
exchanges? The arrangements differing by a circular shift are considered distinct.
2014 IMAC Arhimede, 6
If $a, b, c, d$ are positive numbers, prove that
$$\sum_{cyclic}\frac{a-\sqrt[3]{bcd}}{a+3(b+c+d)}\ge 0$$
2022 OMpD, 4
Let $ABCD$ be a cyclic quadrilateral and $M,N$ be the midpoints of $AB$, $CD$ respectively. The diagonals $AC$ and $BD$ intersect at $L$. Suppose that the circumcircle of $LMN$, with center $T$, intersects the circumcircle of $ABCD$ at two distinct points $X,Y$. If the line $MN$ intersects the line $XY$ at $S$ and the line $XM$ intersects the line $YN$ at $P$, prove that $PL$ is perpendicular to $ST$.
1996 India National Olympiad, 3
Solve the following system for real $a , b, c, d, e$: \[ \left\{ \begin{array}{ccc} 3a & = & ( b + c+ d)^3 \\ 3b & = & ( c + d +e ) ^3 \\ 3c & = & ( d + e +a )^3 \\ 3d & = & ( e + a +b )^3 \\ 3e &=& ( a + b +c)^3. \end{array}\right. \]
Estonia Open Senior - geometry, 1996.1.4
A unit square has a circle of radius $r$ with center at it's midpoint. The four quarter circles are centered on the vertices of the square and are tangent to the central circle (see figure). Find the maximum and minimum possible value of the area of the striped figure in the figure and the corresponding values of $r$ such these, the maximum and minimum are achieved.
[img]https://2.bp.blogspot.com/-DOT4_B5Mx-8/XnmsTlWYfyI/AAAAAAAALgs/TVYkrhqHYGAeG8eFuqFxGDCTnogVbQFUwCK4BGAYYCw/s400/96%2Bestonia%2Bopen%2Bs1.4.png[/img]
2005 Purple Comet Problems, 5
A palindrome is a number that reads the same forwards and backwards such as $3773$ or $42924$. Find the sum of the twelve smallest five digit palindromes.
2021 Junior Macedonian Mathematical Olympiad, Problem 3
Find all positive integers $n$ and prime numbers $p$ such that $$17^n \cdot 2^{n^2} - p =(2^{n^2+3}+2^{n^2}-1) \cdot n^2.$$
[i]Authored by Nikola Velov[/i]
1985 Polish MO Finals, 5
$p(x,y)$ is a polynomial such that $p(cos t, sin t) = 0$ for all real $t$.
Show that there is a polynomial $q(x,y)$ such that $p(x,y) = (x^2 + y^2 - 1) q(x,y)$.
1978 Dutch Mathematical Olympiad, 1
Prove that no integer $x$ and $y$ satisfy: $$3x^2 = 9 + y^3.$$
1960 Miklós Schweitzer, 7
[b]7.[/b] Define the generalized derivative at $x_0$ of the function $f(x)$ by
$\lim_{h \to 0} 2 \frac{ \frac{1}{h} \int_{x_0}^{x_0+h} f(t) dt - f(x_0)}{h}$
Show that there exists a function, continuous everywhere, which is nowhere differentiable in this general sense [b]( R. 8)[/b]
2017 Brazil Team Selection Test, 3
Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\] Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.
2019 Dürer Math Competition (First Round), P5
Let $ABC$ and $A'B'C'$ be similar triangles with different orientation such that their orthocenters coincide. Show that lines $AA′, BB′, CC′ are concurrent or parallel.
2017 AMC 10, 25
Last year Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test?
$\textbf{(A)} \text{ 92} \qquad \textbf{(B)} \text{ 94} \qquad \textbf{(C)} \text{ 96} \qquad \textbf{(D)} \text{ 98} \qquad \textbf{(E)} \text{ 100}$
2010 Contests, 1
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$
[i]Proposed by Pierre Bornsztein, France[/i]
1987 Balkan MO, 3
In the triangle $ABC$ the following equality holds:
\[\sin^{23}{\frac{A}{2}}\cos^{48}{\frac{B}{2}}=\sin^{23}{\frac{B}{2}}\cos^{48}{\frac{A}{2}}\]
Determine the value of $\frac{AC}{BC}$.
2018 PUMaC Team Round, 13
Consider a 10-dimensional \(10 \times 10 \times \cdots \times 10 \) cube consisting of \(10^{10}\) unit cubes, such that one cube \(A\) is centered at the origin, and one cube \(B\) is centered at \((9, 9, 9, 9, 9, 9, 9, 9, 9, 9)\). Paint \(A\) red and remove \(B\), leaving an empty space. Let a move consist of taking a cube adjacent to the empty space and placing it into the empty space, leaving the space originally contained by the cube empty. What is the minimum number of moves required to result in a configuration where the cube centered at \((9, 9, 9, 9, 9, 9, 9, 9, 9, 9)\) is red?