Found problems: 85335
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?
1997 China Team Selection Test, 3
Prove that there exists $m \in \mathbb{N}$ such that there exists an integral sequence $\lbrace a_n \rbrace$ which satisfies:
[b]I.[/b] $a_0 = 1, a_1 = 337$;
[b]II.[/b] $(a_{n + 1} a_{n - 1} - a_n^2) + \frac{3}{4}(a_{n + 1} + a_{n - 1} - 2a_n) = m, \forall$ $n \geq 1$;
[b]III. [/b]$\frac{1}{6}(a_n + 1)(2a_n + 1)$ is a perfect square $\forall$ $n \geq 1$.
2004 China Team Selection Test, 3
$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying:
(1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$.
(2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$.
Find the largest possible value of $ |S|$.
2002 Miklós Schweitzer, 9
Let $M$ be a connected, compact $C^{\infty}$-differentiable manifold, and denote the vector space of smooth real functions on $M$ by $C^{\infty}(M)$. Let the subspace $V\le C^{\infty}(M)$ be invariant under $C^{\infty}$-diffeomorphisms of $M$, that is, let $f\circ h\in V$ for every $f\in V$ and for every $C^{\infty}$-diffeomorphism $h\colon M\rightarrow M$. Prove that if $V$ is different from the subspaces $\{ 0\}$ and $C^{\infty}(M)$ then $V$ only contains the constant functions.
2015 Abels Math Contest (Norwegian MO) Final, 3
The five sides of a regular pentagon are extended to lines $\ell_1, \ell_2, \ell_3, \ell_4$, and $\ell_5$.
Denote by $d_i$ the distance from a point $P$ to $\ell_i$.
For which point(s) in the interior of the pentagon is the product $d_1d_2d_3d_4d_5$ maximal?
2010 Indonesia TST, 4
Let $ ABC$ be a non-obtuse triangle with $ CH$ and $ CM$ are the altitude and median, respectively. The angle bisector of $ \angle BAC$ intersects $ CH$ and $ CM$ at $ P$ and $ Q$, respectively. Assume that \[ \angle ABP\equal{}\angle PBQ\equal{}\angle QBC,\]
(a) prove that $ ABC$ is a right-angled triangle, and
(b) calculate $ \dfrac{BP}{CH}$.
[i]Soewono, Bandung[/i]
1995 Turkey MO (2nd round), 6
Find all surjective functions $f: \mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in \mathbb{N}$ \[f(m)\mid f(n) \mbox{ if and only if }m\mid n.\]
2020 Candian MO, 4#
$S= \{1,4,8,9,16,...\} $is the set of perfect integer power. ( $S=\{ n^k| n, k \in Z, k \ge 2 \}$. )We arrange the elements in $S$ into an increasing sequence $\{a_i\}$ . Show that there are infinite many $n$, such that $9999|a_{n+1}-a_n$
2017 Junior Regional Olympiad - FBH, 3
Find all real numbers $x$ such that: $$ \sqrt{\frac{x-7}{2015}}+\sqrt{\frac{x-6}{2016}}+\sqrt{\frac{x-5}{2017}}=\sqrt{\frac{x-2015}{7}}+\sqrt{\frac{x-2016}{6}}+\sqrt{\frac{x-2017}{5}}$$
2017 CMIMC Team, 3
Suppose Pat and Rick are playing a game in which they take turns writing numbers from $\{1, 2, \dots, 97\}$ on a blackboard. In each round, Pat writes a number, then Rick writes a number; Rick wins if the sum of all the numbers written on the blackboard after $n$ rounds is divisible by 100. Find the minimum positive value of $n$ for which Rick has a winning strategy.
2016 Argentina National Olympiad Level 2, 4
There is a board with $n$ rows and $12$ columns. Each cell of the board contains a $1$ or a $0$. The board has the following properties:
[list=i]
[*]All rows are distinct.
[*]Each row contains exactly $4$ cells with $1$.
[*]For every $3$ rows, there is a column that intersects them in $3$ cells with $0$.
[/list]
Find the largest $n$ for which a board with these properties exists.
2009 Princeton University Math Competition, 5
Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$, $b$ and $c$.
2021 Romania National Olympiad, 4
Determine the smallest non-negative integer $n$ such that
\[\sqrt{(6n+11)(6n+14)(20n+19)}\in\mathbb Q.\]
[i]Mihai Bunget[/i]
2007 Tournament Of Towns, 6
In the quadrilateral $ABCD$, $AB = BC = CD$ and $\angle BMC = 90^\circ$, where $M$ is the midpoint of $AD$. Determine the acute angle between the lines $AC$ and $BD$.