Found problems: 85335
1979 Vietnam National Olympiad, 4
For each integer $n > 0$ show that there is a polynomial $p(x)$ such that $p(2 cos x) = 2 cos nx$.
1998 Denmark MO - Mohr Contest, 5
A neat fruit arrangement on a large round dish is edged with strawberries. Between $100$ and $200$ berries are used for this border. A deliciously hungry child eats first one of the strawberries and then starts going round and round the dish, she eats strawberries in the following way: When she has eaten a berry, she leaves it next lie, then she eats the next, leaves the next, etc. Thus she continues until there is only one strawberry left. This berry is the one that was lying right after the very first thing she ate. How many berries were there originally?
2018 Miklós Schweitzer, 7
Describe all functions $f: \{ 0,1\}^n \to \{ 0,1\}$ which satisfy the equation
\begin{align*}
& f(f(a_{11},a_{12},\dotsc ,a_{1n}),f(a_{21},a_{22},\dotsc ,a_{2n}),\dotsc ,f(a_{n1},a_{n2},\dotsc ,a_{nn}))\\
& = f(f(a_{11},a_{21},\dotsc ,a_{n1}),f(a_{12},a_{22},\dotsc ,a_{n2}),\dotsc ,f(a_{1n},a_{2n},\dotsc ,a_{nn}))\end{align*}
for arbitrary $a_{ij}\in \{ 0,1\}$ where $i,j\in \{1,2,\dotsc ,n\}.$
2019 CMIMC, 14
Consider the following function.
$\textbf{procedure }\textsc{M}(x)$
$\qquad\textbf{if }0\leq x\leq 1$
$\qquad\qquad\textbf{return }x$
$\qquad\textbf{return }\textsc{M}(x^2\bmod 2^{32})$
Let $f:\mathbb N\to\mathbb N$ be defined such that $f(x) = 0$ if $\textsc{M}(x)$ does not terminate, and otherwise $f(x)$ equals the number of calls made to $\textsc{M}$ during the running of $\textsc{M}(x)$, not including the initial call. For example, $f(1) = 0$ and $f(2^{31}) = 1$. Compute the number of ones in the binary expansion of
\[
f(0) + f(1) + f(2) + \cdots + f(2^{32} - 1).
\]
2014 Sharygin Geometry Olympiad, 5
A triangle with angles of $30, 70$ and $80$ degrees is given. Cut it by a straight line into two triangles in such a way that an angle bisector in one of these triangles and a median in the other one drawn from two endpoints of the cutting segment are parallel to each other. (It suffices to find one such cutting.)
(A. Shapovalov )
2023 China Team Selection Test, P19
Let $A,B$ be two fixed points on the unit circle $\omega$, satisfying $\sqrt{2} < AB < 2$. Let $P$ be a point that can move on the unit circle, and it can move to anywhere on the unit circle satisfying $\triangle ABP$ is acute and $AP>AB>BP$. Let $H$ be the orthocenter of $\triangle ABP$ and $S$ be a point on the minor arc $AP$ satisfying $SH=AH$. Let $T$ be a point on the minor arc $AB$ satisfying $TB || AP$. Let $ST\cap BP = Q$.
Show that (recall $P$ varies) the circle with diameter $HQ$ passes through a fixed point.
2021 Azerbaijan IMO TST, 3
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
2015 IMO Shortlist, G8
A [i]triangulation[/i] of a convex polygon $\Pi$ is a partitioning of $\Pi$ into triangles by diagonals having no common points other than the vertices of the polygon. We say that a triangulation is a [i]Thaiangulation[/i] if all triangles in it have the same area.
Prove that any two different Thaiangulations of a convex polygon $\Pi$ differ by exactly two triangles. (In other words, prove that it is possible to replace one pair of triangles in the first Thaiangulation with a different pair of triangles so as to obtain the second Thaiangulation.)
[i]Proposed by Bulgaria[/i]
2000 Baltic Way, 3
Given a triangle $ ABC$ with $ \angle A \equal{} 90^{\circ}$ and $ AB \neq AC$. The points $ D$, $ E$, $ F$ lie on the sides $ BC$, $ CA$, $ AB$, respectively, in such a way that $ AFDE$ is a square. Prove that the line $ BC$, the line $ FE$ and the line tangent at the point $ A$ to the circumcircle of the triangle $ ABC$ intersect in one point.
2003 Nordic, 2
Find all triples of integers ${(x, y, z)}$ satisfying ${x^3 + y^3 + z^3 - 3xyz = 2003}$
2021 All-Russian Olympiad, 1
On the side $BC$ of the parallelogram $ABCD$, points $E$ and $F$ are given ($E$ lies between $B$ and $F$) and the diagonals $AC, BD$ meet at $O$. If it's known that $AE, DF$ are tangent to the circumcircle of $\triangle AOD$, prove that they're tangent to the circumcircle of $\triangle EOF$ as well.
2016 ASDAN Math Tournament, 1
Pooh has an unlimited supply of $1\times1$, $2\times2$, $3\times3$, and $4\times4$ squares. What is the minimum number of squares he needs to use in order to fully cover a $5\times5$ with no $2$ squares overlapping?
2016 IMO Shortlist, A5
Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers.
(a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$.
(b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.
2021 AMC 12/AHSME Fall, 23
What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{ 1, 2, 3, …, 30\}$? (For example the set $\{1, 17, 18, 19, 30\}$ has $2$ pairs of consecutive integers.)
$\textbf{(A)}\ \frac{2}{3} \qquad\textbf{(B)}\ \frac{29}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\
\frac{29}{30} \qquad\textbf{(E)}\ 1$
1992 Cono Sur Olympiad, 1
Find a positive integrer number $n$ such that, if yor put a number $2$ on the left and a number $1$ on the right, the new number is equal to $33n$.
2018 Peru Cono Sur TST, 4
Consider the numbers
$$ S_1 = \frac{1}{1 \cdot 2} + \frac{1}{1 \cdot 3} + \frac{1}{1 \cdot 4} + \dots + \frac{1}{1 \cdot 2018}, $$
$$ S_2 = \frac{1}{2 \cdot 3} + \frac{1}{2 \cdot 4} + \frac{1}{2 \cdot 5} + \dots + \frac{1}{2 \cdot 2018}, $$
$$ S_3 = \frac{1}{3 \cdot 4} + \frac{1}{3 \cdot 5} + \frac{1}{3 \cdot 6} + \dots + \frac{1}{3 \cdot 2018}, $$
$$ \vdots $$
$$ S_{2017} = \frac{1}{2017 \cdot 2018}. $$
Prove that the number $ S_1 + S_2 + S_3 + \dots + S_{2017} $ is not an integer.
2004 Federal Math Competition of S&M, 3
Let $M, N, P$ be arbitrary points on the sides $BC, CA, AB$ respectively of an acute-angled triangle $ABC$. Prove that at least one of the following inequalities is satisfied:
$NP \geq \frac{1}{2}BC; PM \geq \frac{1}{2}CA; MN \geq \frac{1}{2}AB$
2012 All-Russian Olympiad, 4
Initially there are $n+1$ monomials on the blackboard: $1,x,x^2, \ldots, x^n $. Every minute each of $k$ boys simultaneously write on the blackboard the sum of some two polynomials that were written before. After $m$ minutes among others there are the polynomials $S_1=1+x,S_2=1+x+x^2,S_3=1+x+x^2+x^3,\ldots ,S_n=1+x+x^2+ \ldots +x^n$ on the blackboard. Prove that $ m\geq \frac{2n}{k+1} $.
2009 Moldova National Olympiad, 12.1
Calculate $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{cos(x)^7}{e^x+1} dx$.
1958 Polish MO Finals, 5
Prove the theorem:
In a tetrahedron, the plane bisector of any dihedral angle divides the opposite edge into segments proportional to the areas of the tetrahedron faces that form this dihedral angle.
1974 Chisinau City MO, 71
The sides of the triangle $ABC$ lie on the sides of the angle $MAN$. Construct a triangle $ABC$ if the point $O$ of the intersection of its medians is given.
2019 Germany Team Selection Test, 1
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
2023 UMD Math Competition Part I, #2
Peter Rabbit is hopping along the number line, always jumping in the positive $x$ direction. For his first jump, he starts at $0$ and jumps $1$ unit to get to the number $1.$ For his second jump, he jumps $4$ units to get to the number $5.$ He continues jumping by jumping $1$ unit whenever he is on a multiple of $3$ and by jumping $4$ units whenever he is on a number that is not a multiple of $3.$ What number does he land on at the end of his $100$th jump?
$$
\mathrm a. ~ 297\qquad \mathrm b.~298\qquad \mathrm c. ~299 \qquad \mathrm d. ~300 \qquad \mathrm e. ~301
$$
1996 Estonia Team Selection Test, 3
Each face of a cube is divided into $n^2$ equal squares. The vertices of the squares are called [i]nodes[/i], so each face has $(n+1)^2$ nodes.
$(a)$ If $n=2$,does there exist a closed polygonal line whose links are sids of the squares and which passes through each node exactly once?
$(b)$ Prove that, for each $n$, such a polygonal line divides the surface area of the cube into two equal parts
2010 ELMO Problems, 3
Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$.
[i]Amol Aggarwal.[/i]