Found problems: 85335
2007 Iran MO (2nd Round), 1
Prove that for every positive integer $n$, there exist $n$ positive integers such that the sum of them is a perfect square and the product of them is a perfect cube.
2004 National Olympiad First Round, 13
If the tangents of all interior angles of a triangle are integers, what is the sum of these integers?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 9
\qquad\textbf{(E)}\ \text{None of above}
$
2000 Czech and Slovak Match, 2
Let ${ABC}$ be a triangle, ${k}$ its incircle and ${k_a,k_b,k_c}$ three circles orthogonal to ${k}$ passing through ${B}$ and ${C, A}$ and ${C}$ , and ${A}$ and ${B}$ respectively. The circles ${k_a,k_b}$ meet again in ${C'}$ ; in the same way we obtain the points ${B'}$ and ${A'}$ . Prove that the radius of the circumcircle of ${A'B'C'}$ is half the radius of ${k}$ .
1999 Tournament Of Towns, 3
Find all pairs $(x, y)$ of integers satisfying the following condition:
each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ .
(S Zlobin)
2008 Princeton University Math Competition, A7
Joe makes two cubes of sidelengths $9$ and $10$ from $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. These cubes are dipped into white paint. Then two cubes of sidelengths $1$ and $12$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and these cubes are dipped into paint remover. Joe continues to alternately dip cubes of sides $9$ and $10$ into paint and cubes of sides $1$ and $12$ into paint remover ad nauseam.
What is the limit of the expected number of painted unit cube faces immediately after dipping in paint remover?
Kvant 2021, M2650
For which $n{}$ is it possible that a product of $n{}$ consecutive positive integers is equal to a sum of $n{}$ consecutive (not necessarily the same) positive integers?
[i]Boris Frenkin[/i]
2019 Romanian Master of Mathematics Shortlist, C3
Fix an odd integer $n > 1$. For a permutation $p$ of the set $\{1,2,...,n\}$, let S be the number of pairs of indices $(i, j)$, $1 \le i \le j \le n$, for which $p_i +p_{i+1} +...+p_j$ is divisible by $n$. Determine the maximum possible value of $S$.
Croatia
2017 Sharygin Geometry Olympiad, P18
Let $L$ be the common point of the symmedians of triangle $ABC$, and $BH$ be its altitude. It is known that $\angle ALH = 180^o -2\angle A$. Prove that $\angle CLH = 180^o - 2\angle C$.
2024 Portugal MO, 6
Alexandre and Bernado are playing the following game. At the beginning, there are $n$ balls in a bag. At first turn, Alexandre can take one ball from the bag; at second turn, Bernado can take one or two balls from the bag, and so on. So they take turns and in $k$ turn, they can take a number of balls from $1$ to $k$. Wins the one who makes the bag empty.
For each value of $n$, find who has the winning strategy.
2017 Online Math Open Problems, 19
Tessa the hyper-ant is at the origin of the four-dimensional Euclidean space $\mathbb R^4$. For each step she moves to another lattice point that is $2$ units away from the point she is currently on. How many ways can she return to the origin for the first time after exactly $6$ steps?
[i]Proposed by Yannick Yao
2003 District Olympiad, 4
Consider the continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R}, g: [0,1]\longrightarrow\mathbb{R} , $ where $
f $ has a finite limit at $ \infty . $ Show that:
$$ \lim_{n \to \infty} \frac{1}{n}\int_0^n f(x) g\left( \frac{x}{n} \right) dx =\int_0^1 g(x)dx\cdot\lim_{x\to\infty} f(x) . $$
1993 AIME Problems, 8
Let $S$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S$ so that the union of the two subsets is $S$? The order of selection does not matter; for example, the pair of subsets $\{a, c\}$, $\{b, c, d, e, f\}$ represents the same selection as the pair $\{b, c, d, e, f\}$, $\{a, c\}$.
2016 Macedonia National Olympiad, Problem 1
Solve the equation in the set of natural numbers $1+x^z + y^z = LCM(x^z,y^z)$
2004 Switzerland Team Selection Test, 5
A brick has the shape of a cube of size $2$ with one corner unit cube removed. Given a cube of side $2^{n}$ divided into unit cubes from which an arbitrary unit cube is removed, show that the remaining figure can be built using the described bricks.
2011 Regional Competition For Advanced Students, 1
Let $p_1, p_2, \ldots, p_{42}$ be $42$ pairwise distinct prime numbers. Show that the sum \[\sum_{j=1}^{42}\frac{1}{p_j^2+1}\] is not a unit fraction $\frac{1}{n^2}$ of some integer square number.
2022 Iran MO (3rd Round), 2
In the triangle $ABC$, variable points $D, E, F$ are on the sides[lines] $BC, CA, AB$ respectively so the triangle $DFE$ is similar to the triangle $ABC$ in this order. Circumcircles of $BDF$ and $CDE$ intersect respectively the circumcircle of $ABC$ at $P$ and $Q$ for the second time. Prove that the circumcircle of $DPQ$ passes through a fixed point.
2017 Yasinsky Geometry Olympiad, 2
Medians $AM$ and $BE$ of a triangle $ABC$ intersect at $O$. The points $O, M, E, C$ lie on one circle. Find the length of $AB$ if $BE = AM =3$.
1983 AMC 12/AHSME, 10
Segment $AB$ is both a diameter of a circle of radius 1 and a side of an equilateral triangle $ABC$. The circle also intersects $AC$ and $BD$ at points $D$ and $E$, respectively. The length of $AE$ is
$\displaystyle \text{(A)} \ \frac{3}{2} \qquad \text{(B)} \ \frac{5}{3} \qquad \text{(C)} \ \frac{\sqrt 3}{2} \qquad \text{(D)} \ \sqrt{3} \qquad \text{(E)} \ \frac{2 + \sqrt 3}{2}$
2023 Pan-African, 6
Let $ABC$ be an acute triangle with $AB<AC$. Let $D, E,$ and $F$ be the feet of the perpendiculars from $A, B,$ and $C$ to the opposite sides, respectively. Let $P$ be the foot of the perpendicular from $F$ to line $DE$. Line $FP$ and the circumcircle of triangle $BDF$ meet again at $Q$. Show that $\angle PBQ = \angle PAD$.
2015 Argentina National Olympiad, 4
An segment $S$ of length $50$ is covered by several segments of length $1$ , all of them contained in $S$. If any of these unit segments were removed, $S$ would no longer be completely covered. Find the maximum number of unit segments with this property.
Clarification: Assume that the segments include their endpoints.
2007 Sharygin Geometry Olympiad, 18
Determine the locus of vertices of triangles which have prescribed orthocenter and center of circumcircle.
1999 USAMO, 5
The Y2K Game is played on a $1 \times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.
2024 Iran MO (3rd Round), 1
Suppose that $T\in \mathbb N$ is given. Find all functions $f:\mathbb Z \to \mathbb C$ such that, for all $m\in \mathbb Z$ we have $f(m+T)=f(m)$ and:
$$\forall a,b,c \in \mathbb Z: f(a)\overline{f(a+b)f(a+c)}f(a+b+c)=1.$$
Where $\overline{a}$ is the complex conjugate of $a$.
2024 Moldova Team Selection Test, 3
Joe and Penny play a game. Initially there are $5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $k$-th move, any number of stones between $1$ and $k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?
2019 Saudi Arabia JBMO TST, 2
The quadrilateral ABCD is circumscribed by a circle C and K, L, M, N are the tangent points of C with the sides AB, BC, CD, DA. Let S be the point of intersection of the lines KM and LN. If the SKBL quadrilateral is cyclic, prove that the quadrilateral SNDM is also cyclic.