Found problems: 85335
2001 Federal Competition For Advanced Students, Part 2, 1
Prove that $\frac{1}{25} \sum_{k=0}^{2001} \left[ \frac{2^k}{25}\right]$ is a positive integer.
1998 AIME Problems, 14
An $m\times n\times p$ rectangular box has half the volume of an $(m+2)\times(n+2)\times(p+2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\le n\le p.$ What is the largest possible value of $p$?
2014 India Regional Mathematical Olympiad, 2
let $x,y$ be positive real numbers.
prove that
$ 4x^4+4y^3+5x^2+y+1\geq 12xy $
2013 Romania Team Selection Test, 4
Let $n$ be an integer greater than 1. The set $S$ of all diagonals of a $ \left( 4n-1\right) $-gon is partitioned into $k$ sets, $S_{1},S_{2},\ldots ,S_{k},$ so that, for every pair of distinct indices $i$ and $j,$ some diagonal in $S_{i}$ crosses some diagonal in $S_{j};$ that is, the two diagonals share an interior point. Determine the largest possible value of $k $ in terms of $n.$
1981 Putnam, A4
A point $P$ moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it
meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection.
Let $N( t)$ be the number of starting directions from a fixed interior point $P_0$ for which $P$ escapes within $t$ units of time. Find the least constant $a$ for which constants $b$ and $c$ exist such that
$$N(t) \leq at^2 +bt+c$$
for all $t>0$ and all initial points $P_0 .$
2014 Greece National Olympiad, 3
For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once).
Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that we can achieve $\frac{S_1}{S_2}=\frac{39}{64}.$
2023 Taiwan Mathematics Olympiad, 3
Let $O$ be the center of circle $\Gamma$, and $A$, $B$ be two points on $\Gamma$ so that $O, A$ and $B$ are not collinear. Let $M$ be the midpoint of $AB$. Let $P$ and $Q$ be points on $OA$ and $OB$, respectively, so that $P \neq A$ and $P, M, Q$ are collinear. Let $X$ be the intersection of the line passing through $P$ and parallel to $AB$ and the line passing through $Q$ and parallel to $OM$. Let $Y$ be the intersection of the line passing through $X$ and parallel to $OA$ and the line passing through $B$ and orthogonal to $OX$. Prove that: if $X$ is on $\Gamma$, then $Y$ is also on $\Gamma$.
[i]
Proposed by usjl[/i]
2014 Belarus Team Selection Test, 2
Let $x,y,z$ be pairwise distinct real numbers such that $x^2-1/y = y^2 -1/z = z^2 -1/x$.
Given $z^2 -1/x = a$, prove that $(x + y + z)xyz= -a^2$.
(I. Voronovich)
2006 Iran MO (3rd Round), 1
Let $A$ be a family of subsets of $\{1,2,\ldots,n\}$ such that no member of $A$ is contained in another. Sperner’s Theorem states that $|A|\leq{n\choose{\lfloor\frac{n}{2}\rfloor}}$. Find all the families for which the equality holds.
2013-2014 SDML (Middle School), 8
On a windless day, a pigeon can fly from Albatrocity to Finchester and back in $3$ hours and $45$ minutes. However, when there is a $10$ mile per hour win blowing from Albatrocity to Finchester, it takes the pigeon $4$ hours to make the round trip. How many miles is it from Albatrocity to Finchester?
2024 China Western Mathematical Olympiad, 1
For positive integer $n$, note $S_n=1^{2024}+2^{2024}+ \cdots +n^{2024}$.
Prove that there exists infinitely many positive integers $n$, such that $S_n$ isn’t divisible by $1865$ but $S_{n+1}$ is divisible by $1865$
1990 AMC 8, 11
The numbers on the faces of this cube are consecutive whole numbers. The sums of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3));
draw((3,3)--(5,5));
label("$15$",(1.5,1.2),N); label("$11$",(4,2.3),N); label("$14$",(2.5,3.7),N);[/asy]
$ \text{(A)}\ 75\qquad\text{(B)}\ 76\qquad\text{(C)}\ 78\qquad\text{(D)}\ 80\qquad\text{(E)}\ 81 $
2007 Harvard-MIT Mathematics Tournament, 13
Determine the largest integer $n$ such that $7^{2048}-1$ is divisible by $2^n$.
CIME I 2018, 1
A positive integer $n$ is defined as a $\textit{stepstool number}$ if $n$ has one less positive divisor than $n + 1$. For example, $3$ is a stepstool number, as $3$ has $2$ divisors and $4$ has $2 + 1 = 3$ divisors. Find the sum of all stepstool numbers less than $300$.
[i]Proposed by [b]Th3Numb3rThr33[/b][/i]
2018 Iranian Geometry Olympiad, 5
Suppose that $ABCD$ is a parallelogram such that $\angle DAC = 90^o$. Let $H$ be the foot of perpendicular from $A$ to $DC$, also let $P$ be a point along the line $AC$ such that the line $PD$ is tangent to the circumcircle of the triangle $ABD$. Prove that $\angle PBA = \angle DBH$.
Proposed by Iman Maghsoudi
1988 Austrian-Polish Competition, 4
Determine all strictly increasing functions $f: R \to R$ satisfying $f (f(x) + y) = f(x + y) + f (0)$ for all $x,y \in R$.
1991 AMC 8, 17
An auditorium with $20$ rows of seats has $10$ seats in the first row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, then the maximum number of students that can be seated for an exam is
$\text{(A)}\ 150 \qquad \text{(B)}\ 180 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 400 \qquad \text{(E)}\ 460$
2006 Moldova Team Selection Test, 1
Let the point $P$ in the interior of the triangle $ABC$. $(AP, (BP, (CP$ intersect the circumcircle of $ABC$ at $A_{1}, B_{1}, C_{1}$. Prove that the maximal value of the sum of the areas $A_{1}BC$, $B_{1}AC$, $C_{1}AB$ is $p(R-r)$, where $p, r, R$ are the usual notations for the triangle $ABC$.
1991 All Soviet Union Mathematical Olympiad, 537
Four lines in the plane intersect in six points. Each line is thus divided into two segments and two rays. Is it possible for the eight segments to have lengths $1, 2, 3, ... , 8$? Can the lengths of the eight segments be eight distinct integers?
2019 SAFEST Olympiad, 2
Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.
2005 Cuba MO, 2
Determine the quadratic functions $f(x) = ax^2 + bx + c$ for which there exists an interval $(h, k)$ such that for all $x \in (h, k)$ it holds that $f(x)f(x + 1) < 0$ and $f(x)f(x -1) < 0$.
1962 Bulgaria National Olympiad, Problem 3
It is given a cube with sidelength $a$. Find the surface of the intersection of the cube with a plane, perpendicular to one of its diagonals and whose distance from the centre of the cube is equal to $h$.
2012 Kyoto University Entry Examination, 5
Find the domain of the pairs of positive real numbers $(a,\ b)$ such that there is a $\theta\ (0<\theta \leq \pi)$ such that $\cos a\theta =\cos b\theta$, then draw the domain on the coordinate plane.
30 points
2014 Harvard-MIT Mathematics Tournament, 21
Compute the number of ordered quintuples of nonnegative integers $(a_1,a_2,a_3,a_4,a_5)$ such that $0\leq a_1,a_2,a_3,a_4,a_5\leq 7$ and $5$ divides $2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$.
2002 May Olympiad, 5
Let $x$ and $y$ be positive integers we have a table $x\times y$ where $(x + 1)(y + 1)$ points are red(the points are the vertices of the squares). Initially there is one ant in each red point, in a moment the ants walk by the lines of the table with same speed, each turn that an ant arrive in a red point the ant moves $90º$ to some direction.
Determine all values of $x$ and $y$ where is possible that the ants move indefinitely where can't be in any moment two ants in the same red point.