Found problems: 85335
2024 CCA Math Bonanza, T7
Find the number of $4$ digit positive integers $n$ such that the largest power of 2 that divides $n!$ is $2^{n-1}$.
[i]Team #7[/i]
2017 Romania EGMO TST, P1
Consider five points on a circle. For every three of them, we draw the perpendicular from the centroid of the triangle they determine to the line through the remaining two points. Prove that the ten lines thus formed are concurrent.
1987 IMO Longlists, 50
Let $P,Q,R$ be polynomials with real coefficients, satisfying $P^4+Q^4 = R^2$. Prove that there exist real numbers $p, q, r$ and a polynomial $S$ such that $P = pS, Q = qS$ and $R = rS^2$.
[hide="Variants"]Variants. (1) $P^4 + Q^4 = R^4$; (2) $\gcd(P,Q) = 1$ ; (3) $\pm P^4 + Q^4 = R^2$ or $R^4.$[/hide]
2004 Regional Olympiad - Republic of Srpska, 4
A convex $n$-gon $A_1A_2\dots A_n$ $(n>3)$ is divided into triangles by non-intersecting diagonals.
For every vertex the number of sides issuing from it is even, except for the vertices
$A_{i_1},A_{i_2},\dots,A_{i_k}$, where $1\leq i_1<\dots<i_k\leq n$. Prove that $k$ is even and
\[n\equiv i_1-i_2+\dots+i_{k-1}-i_k\pmod3\]
if $k>0$ and
\[n\equiv0\pmod3\mbox{ for }k=0.\]
Note that this leads to generalization of one recent Tournament of towns problem about triangulating of square.
2009 India IMO Training Camp, 11
Find all integers $ n\ge 2$ with the following property:
There exists three distinct primes $p,q,r$ such that
whenever $ a_1,a_2,a_3,\cdots,a_n$ are $ n$ distinct positive integers with the property that at least one of $ p,q,r$ divides $ a_j - a_k \ \forall 1\le j\le k\le n$,
one of $ p,q,r$ divides all of these differences.
2013 IPhOO, 3
A rigid (solid) cylinder is put at the top of a frictionless $25^\circ$-to-the-horizontal incline that is $3.0 \, \text{m}$ high. It is then released so that it rolls down the incline. If $v$ is the speed at the bottom of the incline, what is $v^2$, in $\text{m}^2/\text{s}^2$?
[i](B. Dejean and Ahaan Rungta, 3 points)[/i]
[b]Note[/b]: Since there is no friction, the cylinder cannot roll, and thus the problem is flawed. Two answers were accepted and given full credit.
2004 Purple Comet Problems, 15
Jerry purchased some stock for $ \$14,400$ at the same time that Susan purchased a bond for $ \$6,250$. Jerry’s investment went up $20$ percent the first year, fell $10$ percent the second year, and rose another $20$ percent the third year. Susan’s investment grew at a constant rate of compound interest for three years. If both investments are worth the same after three years, what was the annual percentage increase of Susan’s investment?
MathLinks Contest 1st, 3
For a set $S$, let $|S|$ denote the number of elements in $S$. Let $A$ be a set of positive integers with $|A| = 2001$. Prove that there exists a set $B$ such that all of the following conditions are fulfilled:
a) $B \subseteq A$;
b) $|B| \ge 668$;
c) for any $x, y \in B$ we have $x + y \notin B$.
2004 Croatia National Olympiad, Problem 3
The sequence $(p_n)_{n\in\mathbb N}$ is defined by $p_1=2$ and, for $n\ge2$, $p_n$ is the largest prime factor of $p_1p_2\cdots p_{n-1}+1$. Show that $p_n\ne5$ for all $n$.
1999 Brazil Team Selection Test, Problem 1
For a positive integer n, let $w(n)$ denote the number of distinct prime
divisors of n. Determine the least positive integer k such that
$2^{w(n)} \leq k \sqrt[4]{n}$
for all positive integers n.
2020 China Girls Math Olympiad, 4
Let $p,q$ be primes, where $p>q$. Define $t=\gcd(p!-1,q!-1)$. Prove that $t\le p^{\frac{p}{3}}$.
2006 AMC 12/AHSME, 18
The function $ f$ has the property that for each real number $ x$ in its domain, $ 1/x$ is also in its domain and
\[ f(x) \plus{} f\left(\frac {1}{x}\right) \equal{} x.
\]What is the largest set of real numbers that can be in the domain of $ f$?
$ \textbf{(A) } \{ x | x\ne 0\} \qquad \textbf{(B) } \{ x | x < 0\} \qquad \textbf{(C) }\{ x | x > 0\}\\
\textbf{(D) } \{ x | x\ne \minus{} 1 \text{ and } x\ne 0 \text{ and } x\ne 1\} \qquad \textbf{(E) } \{ \minus{} 1,1\}$
2022 Mediterranean Mathematics Olympiad, 2
(a) Decide whether there exist two decimal digits $a$ and $b$, such that every integer with decimal representation $ab222 ... 231$ is divisible by $73$.
(b) Decide whether there exist two decimal digits $c$ and $d$, such that every integer with decimal representation $cd222... 231$ is divisible by $79$.
IV Soros Olympiad 1997 - 98 (Russia), 11.9
The numbers $a$, $b$ and $c$ satisfy the conditions
$$0 < a \le b \le c\,\,\,,\,\,\, a+b+ c = 7\,\,\,, \,\,\,abc = 9.$$
Within what limits can each of the numbers $a$, $b$ and $c$ vary?
OMMC POTM, 2024 6
Find the remainder modulo $101$ of
$$\left\lfloor \dfrac{1}{(2 \cos \left(\frac{4\pi}{7} \right))^{103}}\right\rfloor$$
1985 Balkan MO, 3
Let $S$ be the set of all positive integers of the form $19a+85b$, where $a,b$ are arbitrary positive integers. On the real axis, the points of $S$ are colored in red and the remaining integer numbers are colored in green. Find, with proof, whether or not there exists a point $A$ on the real axis such that any two points with integer coordinates which are symmetrical with respect to $A$ have necessarily distinct colors.
2011 Peru IMO TST, 2
Let $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively,
\[\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.\]
[i]Proposed by Nairi Sedrakyan, Armenia[/i]
2013 Korea National Olympiad, 6
Let $ O $ be circumcenter of triangle $ABC$. For a point $P$ on segmet $BC$, the circle passing through $ P, B $ and tangent to line $AB $ and the circle passing through $ P, C $ and tangent to line $AC $ meet at point $ Q ( \ne P ) $. Let $ D, E $ be foot of perpendicular from $Q$ to $ AB, AC$. ($D \ne B, E \ne C $) Two lines $DE $ and $ BC $ meet at point $R$. Prove that $ O, P, Q $ are collinear if and only if $ A, R, Q $ are collinear.
2013 IberoAmerican, 6
A [i]beautiful configuration[/i] of points is a set of $n$ colored points, such that if a triangle with vertices in the set has an angle of at least $120$ degrees, then exactly 2 of its vertices are colored with the same color. Determine the maximum possible value of $n$.
2022 BMT, 15
Let $f(x)$ be a function acting on a string of $0$s and $1$s, defined to be the number of substrings of $x$ that have at least one $1$, where a substring is a contiguous sequence of characters in $x$. Let $S$ be the set of binary strings with $24$ ones and $100$ total digits. Compute the maximum possible value of $f(s)$ over all $s\in S$.
For example, $f(110) = 5$ as $\underline{1}10$, $1\underline{1}0$, $\underline{11}0$, $1\underline{10}$, and $\underline{110}$ are all substrings including a $1$. Note that $11\underline{0}$ is not such a substring.
2018 ELMO Shortlist, 2
Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be the midpoint of $\overline{AH}$ and let $T$ be on line $BC$ with $\angle TAO=90^{\circ}$. Let $X$ be the foot of the altitude from $O$ onto line $PT$. Prove that the midpoint of $\overline{PX}$ lies on the nine-point circle* of $\triangle ABC$.
*The nine-point circle of $\triangle ABC$ is the unique circle passing through the following nine points: the midpoint of the sides, the feet of the altitudes, and the midpoints of $\overline{AH}$, $\overline{BH}$, and $\overline{CH}$.
[i]Proposed by Zack Chroman[/i]
2020 June Advanced Contest, 1
A tuple of real numbers $(a_1, a_2, \dots, a_m)$ is called [i]stable [/i]if for each $k \in \{1, 2, \cdots, m-1\}$,
$$ \left \vert \frac{a_1+ a_2 + \cdots + a_k}{k} - a_{k+1} \right \vert < 1. $$
Does there exist a stable $n$-tuple $(x_1, x_2, \dots, x_n)$ such that for any real number $x$, the $(n+1)$-tuple $(x, x_1, x_2, \dots, x_n)$ is not stable?
2007 Hanoi Open Mathematics Competitions, 12
Calculate the sum $\frac{1}{2.7.12} + \frac{1}{7.12.17} + ... + \frac{1}{1997.2002.2007}$.
2021-2022 OMMC, 2
Alex writes down some distinct integers on a blackboard. For each pair of integers, he writes the positive difference of those on a piece of paper. Find the sum of all $n\leq2022$ such that it is possible for the numbers on the paper to contain only the positive integers between $1$ and $n$, inclusive exactly once.
[i]Proposed by Alexander Wang[/i]
2011 USAMTS Problems, 4
Let $ABCDEF$ and $ABC'D'E'F'$ be regular planar hexagons in three-dimensional space with side length $1$, such that $\angle EAE'=60^{\circ}$. Let $P$ be the convex polyhedron whose vertices are $A$, $B$, $C$, $C'$, $D$, $D'$, $E$, $E'$, $F$, and $F'$.
(a) Find the radius $r$ of the largest sphere that can be enclosed in polyhedron $P$.
(b) Let $S$ be a sphere enclosed in polyhedron $P$ with radius $r$ (as derived in part (a)). The set of possible centers of $S$ is a line segment $\overline{XY}$. Find the length $XY$.