Found problems: 85335
2020 Canada National Olympiad, 1
There are $n \ge 3$ distinct positive real numbers. Show that there are at most $n-2$ different integer power of three that can be written as the sum of three distinct elements from these $n$ numbers.
2016 PUMaC Combinatorics A, 7
The Dinky is a train connecting Princeton to the outside world. It runs on an odd schedule: the train arrive once every one-hour block at some uniformly random time (once at a random time between $\text{9am}$ and $\text{10am}$, once at a random time between $\text{10am}$ and $\text{11am}$, and so on). One day, Emilia arrives at the station, at some uniformly random time, and does not know the time. She expects to wait for $y$ minutes for the next train to arrive. After waiting for an hour, a train has still not come. She now expects to wait for $z$ minutes. Find $yz$.
1991 Vietnam National Olympiad, 2
Let $k>1$ be an odd integer. For every positive integer n, let $f(n)$ be the greatest positive integer for which $2^{f(n)}$ divides $k^n-1$. Find $f(n)$ in terms of $k$ and $n$.
2020 Indonesia MO, 8
Determine the smallest natural number $n > 2$, or show that no such natural numbers $n$ exists, that satisfy the following condition: There exists natural numbers $a_1, a_2, \dots, a_n$ such that
\[ \gcd(a_1, a_2, \dots, a_n) = \sum_{k = 1}^{n - 1} \underbrace{\left( \frac{1}{\gcd(a_k, a_{k + 1})} + \frac{1}{\gcd(a_k, a_{k + 2})} + \dots + \frac{1}{\gcd(a_k, a_n)} \right)}_{n - k \ \text{terms}} \]
1971 Swedish Mathematical Competition, 1
Show that
\[
\left(1 + a + a^2\right)^2 < 3\left(1 + a^2 + a^4\right)
\]
for real $a \neq 1$.
1965 Bulgaria National Olympiad, Problem 2
Prove the inequality:
$$(1+\sin^2\alpha)^n+(1+\cos^2\alpha)^n\ge2\left(\frac32\right)^n$$is true for every natural number $n$. When does equality hold?
1994 Czech And Slovak Olympiad IIIA, 4
Let $a_1,a_2,...$ be a sequence of natural numbers such that for each $n$, the product $(a_n - 1)(a_n- 2)...(a_n - n^2)$ is a positive integral multiple of $n^{n^2-1}$. Prove that for any finite set $P$ of prime numbers the following inequality holds:
$$\sum_{p\in P}\frac{1}{\log_p a_p}< 1$$
1989 IMO Longlists, 2
An accurate 12-hour analog clock has an hour hand, a minute hand, and a second hand that are aligned at 12:00 o’clock and make one revolution in 12 hours, 1 hour, and 1 minute, respectively. It is well known, and not difficult to prove, that there is no time when the three hands are equally spaced around the clock, with each separating angle $ \frac{2 \cdot \pi}{3}.$ Let $ f(t), g(t), h(t)$ be the respective absolute deviations of the separating angles from \frac{2 \cdot \pi}{3} at $ t$ hours after 12:00 o’clock. What is the minimum value of $ max\{f(t), g(t), h(t)\}?$
1999 Singapore MO Open, 4
Let $ABCD$ be a quadrilateral with each interior angle less than $180^o$. Show that if $A, B, C, D$ do not lie on a circle, then $AB \cdot CD + AD\cdot BC > AC \cdot BD$
2021 Caucasus Mathematical Olympiad, 1
Let $a$, $b$, $c$ be real numbers such that $a^2+b=c^2$, $b^2+c=a^2$, $c^2+a=b^2$. Find all possible values of $abc$.
1979 Chisinau City MO, 181
Prove that if every line connecting any two points of some finite set of points of the plane contains at least one more point of this set, then all points of the set lie on one straight line.
2003 Polish MO Finals, 4
A prime number $p$ and integers $x, y, z$ with $0 < x < y < z < p$ are given. Show that if the numbers $x^3, y^3, z^3$ give the same remainder when divided by $p$, then $x^2 + y^2 + z^2$ is divisible by $x + y + z.$
LMT Team Rounds 2010-20, A21 B23
The LHS Math Team wants to play Among Us. There are so many people who want to play that they are going to form several games. Each game has at most 10 people. People are $\textit{happy}$ if they are in a game that has at least 8 people in it. What is the largest possible number of people who would like to play Among Us such that it is impossible to make everyone $\textit{happy}$?
[i]Proposed by Sammy Charney[/i]
2014 AMC 10, 8
Which of the following numbers is a perfect square?
$ \textbf{(A)}\ \dfrac{14!15!}2\qquad\textbf{(B)}\ \dfrac{15!16!}2\qquad\textbf{(C)}\ \dfrac{16!17!}2\qquad\textbf{(D)}\ \dfrac{17!18!}2\qquad\textbf{(E)}\ \dfrac{18!19!}2 $
2013 Junior Balkan Team Selection Tests - Romania, 4
Find all integers $n \ge 2$ with the property:
there is a permutation $(a_1,a2,..., a_n)$ of the set $\{1, 2,...,n\}$ so that the numbers $a_1 + a_2 +...+ a_k, k = 1, 2,...,n$ have diffferent remainders when divided by $n$
2019 CHMMC (Fall), 9
Consider a rectangle with length $6$ and height $4$. A rectangle with length $3$ and height $1$ is placed inside the larger rectangle such that it is distance $1$ from the bottom and leftmost sides of the larger rectangle.
We randomly select one point from each side of the larger rectangle, and connect these $4$ points to form a quadrilateral. What is the probability that the smaller rectangle is strictly contained within that quadrilateral?
2019 Thailand Mathematical Olympiad, 10
Prove that there are infinitely many positive odd integer $n$ such that $n!+1$ is composite number.
2020 Kosovo National Mathematical Olympiad, 2
Find all positive integers $x$, $y$ such that $2^x+5^y+2$ is a perfect square.
2015 İberoAmerican, 3
Let $\alpha$ and $\beta$ be the roots of $x^{2} - qx + 1$, where $q$ is a rational number larger than $2$. Let $s_1 = \alpha + \beta$, $t_1 = 1$, and for all integers $n \geq 2$:
$s_n = \alpha^n + \beta^n$
$t_n = s_{n-1} + 2s_{n-2} + \cdot \cdot \cdot + (n - 1)s_{1} + n$
Prove that, for all odd integers $n$, $t_n$ is the square of a rational number.
2009 Balkan MO Shortlist, G6
Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.
2018 Junior Balkan Team Selection Tests - Romania, 1
Prove that the equation $x^2+y^2+z^2 = x+y+z+1$ has no rational solutions.
2024 IMC, 5
Let $n>d$ be positive integers. Choose $n$ independent, uniformly distributed random points $x_1,\dots,x_n$ in the unit ball $B \subset \mathbb{R}^d$ centered at the origin. For a point $p \in B$ denote by $f(p)$ the probability that the convex hull of $x_1,\dots,x_n$ contains $p$. Prove that if $p,q \in B$ and the distance of $p$ from the origin is smaller than the distance of $q$ from the origin, then $f(p) \ge f(q)$.
1998 Korea - Final Round, 2
Let $D$,$E$,$F$ be points on the sides $BC$,$CA$,$AB$ respectively of a triangle $ABC$. Lines $AD$,$BE$,$CF$ intersect the circumcircle of $ABC$ again at $P$,$Q$,$R$, respectively.Show that:
\[\frac{AD}{PD}+\frac{BE}{QE}+\frac{CF}{RF}\geq 9\]
and find the cases of equality.
1994 AIME Problems, 10
In triangle $ABC,$ angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D.$ The lengths of the sides of $\triangle ABC$ are integers, $BD=29^3,$ and $\cos B=m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
MathLinks Contest 4th, 3.3
Let $ABC$ be a triangle, and let $C$ be its circumcircle. Let $T$ be the circle tangent to $AB, AC$ and $C$ internally in the points $F, E$ and $D$ respectively. Let $P, Q$ be the intersection points between the line $EF$ and the lines $DB$ and $DC$ respectively. Prove that if $DP = DQ$ then the triangle $ABC$ is isosceles.