Found problems: 85335
2006 AIME Problems, 11
A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k,\thinspace 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules:
$\bullet$ Any cube may be the bottom cube in the tower.
$\bullet$ The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$
Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by 1000?
MOAA Gunga Bowls, 2021.2
Add one pair of brackets to the expression
\[1+2\times 3+4\times 5+6\]
so that the resulting expression has a valid mathematical value, e.g., $1+2\times (3 + 4\times 5)+6=53$. What is the largest possible value that one can make?
[i]Proposed by Nathan Xiong[/i]
2010 All-Russian Olympiad, 3
Let us call a natural number $unlucky$ if it cannot be expressed as $\frac{x^2-1}{y^2-1} $ with natural numbers $x,y >1$. Is the number of $unlucky$ numbers finite or infinite?
1989 IMO Shortlist, 16
The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions:
[b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$
[b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\]
Prove that $ c \leq \frac{1}{4n}.$
2018 China Team Selection Test, 4
Suppose $A_1,A_2,\cdots ,A_n \subseteq \left \{ 1,2,\cdots ,2018 \right \}$ and $\left | A_i \right |=2, i=1,2,\cdots ,n$, satisfying that $$A_i + A_j, \; 1 \le i \le j \le n ,$$ are distinct from each other. $A + B = \left \{ a+b|a\in A,\,b\in B \right \}$. Determine the maximal value of $n$.
2016 Polish MO Finals, 6
Let $I$ be an incenter of $\triangle ABC$. Denote $D, \ S \neq A$ intersections of $AI$ with $BC, \ O(ABC)$ respectively. Let $K, \ L$ be incenters of $\triangle DSB, \ \triangle DCS$. Let $P$ be a reflection of $I$ with the respect to $KL$. Prove that $BP \perp CP$.
1994 Miklós Schweitzer, 5
Let H be a $G_{\delta}$ subset of $\mathbb R$ whose closure has a positive Lebesgue measure. Prove that the set $H + H + H + H = \{ x + y + z + u : x , y , z , u \in H \}$ contains an interval.
2011 VJIMC, Problem 1
(a) Is there a polynomial $P(x)$ with real coefficients such that $P\left(\frac1k\right)=\frac{k+2}k$ for all positive integers $k$?
(b) Is there a polynomial $P(x)$ with real coefficients such that $P\left(\frac1k\right)=\frac1{2k+1}$ for all positive integers $k$?
2017 Danube Mathematical Olympiad, 3
Let $O,H$ be the circumcenter and the orthocenter of triangle $ABC$. Let $F$ be the foot of the perpendicular from C onto AB, and $M$ the midpoint of $CH$. Let N be the foot of the perpendicular from C onto the parallel through H at $OM$. Let $D$ be on $AB$ such that $CA=CD$. Let $BN$ intersect $CD$ at $P$. Let $PH$ intersect $CA$ at $Q$. Prove that $QF\perp OF$.
2019 CMIMC, 10
Let $\triangle ABC$ be a triangle with side lengths $a$, $b$, and $c$. Circle $\omega_A$ is the $A$-excircle of $\triangle ABC$, defined as the circle tangent to $BC$ and to the extensions of $AB$ and $AC$ past $B$ and $C$ respectively. Let $\mathcal{T}_A$ denote the triangle whose vertices are these three tangency points; denote $\mathcal{T}_B$ and $\mathcal{T}_C$ similarly. Suppose the areas of $\mathcal{T}_A$, $\mathcal{T}_B$, and $\mathcal{T}_C$ are $4$, $5$, and $6$ respectively. Find the ratio $a:b:c$.
1998 China Team Selection Test, 2
$n \geq 5$ football teams participate in a round-robin tournament. For every game played, the winner receives 3 points, the loser receives 0 points, and in the event of a draw, both teams receive 1 point. The third-from-bottom team has fewer points than all the teams ranked before it, and more points than the last 2 teams; it won more games than all the teams before it, but fewer games than the 2 teams behind it. Find the smallest possible $n$.
2009 USA Team Selection Test, 9
Prove that for positive real numbers $x$, $y$, $z$, \[ x^3(y^2+z^2)^2 + y^3(z^2+x^2)^2+z^3(x^2+y^2)^2 \geq xyz\left[xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\right].\] [i]Zarathustra (Zeb) Brady.[/i]
2018 Pan African, 1
Find all functions $f : \mathbb Z \to \mathbb Z$ such that $$(f(x + y))^2 = f(x^2) + f(y^2)$$ for all $x, y \in \mathbb Z$.
2005 Tuymaada Olympiad, 1
The positive integers $1,2,...,121$ are arranged in the squares of a $11 \times 11$ table. Dima found the product of numbers in each row and Sasha found the product of the numbers in each column. Could they get the same set of $11$ numbers?
[i]Proposed by S. Berlov[/i]
2019 India PRMO, 29
Let $ABC$ be an acute angled triangle with $AB=15$ and $BC=8$. Let $D$ be a point on $AB$ such that $BD=BC$. Consider points $E$ on $AC$ such that $\angle DEB=\angle BEC$. If $\alpha$ denotes the product of all possible values of $AE$, find $\lfloor \alpha \rfloor$ the integer part of $\alpha$.
1996 Tournament Of Towns, (504) 1
Do there exist $10$ consecutive positive integers such that the sum of their squares is equal to the sum of squares of the next $9$ integers?
(Inspired by a diagram in an old text book)
2010 Contests, 2
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying
\[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\]
[i]Marko Radovanović, Serbia[/i]
1940 Moscow Mathematical Olympiad, 067
Which is greater: $300!$ or $100^{300}$?
2013 Bosnia And Herzegovina - Regional Olympiad, 1
If $x$ and $y$ are real numbers such that $x^{2013}+y^{2013}>x^{2012}+y^{2012}$, prove that $x^{2014}+y^{2014}>x^{2013}+y^{2013}$
2000 German National Olympiad, 5
(a) Let be given $2n$ distinct points on a circumference, $n$ of which are red and $n$ are blue. Prove that one can join these points pairwise by $n$ segments so that no two segments intersect and the endpoints of each segments have different colors.
(b) Show that the statement from (a) remains valid if the points are in an arbitrary position in the plane so that no three of them are collinear.
1990 Hungary-Israel Binational, 3
Prove that: \[ \frac{1989}{2}\minus{}\frac{1988}{3}\plus{}\frac{1987}{4}\minus{}\cdots\minus{}\frac{2}{1989}\plus{}\frac{1}{1990}\equal{}\frac{1}{996}\plus{}\frac{3}{997}\plus{}\frac{5}{998}\plus{}\cdots\plus{}\frac{1989}{1990}\]
2000 AMC 10, 15
Two non-zero real numbers, $a$ and $b$, satisfy $ab=a-b$. Which of the following is a possible value of $\frac ab+\frac ba-ab$?
$\text{(A)}\ -2\qquad\text{(B)}\ -\frac12\qquad\text{(C)}\ \frac13\qquad\text{(D)}\ \frac12\qquad\text{(E)}\ 2$
2022 Yasinsky Geometry Olympiad, 2
In the acute triangle $ABC$, the sum of the distances from the vertices $B$ and $C$ to of the orthocenter $H$ is equal to $4r,$ where $r$ is the radius of the circle inscribed in this triangle. Find the perimeter of triangle $ABC$ if it is known that $BC=a$.
(Gryhoriy Filippovskyi)
2017 HMNT, 5
Given that $a,b,c$ are integers with $abc = 60$, and that complex number $\omega \neq 1$ satisfies $\omega^3=1$, find the minimum possible value of $| a + b\omega + c\omega^2|$.
2023 Moldova EGMO TST, 4
Find all triplets of prime numbers $(m, n, p)$, that satisfy the system of equations: $$\left\{\begin{matrix} 2m-n+13p=2072,\\3m+11n+13p=2961.\end{matrix}\right.$$