Found problems: 85335
2002 AMC 10, 7
The dimensions of a rectangular box in inches are all positive integers and the volume of the box is $2002\text{ in}^3$. Find the minimum possible sum in inches of the three dimensions.
$\textbf{(A) }36\qquad\textbf{(B) }38\qquad\textbf{(C) }42\qquad\textbf{(D) }44\qquad\textbf{(E) }92$
2018 AIME Problems, 12
For every subset $T$ of $U = \{ 1,2,3,\ldots,18 \}$, let $s(T)$ be the sum of the elements of $T$, with $s(\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets of $U$, the probability that $s(T)$ is divisible by $3$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.
2013 EGMO, 6
Snow White and the Seven Dwarves are living in their house in the forest. On each of $16$ consecutive days, some of the dwarves worked in the diamond mine while the remaining dwarves collected berries in the forest. No dwarf performed both types of work on the same day. On any two different (not necessarily consecutive) days, at least three dwarves each performed both types of work. Further, on the first day, all seven dwarves worked in the diamond mine.
Prove that, on one of these $16$ days, all seven dwarves were collecting berries.
2006 IberoAmerican Olympiad For University Students, 3
Let $p_1(x)=p(x)=4x^3-3x$ and $p_{n+1}(x)=p(p_n(x))$ for each positive integer $n$. Also, let $A(n)$ be the set of all the real roots of the equation $p_n(x)=x$.
Prove that $A(n)\subseteq A(2n)$ and that the product of the elements of $A(n)$ is the average of the elements of $A(2n)$.
2021 Saudi Arabia IMO TST, 5
Let $ABC$ be a non isosceles triangle with incenter $I$ . The circumcircle of the triangle $ABC$ has radius $R$. Let $AL$ be the external angle bisector of $\angle BAC $with $L \in BC$. Let $K$ be the point on perpendicular bisector of $BC$ such that $IL \perp IK$.Prove that $OK=3R$.
2013 ELMO Shortlist, 7
A $2^{2014} + 1$ by $2^{2014} + 1$ grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer $n$ greater than $2$, there do not exist pairwise distinct black squares $s_1$, $s_2$, \dots, $s_n$ such that $s_i$ and $s_{i+1}$ share an edge for $i=1,2, \dots, n$ (here $s_{n+1}=s_1$).
What is the maximum possible number of filled black squares?
[i]Proposed by David Yang[/i]
2002 Bundeswettbewerb Mathematik, 2
Each lottery ticket has a 9-digit numbers, which uses only the digits 1, 2, 3. Each ticket is colored red, blue or green. If two tickets have numbers which differ in all nine places, then the tickets have different colors. Ticket 122222222 is red, and ticket 222222222 is green. What color is ticket 123123123?
(a) Green (b) Red (c) Blue (d) Data insufficient
2014 Harvard-MIT Mathematics Tournament, 10
Fix a positive real number $c>1$ and positive integer $n$. Initially, a blackboard contains the numbers $1,c,\ldots, c^{n-1}$. Every minute, Bob chooses two numbers $a,b$ on the board and replaces them with $ca+c^2b$. Prove that after $n-1$ minutes, the blackboard contains a single number no less than \[\left(\dfrac{c^{n/L}-1}{c^{1/L}-1}\right)^L,\] where $\phi=\tfrac{1+\sqrt 5}2$ and $L=1+\log_\phi(c)$.
2023 UMD Math Competition Part I, #19
Three positive real numbers $a, b, c$ satisfy $a^b = 343, b^c = 10, a^c = 7.$ Find $b^b.$
$$
\mathrm a. ~ 1000\qquad \mathrm b.~900\qquad \mathrm c. ~1200 \qquad \mathrm d. ~4000 \qquad \mathrm e. ~100
$$
2008 Germany Team Selection Test, 2
[b](i)[/b] Determine the smallest number of edges which a graph of $ n$ nodes may have given that adding an arbitrary new edge would give rise to a 3-clique (3 nodes joined pairwise by edges).
[b](ii)[/b] Determine the smallest number of edges which a graph of $ n$ nodes may have given that adding an arbitrary new edge would give rise to a 4-clique (4 nodes joined pairwise by edges).
MBMT Team Rounds, 2020.23
Let $ABCD$ be a cyclic quadrilateral so that $\overline{AC} \perp \overline{BD}$. Let $E$ be the intersection of $\overline{AC}$ and $\overline{BD}$, and let $F$ be the foot of the altitude from $E$ to $\overline{AB}$. Let $\overline{EF}$ intersect $\overline{CD}$ at $G$, and let the foot of the perpendiculars from $G$ to $\overline{AC}$ and $\overline{BD}$ be $H, I$ respectively. If $\overline{AB} = \sqrt{5}, \overline{BC} = \sqrt{10}, \overline{CD} = 3\sqrt{5}, \overline{DA} = 2\sqrt{10}$, find the length of $\overline{HI}$.
[i]Proposed by Timothy Qian[/i]
1986 IMO Longlists, 57
In a triangle $ABC$, the incircle touches the sides $BC, CA, AB$ in the points $A',B', C'$, respectively; the excircle in the angle $A$ touches the lines containing these sides in $A_1,B_1, C_1$, and similarly, the excircles in the angles $B$ and $C$ touch these lines in $A_2,B_2, C_2$ and $A_3,B_3, C_3$. Prove that the triangle $ABC$ is right-angled if and only if one of the point triples $(A',B_3, C'),$ $ (A_3,B', C_3), (A',B', C_2), (A_2,B_2, C'), (A_2,B_1, C_2), (A_3,B_3, C_1),$ $ (A_1,B_2, C_1), (A_1,B_1, C_3)$ is collinear.
2021 Malaysia IMONST 1, 5
How many integers $n$ (with $1 \le n \le 2021$) have the property that $8n + 1$ is a perfect square?
2024 CCA Math Bonanza, I13
Call a sequence $a_0,a_1,a_2,\dots$ of positive integers defined by $a_k = 25 a_{k - 1} + 96$ for all $k > 0$ a \textit{valid} sequence. Call the \textit{goodness} of a \textit{valid} sequence the maximum value of $\gcd(a_k, a_{k+2024})$ over all $k$. Call a \textit{valid} sequence \textit{best} if it has the maximal \textit{goodness} across all possible \textit{valid} sequences. Find the second largest $a_0$ across all \textit{best} sequences.
[i]Individual #13[/i]
2010 May Olympiad, 1
A closed container in the shape of a rectangular parallelepiped contains $1$ liter of water. If the container rests horizontally on three different sides, the water level is $2$ cm, $4$ cm and $5$ cm. Calculate the volume of the parallelepiped.
PEN H Problems, 88
(Leo Moser) Show that the Diophantine equation \[\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}+\frac{1}{x_{1}x_{2}\cdots x_{n}}= 1\] has at least one solution for every positive integers $n$.
2005 Morocco TST, 4
Let $ABCD$ be a cyclic qudrilaterlal such that $AB.BC=2.CD.DA$
Prove that $8.BD^2 \leq 9.AC^2$
2019 BMT Spring, 3
A cylinder with radius $5$ and height $1$ is rolling on the (unslanted) floor. Inside the cylinder, there is water that has constant height $\frac{15}{2}$ as the cylinder rolls on the floor. What is the volume of the water?
2005 Croatia National Olympiad, 3
If $k, l, m$ are positive integers with $\frac{1}{k}+\frac{1}{l}+\frac{1}{m}<1$, find the maximum possible value of $\frac{1}{k}+\frac{1}{l}+\frac{1}{m}$.
2021 All-Russian Olympiad, 2
Let $P(x)$ be a nonzero polynomial of degree $n>1$ with nonnegative coefficients such that function $y=P(x)$ is odd. Is that possible thet for some pairwise distinct points $A_{1}, A_{2}, \dots A_{n}$ on the graph $G: y = P(x)$ the following conditions hold: tangent to $G$ at $A_{1}$ passes through $A_{2}$, tangent to $G$ at $A_{2}$ passes through $A_{3}$, $\dots$, tangent to $G$ at $A_{n}$ passes through $A_{1}$?
2018 Bulgaria National Olympiad, 5.
Given a polynomial $P(x)=a_{d}x^{d}+ \ldots +a_{2}x^{2}+a_{0}$ with positive integers for coefficients and degree $d\geq 2$. Consider the sequence defined by $$b_{1}=a_{0} ,b_{n+1}=P(b_{n}) $$ for $n \geq 1$ . Prove that for all $n \geq 2$ there exists a prime $p$ such that $p$ divides $b_{n}$ but does not divide $b_{1}b_{2} \ldots b_{n-1}$.
2024/2025 TOURNAMENT OF TOWNS, P1
Find the minimum positive integer such that some four of its natural divisors sum up to $2025$.
2011 AMC 10, 22
A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
$ \textbf{(A)}\ 5\sqrt{2}-7 \qquad
\textbf{(B)}\ 7-4\sqrt{3} \qquad
\textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad
\textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad
\textbf{(E)}\ \frac{\sqrt{3}}{9} $
2016 HMNT, 3
Let $V$ be a rectangular prism with integer side lengths. The largest face has area $240$ and the smallest face has area $48$. A third face has area $x$, where $x$ is not equal to $48$ or $240$. What is the sum of all possible values of $x$?
PEN E Problems, 3
Find the sum of all distinct positive divisors of the number $104060401$.