Found problems: 85335
2011 Danube Mathematical Competition, 1
Let $ABCM$ be a quadrilateral and $D$ be an interior point such that $ABCD$ is a parallelogram. It is known that $\angle AMB =\angle CMD$. Prove that $\angle MAD =\angle MCD$.
2013 Korea National Olympiad, 7
For positive integer $k$, define integer sequence $\{ b_n \}, \{ c_n \} $ as follows:
\[ b_1 = c_1 = 1 \]
\[ b_{2n} = kb_{2n-1} + (k-1)c_{2n-1}, c_{2n} = b_{2n-1} + c_{2n-1} \]
\[ b_{2n+1} = b_{2n} + (k-1)c_{2n}, c_{2n+1} = b_{2n} + kc_{2n} \]
Let $a_k = b_{2014} $. Find the value of
\[ \sum_{k=1}^{100} { (a_k - \sqrt{{a_k}^2-1} )^{ \frac{1}{2014}} }\]
2003 AMC 8, 7
Blake and Jenny each took four $100$ point tests. Blake averaged $78$ on the four tests. Jenny scored $10$ points higher than Blake on the first test, $10$ points lower on the second test, and $20$ points higher on both the third and fourth test.
What is the difference between Blake's average on the four tests and Jenny's average on the four tests?
$\textbf{(A)}\ 10 \qquad
\textbf{(B)}\ 15 \qquad
\textbf{(C)}\ 20 \qquad
\textbf{(D)}\ 25 \qquad
\textbf{(E)}\ 40$
LMT Team Rounds 2010-20, 2020.S15
Let $\phi(k)$ denote the number of positive integers less than or equal to $k$ that are relatively prime to $k$. For example, $\phi(2)=1$ and $\phi(10)=4$. Compute the number of positive integers $n \leq 2020$ such that $\phi(n^2)=2\phi(n)^2$.
2004 Baltic Way, 14
We say that a pile is a set of four or more nuts. Two persons play the following game. They start with one pile of $n \geq 4$ nuts. During a move a player takes one of the piles that they have and split it into two nonempty sets (these sets are not necessarily piles, they can contain arbitrary number of nuts). If the player cannot move, he loses. For which values of $n$ does the first player have a winning strategy?
1994 National High School Mathematics League, 1
In the equation $x^2+z_1x+z_2+m=0$, $z_1,z_2,m$ are complex numbers, and $z_1^2-4z_2=16+20\text{i}$. Two roots of the equations are $\alpha,\beta$. If $|\alpha-\beta|=2\sqrt7$, find the maximum and minumum value of $|m|$.
2017 HMNT, 8
Find the number of quadruples $(a, b, c, d)$ of integers with absolute value at most $5$ such that $$(a^2 + b^2 + c^2 + d^2)^2 = (a + b + c + d)(a - b + c -d)((a - c)^2 + (b - d)^2).$$
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?