Found problems: 85335
2020 HMNT (HMMO), 4
Points $G$ and $N$ are chosen on the interiors of sides $ED$ and $DO$ of unit square $DOME$, so that pentagon $GNOME$ has only two distinct side lengths. The sum of all possible areas of quadrilateral $NOME$ can be expressed as $\frac{a-b\sqrt{c}}{d}$, where $a,b,c,d$ are positive integers such that $\gcd(a,b,d) = 1$ and $c$ is square-free (i.e. no perfect square greater than $1$ divides $c$). Compute $1000a+100b+10c+d$.
2016 Japan MO Preliminary, 7
Let $a, b, c, d$ be real numbers satisfying the system of equation
$\[(a+b)(c+d)=2 \\
(a+c)(b+d)=3 \\
(a+d)(b+c)=4\]$
Find the minimum value of $a^2+b^2+c^2+d^2$.
2010 ELMO Shortlist, 1
For a permutation $\pi$ of $\{1,2,3,\ldots,n\}$, let $\text{Inv}(\pi)$ be the number of pairs $(i,j)$ with $1 \leq i < j \leq n$ and $\pi(i) > \pi(j)$.
[list=1]
[*] Given $n$, what is $\sum \text{Inv}(\pi)$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?
[*] Given $n$, what is $\sum \left(\text{Inv}(\pi)\right)^2$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?[/list]
[i]Brian Hamrick.[/i]
2012 AIME Problems, 3
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible comittees that can be formed subject to these requirements.
2014 Contests, 1
Compute $1+2\cdot3^4$.
[i]Proposed by Evan Chen[/i]
2019 Middle European Mathematical Olympiad, 8
Let $N$ be a positive integer such that the sum of the squares of all positive divisors of $N$ is equal to the product $N(N+3)$. Prove that there exist two indices $i$ and $j$ such that $N=F_iF_j$ where $(F_i)_{n=1}^{\infty}$ is the Fibonacci sequence defined as $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$.
[i]Proposed by Alain Rossier, Switzerland[/i]
2018 USAMTS Problems, 2:
Let $n>1$ be an integer. There are $n$ orangutoads, conveniently numbered $1,2,\dots{},n$, each sitting at an integer position on the number line. They take turns moving in the order $1,2,\dots{},n$, and then going back to $1$ to start the process over; they stop if any orangutoad is ever unable to move. To move, an orangutoad chooses another orangutoad who is at least $2$ units away from her towards them by a a distance of $1$ unit. (Multiple orangutoads can be at the same position.) Show that eventually some orangutoad will be unable to move.
2016 Moldova Team Selection Test, 11
Let $ABCD$ be a cyclic quadrilateral. Circle with diameter $AB$ intersects $CA$, $CB$, $DA$, and $DB$ in $E$, $F$, $G$, and $H$, respectively (all different from $A$ and $B$). The lines $EF$ and $GH$ intersect in $I$. Prove that the bisector of $\angle GIF$ and the line $CD$ are perpendicular.
2009 Oral Moscow Geometry Olympiad, 3
In the triangle $ABC$, $AA_1$ and $BB_1$ are altitudes. On the side $AB$ , points $M$ and $K$ are selected so that $B_1K \parallel BC$ and $A_1M \parallel AC$. Prove that the angle $AA_1K$ is equal to the angle $BB_1M$.
(D. Prokopenko)
2021 MOAA, 10
For how many nonempty subsets $S \subseteq \{1, 2, \ldots , 10\}$ is the sum of all elements in $S$ even?
[i]Proposed by Andrew Wen[/i]
EGMO 2017, 4
Let $n\geq1$ be an integer and let $t_1<t_2<\dots<t_n$ be positive integers. In a group of $t_n+1$ people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following two conditions to hold at the same time:
(i) The number of games played by each person is one of $t_1,t_2,\dots,t_n$.
(ii) For every $i$ with $1\leq i\leq n$, there is someone who has played exactly $t_i$ games of chess.
2004 Federal Math Competition of S&M, 1
In a triangle $ABC$ of the area $S$, point $H$ is the orthocenter, $D,E,F$ are the feet of the altitudes from $A,B,C$, and $P,Q,R$ are the reflections of $A,B,C$ in $BC,CA,AB$, respectively. The triangles $DEF$ and $PQR$ have the same area $T$. Given that $T > \frac{3}{5}S$, prove that $T = S$.
2009 Indonesia TST, 3
Find all triples $ (x,y,z)$ of positive real numbers which satisfy
$ 2x^3 \equal{} 2y(x^2 \plus{} 1) \minus{} (z^2 \plus{} 1)$;
$ 2y^4 \equal{} 3z(y^2 \plus{} 1) \minus{} 2(x^2 \plus{} 1)$;
$ 2z^5 \equal{} 4x(z^2 \plus{} 1) \minus{} 3(y^2 \plus{} 1)$.
1952 AMC 12/AHSME, 4
The cost $ C$ of sending a parcel post package weighing $ P$ pounds, $ P$ and integer, is $ 10$ cents for the first pound and $ 3$ cents for each additional pound. The formula for the cost is:
$ \textbf{(A)}\ C \equal{} 10 \plus{} 3P \qquad\textbf{(B)}\ C \equal{} 10P \plus{} 3 \qquad\textbf{(C)}\ C \equal{} 10 \plus{} 3(P \minus{} 1)$
$ \textbf{(D)}\ C \equal{} 9 \plus{} 3P \qquad\textbf{(E)}\ C \equal{} 10P \minus{} 7$
1991 Baltic Way, 6
Solve the equation $[x] \cdot \{x\} = 1991x$. (Here $[x]$ denotes the greatest integer less than or equal to $x$, and $\{x\}=x-[x]$.)
1998 National High School Mathematics League, 14
Function $f(x)=ax^2+8x+3(a<0)$. For any given nerative number $a$, define the largest positive number $l(a)$: $|f(x)|\leq5$ for all $x\in[0,l(a)]$.
Find the largest $l(a)$, and $a$ when $l(a)$ takes its maximum value.
2025 Ukraine National Mathematical Olympiad, 10.2
Given $12$ segments, it is known that they can be divided into $4$ groups of $3$ segments each in such a way that a triangle can be formed from the segments of each triplet. Is it always possible to divide these $12$ segments into $3$ groups of $4$ segments each in such a way that a quadrilateral can be formed from the segments of each quartet?
[i]Proposed by Mykhailo Shtandenko[/i]
2021 CMIMC, 2
Let $p_1, p_2, p_3, p_4, p_5, p_6$ be distinct primes greater than $5$. Find the minimum possible value of $$p_1 + p_2 + p_3 + p_4 + p_5 + p_6 - 6\min\left(p_1, p_2, p_3, p_4, p_5, p_6\right)$$
[i]Proposed by Oliver Hayman[/i]
2008 Costa Rica - Final Round, 5
Let $ p$ be a prime number such that $ p\minus{}1$ is a perfect square. Prove that the equation
$ a^{2}\plus{}(p\minus{}1)b^{2}\equal{}pc^{2}$
has infinite many integer solutions $ a$, $ b$ and $ c$ with $ (a,b,c)\equal{}1$
1996 AIME Problems, 14
In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.
2005 Colombia Team Selection Test, 1
Let $a,b,c$ be integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$ prove that $abc$ is a perfect cube!
2010 Kazakhstan National Olympiad, 3
Call $A \in \mathbb{N}^0$ be $number of year$ if all digits of $A$ equals $0$, $1$ or $2$ (in decimal representation).
Prove that exist infinity $N \in \mathbb{N}$, such that $N$ can't presented as $A^2+B$ where $A \in \mathbb{N}^0 ; B$- $number of year$.
2013 ELMO Shortlist, 8
We define the [i]Fibonacci sequence[/i] $\{F_n\}_{n\ge0}$ by $F_0=0$, $F_1=1$, and for $n\ge2$, $F_n=F_{n-1}+F_{n-2}$; we define the [i]Stirling number of the second kind[/i] $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets.
For every positive integer $n$, let $t_n = \sum_{k=1}^{n} S(n,k) F_k$. Let $p\ge7$ be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all $n\ge1$.
[i]Proposed by Victor Wang[/i]
2022 Harvard-MIT Mathematics Tournament, 4
Compute the number of nonempty subsets $S \subseteq\{-10,-9,-8, . . . , 8, 9, 10\}$ that satisfy $$|S| +\ min(S) \cdot \max (S) = 0.$$
2005 Singapore MO Open, 4
Place 2005 points on the circumference of a circle. Two points $P,Q$ are said to form a pair of neighbours if the chord $PQ$ subtends an angle of at most 10 degrees at the centre. Find the smallest number of pairs of neighbours.