Found problems: 85335
2007 AIME Problems, 4
The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 whoosits. In two hours, 60 workers can produce 240 widgets and 300 whoosits. In three hours, 50 workers can produce 150 widgets and m whoosits. Find m.
2009 Indonesia TST, 1
Ati has $ 7$ pots of flower, ordered in $ P_1,P_2,P_3,P_4,P_5,P_6,P_7$. She wants to rearrange the position of those pots to $ B_1,B_2,B_2,B_3,B_4,B_5,B_6,B_7$ such that for every positive integer $ n<7$, $ B_1,B_2,\dots,B_n$ is not the permutation of $ P_1,P_2,\dots,P_7$. In how many ways can Ati do this?
2015 239 Open Mathematical Olympiad, 6
Positive real numbers $a,b,c$ satisfy $$2a^3b+2b^3c+2c^3a=a^2b^2+b^2c^2+c^2a^2.$$
Prove that $$2ab(a-b)^2+2bc(b-c)^2+2ca(c-a)^2 \geq(ab+bc+ca)^2.$$
LMT Team Rounds 2010-20, 2020.S22
The numbers one through eight are written, in that order, on a chalkboard. A mysterious higher power in possession of both an eraser and a piece of chalk chooses three distinct numbers $x$, $y$, and $z$ on the board, and does the following. First, $x$ is erased and replaced with $y$, after which $y$ is erased and replaced with $z$, and finally $z$ is erased and replaced with $x$. The higher power repeats this process some finite number of times. For example, if $(x,y,z)=(2,4,5)$ is chosen, followed by $(x,y,z)=(1,4,3)$, the board would change in the following manner:
\[12345678 \rightarrow 14352678 \rightarrow 43152678\]
Compute the number of possible final orderings of the eight numbers.
2012-2013 SDML (Middle School), 1
On planet Polyped, every creature has either $6$ legs or $10$ legs. In a room with $20$ creatures and $156$ legs, how many of the creatures have $6$ legs?
Kyiv City MO Juniors Round2 2010+ geometry, 2019.9.31
A circle $k$ of radius $r$ is inscribed in $\vartriangle ABC$, tangent to the circle $k$, which are parallel respectively to the sides $AB, BC$ and $CA$ intersect the other sides of $\vartriangle ABC$ at points $M, N; P, Q$ and $L, T$ ($P, T \in AB$, $L, N \in BC$ and $M, Q\in AC$). Denote by $r_1,r_2,r_3$ the radii of inscribed circles in triangles $MNC, PQA$ and $LTB$. Prove that $r_1+r_2+r_3=r$.
2016 ASDAN Math Tournament, 5
In the following diagram, the square and pentagon are both regular and share segment $AB$ as designated. What is the measure of $\angle CBD$ in degrees?
2021 Alibaba Global Math Competition, 5
For the complex-valued function $f(x)$ which is continuous and absolutely integrable on $\mathbb{R}$, define the function $(Sf)(x)$ on $\mathbb{R}$: $(Sf)(x)=\int_{-\infty}^{+\infty}e^{2\pi iux}f(u)du$.
(a) Find the expression for $S(\frac{1}{1+x^2})$ and $S(\frac{1}{(1+x^2)^2})$.
(b) For any integer $k$, let $f_k(x)=(1+x^2)^{-1-k}$. Assume $k\geq 1$, find constant $c_1$, $c_2$ such that the function $y=(Sf_k)(x)$ satisfies the ODE with second order: $xy''+c_1y'+c_2xy=0$.
2023 Math Prize for Girls Problems, 8
For a positive integer $n$, let $p(n)$ denote the number of distinct prime numbers that divide evenly into $n$. Determine the number of solutions, in positive integers $n$, to the inequality $\log_4 n \le p(n)$.
2020 Olympic Revenge, 3
Let $ABC$ be a triangle and $\omega$ its circumcircle. Let $D$ and $E$ be the feet of the angle bisectors relative to $B$ and $C$, respectively. The line $DE$ meets $\omega$ at $F$ and $G$. Prove that the tangents to $\omega$ through $F$ and $G$ are tangents to the excircle of $\triangle ABC$ opposite to $A$.
2024 BAMO, 5
An underground burrow consists of an infinite sequence of rooms labeled by the integers $(\dots, -3, -2, -1, 0, 1, 2, 3,\dots)$. Initially, some of the rooms are occupied by one or more rabbits. Each rabbit wants to be alone. Thus, if there are two or more rabbits in the same room (say, room $m$), half of the rabbits (rounding down) will flee to room $m-1$, and half (also rounding down) to room $m+1$. Once per minute, this happens simultaneously in all rooms that have two or more rabbits. For example, if initially all rooms are empty except for $5$ rabbits in room $\#12$ and $2$ rabbits in room $\#13$, then after one minute, rooms $\text{\#11--\#14}$ will contain $2$, $2$, $2$, and $1$ rabbits, respectively, and all other rooms will be empty.
Now suppose that initially there are $k+1$ rabbits in room $k$ for each $k=0, 1, 2, \ldots, 9, 10$, and all other rooms are empty.
[list=a]
[*]Show that eventually the rabbits will stop moving.
[*] Determine which rooms will be occupied when this occurs.
[/list]
2015 Costa Rica - Final Round, G1
Points $A, B, C$ are vertices of an equilateral triangle inscribed in a circle. Point $D$ lies on the shorter arc $\overarc {AB}$ . Prove that $AD + BD = DC$.
2011 Baltic Way, 16
Let $a$ be any integer. Define the sequence $x_0,x_1,\ldots$ by $x_0=a$, $x_1=3$, and for all $n>1$
\[x_n=2x_{n-1}-4x_{n-2}+3.\]
Determine the largest integer $k_a$ for which there exists a prime $p$ such that $p^{k_a}$ divides $x_{2011}-1$.
2021 Thailand Online MO, P6
Let $m<n$ be two positive integers and $x_m<x_{m+1}<\cdots<x_n$ be a sequence of rational numbers. Suppose that $kx_k$ is an integer for all integers $k$ which $m\leq k\leq n$. Prove that
$$x_n-x_m\geq \frac{1}{m}-\frac{1}{n}.$$
2023 China Girls Math Olympiad, 7
Let $p$ be an odd prime. Suppose that positive integers $a,b,m,r$ satisfy $p\nmid ab$ and $ab > m^2$. Prove that there exists at most one pair of coprime positive integers $(x,y)$ such that $ax^2+by^2=mp^r$.
2006 IMC, 5
Show that there are an infinity of integer numbers $m,n$, with $gcd(m,n)=1$ such that the equation $(x+m)^{3}=nx$ has 3 different integer sollutions.
Novosibirsk Oral Geo Oly VIII, 2021.1
Cut the $9 \times 10$ grid rectangle along the grid lines into several squares so that there are exactly two of them with odd sidelengths.
2009 Stanford Mathematics Tournament, 3
Given a regular pentagon, find the ratio of its diagonal, $d$, to its side, $a$
2019 Turkey EGMO TST, 3
Let $\omega$ be the circumcircle of $\Delta ABC$, where $|AB|=|AC|$. Let $D$ be any point on the minor arc $AC$. Let $E$ be the reflection of point $B$ in line $AD$. Let $F$ be the intersection of $\omega$ and line $BE$ and Let $K$ be the intersection of line $AC$ and the tangent at $F$. If line $AB$ intersects line $FD$ at $L$, Show that $K,L,E$ are collinear points
1980 AMC 12/AHSME, 20
A box contains 2 pennies, 4 nickels, and 6 dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least 50 cents?
$\text{(A)} \ \frac{37}{924} \qquad \text{(B)} \ \frac{91}{924} \qquad \text{(C)} \ \frac{127}{924} \qquad \text{(D)} \ \frac{132}{924} \qquad \text{(E)} \ \text{none of these}$
2013 Polish MO Finals, 3
Given is a quadrilateral $ABCD$ in which we can inscribe circle. The segments $AB, BC, CD$ and $DA$ are the diameters of the circles $o1, o2, o3$ and $o4$, respectively. Prove that there exists a circle tangent to all of the circles $o1, o2, o3$ and $o4$.
1974 AMC 12/AHSME, 8
What is the smallest prime number dividing the sum $ 3^{11} \plus{} 5^{13}$?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 3^{11} \plus{} 5^{13}\qquad\textbf{(E)}\ \text{none of these}$
2022 MMATHS, 6
Prair writes the letters $A,B,C,D$, and $E$ such that neither vowel are written first, and they are not adjacent; such that there exists at least one pair of adjacent consonants; and such that exactly five pairs of letters are in alphabetical order. How many possible ways could Prair have ordered the letters?
Kvant 2023, M2740
Let $a, b, c$ be positive integers such that no number divides some other number. If $ab-b+1 \mid abc+1$, prove that $c \geq b$.
1957 AMC 12/AHSME, 30
The sum of the squares of the first $ n$ positive integers is given by the expression $ \frac{n(n \plus{} c)(2n \plus{} k)}{6}$, if $ c$ and $ k$ are, respectively:
$ \textbf{(A)}\ {1}\text{ and }{2} \qquad
\textbf{(B)}\ {3}\text{ and }{5}\qquad
\textbf{(C)}\ {2}\text{ and }{2}\qquad
\textbf{(D)}\ {1}\text{ and }{1}\qquad
\textbf{(E)}\ {2}\text{ and }{1}$