Found problems: 85335
1941 Moscow Mathematical Olympiad, 083
Consider $\vartriangle ABC$ and a point $M$ inside it. We move $M$ parallel to $BC$ until $M$ meets $CA$, then parallel to $AB$ until it meets $BC$, then parallel to $CA$, and so on. Prove that $M$ traverses a self-intersecting closed broken line and find the number of its straight segments.
1980 AMC 12/AHSME, 13
A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to $(1,0)$. Then it makes a $90^\circ$ counterclockwise and travels $\frac 12$ a unit to $\left(1, \frac 12 \right)$. If it continues in this fashion, each time making a $90^\circ$ degree turn counterclockwise and traveling half as far as the previous move, to which of the following points will it come closest?
$\text{(A)} \ \left(\frac 23, \frac 23 \right) \qquad \text{(B)} \ \left( \frac 45, \frac 25 \right) \qquad \text{(C)} \ \left( \frac 23, \frac 45 \right) \qquad \text{(D)} \ \left(\frac 23, \frac 13 \right) \qquad \text{(E)} \ \left(\frac 25, \frac 45 \right)$
2014 Silk Road, 1
What is the maximum number of coins can be arranged in cells of the table $n \times n$ (each cell is not more of the one coin) so that any coin was not simultaneously below and to the right than any other?
2021 AMC 10 Fall, 14
How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations?
\begin{align*}
x^2+3y&=9\\
(|x|+|y|-4)^2&=1\\
\end{align*}
$\textbf{(A)}\: 1\qquad\textbf{(B)} \: 2\qquad\textbf{(C)} \: 3\qquad\textbf{(D)} \: 5\qquad\textbf{(E)} \: 7$
2025 Taiwan Mathematics Olympiad, 3
For any pair of coprime positive integers $a$ and $b$, define $f(a, b)$ to be the smallest nonnegative integer $k$ such that $b \mid ak+1$. Prove that if a and b are coprime positive integers satisfying
$$f(a, b) - f(b, a) = 2,$$
then there exists a prime number $p$ such that $p^2\mid a + b$.
[i]Proposed by usjl[/i]
2016-2017 SDML (Middle School), 12
What is the area of the region enclosed by the graph of the equations $x^2 - 14x + 3y + 70 = 21 + 11y - y^2$ that lies below the line $y = x-3$?
$\text{(A) }6\pi\qquad\text{(B) }7\pi\qquad\text{(C) }8\pi\qquad\text{(D) }9\pi\qquad\text{(E) }10\pi$
1984 AMC 12/AHSME, 7
When Dave walks to school, he averages 90 steps per minute, each of his steps 75cm long. It takes him 16 minutes to get to school. His brother, Jack, going to the same school by the same route, averages 100 steps per minute, but his steps are only 60 cm long. How long does it take Jack to get to school?
$\textbf{(A) }14 \frac{2}{9}\qquad
\textbf{(B) }15\qquad
\textbf{(C) }18\qquad
\textbf{(D) }20\qquad
\textbf{(E) }22 \frac{2}{9}$
2025 AMC 8, 3
Buffalo Shuffle-o is a card game in which all the cards are distributed evenly among all players at the start of the game. When Annika and $3$ of her friends play Buffalo Shuffle-o, each player is dealt $15$ cards. Suppose $2$ more friends join the next game. How many cards will be dealt to each player?
$\textbf{(A) } 8\qquad\textbf{(B) } 9\qquad\textbf{(C) } 10\qquad\textbf{(D) } 11\qquad\textbf{(E) } 12$
ngl easily silliable
2013 Harvard-MIT Mathematics Tournament, 19
An isosceles trapezoid $ABCD$ with bases $AB$ and $CD$ has $AB=13$, $CD=17$, and height $3$. Let $E$ be the intersection of $AC$ and $BD$. Circles $\Omega$ and $\omega$ are circumscribed about triangles $ABE$ and $CDE$. Compute the sum of the radii of $\Omega$ and $\omega$.
2025 Serbia Team Selection Test for the BMO 2025, 5
In Mexico, there live $n$ Mexicans, some of whom know each other. They decided to play a game. On the first day, each Mexican wrote a non-negative integer on their forehead. On each following day, they changed their number according to the following rule: On day $i+1$, each Mexican writes on their forehead the smallest non-negative integer that did not appear on the forehead of any of their acquaintances on day $i$. It is known that on some day every Mexican wrote the same number as on the previous day, after which they decided to stop the game. Determine the maximum number of days this game could have lasted.
[i]Proposed by Pavle Martinović[/i]
2024 JHMT HS, 8
Let $N_7$ be the answer to problem 7.
Each side of a regular $N_7$-gon is colored with a single color from a set of two given colors. Two colorings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. Compute the number of possible different colorings.
2006 MOP Homework, 1
Let $n$ be an integer greater than $1$, and let $a_1$, $a_2$, ..., $a_n$ be not all identical positive integers. Prove that there are infinitely many primes $p$ such that $p$ divides $a_1^k+a_2^k+...+a_n^k$ for some positive integer $k$.
2015 Online Math Open Problems, 27
Let $ABCD$ be a quadrilateral satisfying $\angle BCD=\angle CDA$. Suppose rays $AD$ and $BC$ meet at $E$, and let $\Gamma$ be the circumcircle of $ABE$. Let $\Gamma_1$ be a circle tangent to ray $CD$ past $D$ at $W$, segment $AD$ at $X$, and internally tangent to $\Gamma$. Similarly, let $\Gamma_2$ be a circle tangent to ray $DC$ past $C$ at $Y$, segment $BC$ at $Z$, and internally tangent to $\Gamma$. Let $P$ be the intersection of $WX$ and $YZ$, and suppose $P$ lies on $\Gamma$. If $F$ is the $E$-excenter of triangle $ABE$, and $AB=544$, $AE=2197$, $BE=2299$, then find $m+n$, where $FP=\tfrac{m}{n}$ with $m,n$ relatively prime positive integers.
[i]Proposed by Michael Kural[/i]
2023 ELMO Shortlist, N5
An ordered pair \((k,n)\) of positive integers is [i]good[/i] if there exists an ordered quadruple \((a,b,c,d)\) of positive integers such that \(a^3+b^k=c^3+d^k\) and \(abcd=n\). Prove that there exist infinitely many positive integers \(n\) such that \((2022,n)\) is not good but \((2023,n)\) is good.
[i]Proposed by Luke Robitaille[/i]
2010 All-Russian Olympiad, 4
There are 100 apples on the table with total weight of 10 kg. Each apple weighs no less than 25 grams. The apples need to be cut for 100 children so that each of the children gets 100 grams. Prove that you can do it in such a way that each piece weighs no less than 25 grams.
2017 Peru IMO TST, 12
Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\] Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.
1991 Vietnam National Olympiad, 3
Three mutually perpendicular rays $O_x,O_y,O_z$ and three points $A,B,C$ on $O_x,O_y,O_z$, respectively. A variable sphere є through $A, B,C$ meets $O_x,O_y,O_z$ again at $A', B',C'$, respectively. Let $M$ and $M'$ be the centroids of triangles $ABC$ and $A'B'C'$. Find the locus of the midpoint of $MM'$.
2009 BAMO, 1
A square grid of $16$ dots (see the figure) contains the corners of nine $1\times1$ squares, four $2\times 2$ squares, and one $3\times3$ square, for a total of $14$ squares whose sides are parallel to the sides of the grid. What is the smallest possible number of dots you can remove so that, after removing those dots, each of the $14$ squares is missing at least one corner?
Justify your answer by showing both that the number of dots you claim is sufficient and by explaining why no smaller number of dots will work.
[img]https://cdn.artofproblemsolving.com/attachments/0/9/bf091a769dbec40eceb655f5588f843d4941d6.png[/img]
1998 Israel National Olympiad, 5
(a) Find two real numebrs $a,b$ such that $|ax+b-\sqrt{x}| \le \frac{1}{24}$ for $1 \le x \le 4$.
(b) Prove that the constant $\frac{1}{24}$ cannot be replaced by a smaller one.
2025 Ukraine National Mathematical Olympiad, 9.6
The sum of $10$ positive integer numbers is equal to $300$. The product of their factorials is a perfect tenth power of some positive integer. Prove that all $10$ numbers are equal to each other.
[i]Proposed by Pavlo Protsenko[/i]
1970 Putnam, A1
Show that the power series for the function
$$e^{ax} \cos bx,$$
where $a,b >0$, has either no zero coefficients or infinitely many zero coefficients.
2018 Saint Petersburg Mathematical Olympiad, 6
Let $a,b,c,d>0$ . Prove that $a^4+b^4+c^4+d^4 \geq 4abcd+4(a-b)^2 \sqrt{abcd}$
2022 IOQM India, 11
In how many ways can four married couples sit in a merry-go-round with identical seats such that men and women occupy alternate seats and no husband seats next to his wife?
1996 Moscow Mathematical Olympiad, 3
At the nodes of graph paper, gardeners live; everywhere around them grow flowers. Each flower is to be taken care of by the three gardeners nearest to it. One of the gardeners wishes to know which are the flowers (s)he has to take care of. Sketch the plot of these gardeners.
Proposed by I. F. Sharygin
2023 Germany Team Selection Test, 3
Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple:
\begin{align*}
\mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\
\mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022}))
\end{align*}
and then write this tuple on the blackboard.
It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?