Found problems: 85335
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?
$\text{(A)} \ \frac{37}{924} \qquad \text{(B)} \ \frac{91}{924} \qquad \text{(C)} \ \frac{127}{924} \qquad \text{(D)} \ \frac{132}{924} \qquad \text{(E)} \ \text{none of these}$
1985 Tournament Of Towns, (085) 1
$a, b$ and $c$ are sides of a triangle, and $\gamma$ is its angle opposite $c$.
Prove that $c \ge (a + b) \sin \frac{\gamma}{2}$
(V. Prasolov )
2013 Purple Comet Problems, 29
You can tile a $2 \times5$ grid of squares using any combination of three types of tiles: single unit squares, two side by side unit squares, and three unit squares in the shape of an L. The diagram below shows the grid, the available tile shapes, and one way to tile the grid. In how many ways can the grid be tiled?
[asy]
import graph; size(15cm);
pen dps = linewidth(1) + fontsize(10); defaultpen(dps);
draw((-3,3)--(-3,1));
draw((-3,3)--(2,3));
draw((2,3)--(2,1));
draw((-3,1)--(2,1));
draw((-3,2)--(2,2));
draw((-2,3)--(-2,1));
draw((-1,3)--(-1,1));
draw((0,3)--(0,1));
draw((1,3)--(1,1));
draw((4,3)--(4,2));
draw((4,3)--(5,3));
draw((5,3)--(5,2));
draw((4,2)--(5,2));
draw((5.5,3)--(5.5,1));
draw((5.5,3)--(6.5,3));
draw((6.5,3)--(6.5,1));
draw((5.5,1)--(6.5,1));
draw((7,3)--(7,1));
draw((7,1)--(9,1));
draw((7,3)--(8,3));
draw((8,3)--(8,2));
draw((8,2)--(9,2));
draw((9,2)--(9,1));
draw((11,3)--(11,1));
draw((11,3)--(16,3));
draw((16,3)--(16,1));
draw((11,1)--(16,1));
draw((12,3)--(12,2));
draw((11,2)--(12,2));
draw((12,2)--(13,2));
draw((13,2)--(13,1));
draw((14,3)--(14,1));
draw((14,2)--(15,2));
draw((15,3)--(15,1));[/asy]
2016 Romania Team Selection Test, 2
Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$
1971 IMO Longlists, 51
Suppose that the sides $AB$ and $DC$ of a convex quadrilateral $ABCD$ are not parallel. On the sides $BC$ and $AD$, pairs of points $(M,N)$ and $(K,L)$ are chosen such that $BM=MN=NC$ and $AK=KL=LD$. Prove that the areas of triangles $OKM$ and $OLN$ are different, where $O$ is the intersection point of $AB$ and $CD$.
2007 Oral Moscow Geometry Olympiad, 5
Given triangle $ABC$. Points $A_1,B_1$ and $C_1$ are symmetric to its vertices with respect to opposite sides. $C_2$ is the intersection point of lines $AB_1$ and $BA_1$. Points$ A_2$ and $B_2$ are defined similarly. Prove that the lines $A_1 A_2, B_1 B_2$ and $C_1 C_2$ are parallel.
(A. Zaslavsky)
1979 Bulgaria National Olympiad, Problem 5
A convex pentagon $ABCDE$ satisfies $AB=BC=CA$ and $CD=DE=EC$. Let $S$ be the center of the equilateral triangle $ABC$ and $M$ and $N$ be the midpoints of $BD$ and $AE$, respectively. Prove that the triangles $SME$ and $SND$ are similar.
2015 Regional Olympiad of Mexico Center Zone, 2
In the triangle $ABC$, we have that $\angle BAC$ is acute. Let $\Gamma$ be the circle that passes through $A$ and is tangent to the side $BC$ at $C$. Let $M$ be the midpoint of $BC$ and let $D$ be the other point of intersection of $\Gamma$ with $AM$. If $BD$ cuts back to$ \Gamma$ at $E$, show that $AC$ is the bisector of $\angle BAE$.
2014 Junior Regional Olympiad - FBH, 2
Find value of $$\frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx}$$ if $x$, $y$ and $z$ are real numbers usch that $xyz=1$
1987 All Soviet Union Mathematical Olympiad, 442
It is known that, having $6$ weighs, it is possible to balance the scales with loads, which weights are successing natural numbers from $1$ to $63$. Find all such sets of weighs.
2019 IFYM, Sozopol, 6
Find all odd numbers $n\in \mathbb{N}$, for which the number of all natural numbers, that are no bigger than $n$ and coprime with $n$, divides $n^2+3$.
2023 Bulgarian Spring Mathematical Competition, 9.4
Given is a directed graph with $28$ vertices, such that there do not exist vertices $u, v$, such that $u \rightarrow v$ and $v \rightarrow u$. Every $16$ vertices form a directed cycle. Prove that among any $17$ vertices, we can choose $15$ which form a directed cycle.
2007 AIME Problems, 13
A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $\overline{AE}$, $\overline{BC}$, and $\overline{CD}$. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$.
MBMT Guts Rounds, 2018
[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide]
[u]Set 1[/u]
[b]C.1 / G.1[/b] Daniel is exactly one year younger than his friend David. If David was born in the year $2008$, in what year was Daniel born?
[b]C.2 / G.3[/b] Mr. Pham flips three coins. What is the probability that no two coins show the same side?
[b]C.3 / G.2[/b] John has a sheet of white paper which is $3$ cm in height and $4$ cm in width. He wants to paint the sky blue and the ground green so the entire paper is painted. If the ground takes up a third of the page, how much space (in cm$^2$) does the sky take up?
[b]C.4 / G.5[/b] Jihang and Eric are busy fidget spinning. While Jihang spins his fidget spinner at $15$ revolutions per second, Eric only manages $10$ revolutions per second. How many total revolutions will the two have made after $5$ continuous seconds of spinning?
[b]C.5 / G.4[/b] Find the last digit of $1333337777 \cdot 209347802 \cdot 3940704 \cdot 2309476091$.
[u]Set 2[/u]
[b]C.6[/b] Evan, Chloe, Rachel, and Joe are splitting a cake. Evan takes $\frac13$ of the cake, Chloe takes $\frac14$, Rachel takes $\frac15$, and Joe takes $\frac16$. There is $\frac{1}{x}$ of the original cake left. What is $x$?
[b]C.7[/b] Pacman is a $330^o$ sector of a circle of radius $4$. Pacman has an eye of radius $1$, located entirely inside Pacman. Find the area of Pacman, not including the eye.
[b]C.8[/b] The sum of two prime numbers $a$ and $b$ is also a prime number. If $a < b$, find $a$.
[b]C.9[/b] A bus has $54$ seats for passengers. On the first stop, $36$ people get onto an empty bus. Every subsequent stop, $1$ person gets off and $3$ people get on. After the last stop, the bus is full. How many stops are there?
[b]C.10[/b] In a game, jumps are worth $1$ point, punches are worth $2$ points, and kicks are worth $3$ points. The player must perform a sequence of $1$ jump, $1$ punch, and $1$ kick. To compute the player’s score, we multiply the 1st action’s point value by $1$, the $2$nd action’s point value by $2$, the 3rd action’s point value by $3$, and then take the sum. For example, if we performed a punch, kick, jump, in that order, our score would be $1 \times 2 + 2 \times 3 + 3 \times 1 = 11$. What is the maximal score the player can get?
[u]Set 3[/u]
[b]C.11[/b] $6$ students are sitting around a circle, and each one randomly picks either the number $1$ or $2$. What is the probability that there will be two people sitting next to each other who pick the same number?
[b]C.12 / G. 8[/b] You can buy a single piece of chocolate for $60$ cents. You can also buy a packet with two pieces of chocolate for $\$1.00$. Additionally, if you buy four single pieces of chocolate, the fifth one is free. What is the lowest amount of money you have to pay for $44$ pieces of chocolate? Express your answer in dollars and cents (ex. $\$3.70$).
[b]C.13 / G.12[/b] For how many integers $k$ is there an integer solution $x$ to the linear equation $kx + 2 = 14$?
[b]C.14 / G.9[/b] Ten teams face off in a swim meet. The boys teams and girls teams are ranked independently, each team receiving some number of positive integer points, and the final results are obtained by adding the points for the boys and the points for the girls. If Blair’s boys got $7$th place while the girls got $5$th place (no ties), what is the best possible total rank for Blair?
[b]C.15 / G.11[/b] Arlene has a square of side length $1$, an equilateral triangle with side length $1$, and two circles with radius $1/6$. She wants to pack her four shapes in a rectangle without items piling on top of each other. What is the minimum possible area of the rectangle?
PS. You should use hide for answers. C16-30/G10-15, G25-30 have been posted [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here[/url] and G16-25 [url=https://artofproblemsolving.com/community/c3h2790679p24540159]here [/url] . Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2010 AMC 12/AHSME, 18
A frog makes $ 3$ jumps, each exactly $ 1$ meter long. The directions of the jumps are chosen independently and at random. What is the probability the frog's final position is no more than $ 1$ meter from its starting position?
$ \textbf{(A)}\ \frac {1}{6} \qquad \textbf{(B)}\ \frac {1}{5} \qquad \textbf{(C)}\ \frac {1}{4} \qquad \textbf{(D)}\ \frac {1}{3} \qquad \textbf{(E)}\ \frac {1}{2}$
ICMC 7, 5
[list=a]
[*]Is there a non-linear integer-coefficient polynomial $P(x)$ and an integer $N{}$ such that all integers greater than $N{}$ may be written as the greatest common divisor of $P(a){}$ and $P(b){}$ for positive integers $a>b$?
[*]Is there a non-linear integer-coefficient polynomial $Q(x)$ and an integer $M{}$ such that all integers greater than $M{}$ may be written as $Q(a) - Q(b)$ for positive integers $a>b$?
[/list][i]Proposed by Dylan Toh[/i]
2009 South africa National Olympiad, 3
Ten girls, numbered from 1 to 10, sit at a round table, in a random order. Each girl then receives a new number, namely the sum of her own number and those of her two neighbours. Prove that some girl receives a new number greater than 17.
2002 District Olympiad, 1
a) Evaluate
\[\lim_{n\to \infty} \underbrace{\sqrt{a+\sqrt{a+\ldots+\sqrt{a+\sqrt{b}}}}}_{n\ \text{square roots}}\]
with $a,b>0$.
b)Let $(a_n)_{n\ge 1}$ and $(x_n)_{n\ge 1}$ such that $a_n>0$ and
\[x_n=\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}},\ \forall n\in \mathbb{N}^*\]
Prove that:
1) $(x_n)_{n\ge 1}$ is bounded if and only if $(a_n)_{n\ge 1}$ is bounded.
2) $(x_n)_{n\ge 1}$ is convergent if and only if $(a_n)_{n\ge 1}$ is convergent.
[i]Valentin Matrosenco[/i]
1984 AMC 12/AHSME, 6
In a certain school, there are three times as many boys as girls and nine times as many girls as teachers. Using the letters $b,g,t$ to represent the number of boys, girls, and teachers, respectively, then the total number of boys, girls, and teachers can be represented by the expression
$\textbf{(A) }31b\qquad
\textbf{(B) }\frac{37b}{27}\qquad
\textbf{(C) }13g\qquad
\textbf{(D) }\frac{37g}{27}\qquad
\textbf{(E) }\frac{37t}{27}$
2004 Irish Math Olympiad, 3
$AB$ is a chord of length $6$ of a circle centred at $O$ and of radius $5$. Let $PQRS$ denote the square inscribed in the sector $OAB$ such that $P$ is on the radius $OA$, $S$ is on the radius $OB$ and $Q$ and $R$ are points on the arc of the circle between $A$ and $B$. Find the area of $PQRS$.
2017 Iran MO (3rd round), 2
Two persons are playing the following game on a $n\times m$ table, with drawn lines:
Person $\#1$ starts the game. Each person in their move, folds the table on one of its lines. The one that could not fold the table on their turn loses the game.
Who has a winning strategy?
2016 IMAR Test, 4
A positive integer $m$ is perfect if the sum of all its positive divisors, $1$ and $m$ inclusive, is equal to $2m$. Determine the positive integers $n$ such that $n^n + 1$ is a perfect number.
2000 Belarus Team Selection Test, 6.2
A positive integer $A_k...A_1A_0$ is called monotonic if $A_k \le ..\le A_1 \le A_0$.
Show that for any $n \in N$ there is a monotonic perfect square with $n$ digits.
2014 Junior Balkan Team Selection Tests - Romania, 2
Determine all real numbers $x, y, z \in (0, 1)$ that satisfy simultaneously the conditions:
$(x^2 + y^2)\sqrt{1- z^2}\ge z$
$(y^2 + z^2)\sqrt{1- x^2}\ge x$
$(z^2 + x^2)\sqrt{1- y^2}\ge y$
1997 USAMO, 2
Let $ABC$ be a triangle. Take points $D$, $E$, $F$ on the perpendicular bisectors of $BC$, $CA$, $AB$ respectively. Show that the lines through $A$, $B$, $C$ perpendicular to $EF$, $FD$, $DE$ respectively are concurrent.