Found problems: 85335
1965 Spain Mathematical Olympiad, 6
We have an empty equilateral triangle with length of a side $l$. We put the triangle, horizontally, over a sphere of radius $r$. Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of $l$ and $r$)?
2005 All-Russian Olympiad, 2
Find the number of subsets $A\subset M=\{2^0,\,2^1,\,2^2,\dots,2^{2005}\}$ such that equation $x^2-S(A)x+S(B)=0$ has integral roots, where $S(M)$ is the sum of all elements of $M$, and $B=M\setminus A$ ($A$ and $B$ are not empty).
2021 Purple Comet Problems, 10
Find the value of $n$ such that the two inequalities
$$|x + 47| \le n \,\,\, and \,\,\, \frac{1}{17} \le \frac{4}{3 - x} \le \frac{1}{8}$$
have the same solutions.
2016 Regional Olympiad of Mexico Center Zone, 2
There are seven piles with $2014$ pebbles each and a pile with $2008$ pebbles. Ana and Beto play in turns and Ana always plays first. One move consists of removing pebbles from all the piles. From each pile is removed a different amount of pebbles, between $1$ and $8$ pebbles. The first player who cannot make a move loses.
a) Who has a winning strategy?
b) If there were seven piles with $2015$ pebbles each and a pile with $2008$ pebbles, who has a winning strategy?
1998 China National Olympiad, 2
Given a positive integer $n>1$, determine with proof if there exist $2n$ pairwise different positive integers $a_1,\ldots ,a_n,b_1,\ldots b_n$ such that $a_1+\ldots +a_n=b_1+\ldots +b_n$ and
\[n-1>\sum_{i=1}^{n}\frac{a_i-b_i}{a_i+b_i}>n-1-\frac{1}{1998}.\]
2014 Thailand TSTST, 2
Prove that the equation $x^8 = n! + 1$ has finitely many solutions in positive integers.
2014 Purple Comet Problems, 7
Andrea is three times as old as Jim was when Jim was twice as old as he was when the sum of their ages was $47$. If Andrea is $29$ years older than Jim, what is the sum of their ages now?
1971 Bundeswettbewerb Mathematik, 4
Let $P$ and $Q$ be two horizontal neighbouring squares on a $n \times n$ chess board, $P$ on the left and $Q$ on the right. On the left square $P$ there is a stone that shall be moved around the board. The following moves are allowed:
1) move it one square upwards
2) move it one square to the right
3) move it one square down and one square to the left (diagonal movement)
Example: you can get from $e5$ to $f5$, $e6$ and $d4$.
Show that for no $n$ there is tour visting every square exactly once and ending in $Q$.
2019 NMTC Junior, 2
Given positive real numbers $a, b, c, d$ such that $cd=1$. Prove that there exists at least one positive integer $m$ such that $$ab\le m^2\le (a+c) (b+d). $$
2009 National Olympiad First Round, 30
How many of
$ 11^2 \plus{} 13^2 \plus{} 17^2$, $ 24^2 \plus{} 25^2 \plus{} 26^2$, $ 12^2 \plus{} 24^2 \plus{} 36^2$, $ 11^2 \plus{} 12^2 \plus{} 132^2$ are perfect square ?
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 d)1 \qquad\textbf{(E)}\ 0$
2016 Bulgaria JBMO TST, 2
a, b, c are positive real numbers and a+b+c=k. Find the minimum value of $ b^2/(ka+bc)^1/2+c^2/(kb+ac)^1/2+a^2/(kc+ab)^1/2 $
2022 Math Prize for Girls Problems, 5
Given a real number $a$, the [i]floor[/i] of $a$, written $\lfloor a \rfloor$, is the greatest integer less than or equal to $a$. For how many real numbers $x$ such that $1 \le x \le 20$ is
\[
x^2 + \lfloor 2x \rfloor = \lfloor x^2 \rfloor + 2x \, ?
\]
2018 Math Prize for Girls Problems, 14
Let $f(x)$ be the polynomial $\prod_{k=1}^{50} \bigl( x - (2k-1) \bigr)$. Let $c$ be the coefficient of $x^{48}$ in $f(x)$. When $c$ is divided by 101, what is the remainder? (The remainder is an integer between 0 and 100.)
1991 IMO Shortlist, 11
Prove that $ \sum_{k \equal{} 0}^{995} \frac {( \minus{} 1)^k}{1991 \minus{} k} {1991 \minus{} k \choose k} \equal{} \frac {1}{1991}$
1975 Polish MO Finals, 5
Show that it is possible to circumscribe a circle of radius $R$ about, and inscribe a circle of radius $r$ in some triangle with one angle equal to $a$, if and only if $$\frac{2R}{r} \ge \dfrac{1}{ \sin \frac{a}{2} \left(1- \sin \frac{a}{2} \right)}$$
1958 AMC 12/AHSME, 12
If $ P \equal{} \frac{s}{(1 \plus{} k)^n}$ then $ n$ equals:
$ \textbf{(A)}\ \frac{\log{\left(\frac{s}{P}\right)}}{\log{(1 \plus{} k)}}\qquad
\textbf{(B)}\ \log{\left(\frac{s}{P(1 \plus{} k)}\right)}\qquad
\textbf{(C)}\ \log{\left(\frac{s \minus{} P}{1 \plus{} k}\right)}\qquad \\
\textbf{(D)}\ \log{\left(\frac{s}{P}\right)} \plus{} \log{(1 \plus{} k)}\qquad
\textbf{(E)}\ \frac{\log{(s)}}{\log{(P(1 \plus{} k))}}$
2012 Serbia National Math Olympiad, 2
Find all natural numbers $a$ and $b$ such that \[a|b^2, \quad b|a^2 \mbox{ and } a+1|b^2+1.\]
2011 ELMO Shortlist, 2
Let $p\ge5$ be a prime. Show that
\[\sum_{k=0}^{(p-1)/2}\binom{p}{k}3^k\equiv 2^p - 1\pmod{p^2}.\]
[i]Victor Wang.[/i]
1990 IMO Longlists, 15
Given a triangle $ ABC$. Let $ G$, $ I$, $ H$ be the centroid, the incenter and the orthocenter of triangle $ ABC$, respectively. Prove that $ \angle GIH > 90^{\circ}$.
2021 Stanford Mathematics Tournament, R4
[b]p13.[/b] Emma has the five letters: $A, B, C, D, E$. How many ways can she rearrange the letters into words? Note that the order of words matter, ie $ABC DE$ and $DE ABC$ are different.
[b]p14.[/b] Seven students are doing a holiday gift exchange. Each student writes their name on a slip of paper and places it into a hat. Then, each student draws a name from the hat to determine who they will buy a gift for. What is the probability that no student draws himself/herself?
[b]p15.[/b] We model a fidget spinner as shown below (include diagram) with a series of arcs on circles of radii $1$. What is the area swept out by the fidget spinner as it’s turned $60^o$ ?
[img]https://cdn.artofproblemsolving.com/attachments/9/8/db27ffce2af68d27eee5903c9f09a36c2a6edf.png[/img]
[b]p16.[/b] Let $a,b,c$ be the sides of a triangle such that $gcd(a, b) = 3528$, $gcd(b, c) = 1008$, $gcd(a, c) = 504$. Find the value of $a * b * c$. Write your answer as a prime factorization.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2017-2018 SDML (Middle School), 12
If $n$ is an integer such that $2 \leq n \leq 2017$, for how many values of $n$ is $\left(1 + \frac{1}{2}\right)\left(1 + \frac{1}{3}\right)\cdots\left(1 + \frac{1}{n}\right)$ equal to a positive integer?
$\mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 1 \qquad \mathrm {(C) \ } 1007 \qquad \mathrm{(D) \ } 1008 \qquad \mathrm{(E) \ } 2016$
2000 All-Russian Olympiad, 7
Let $E$ be a point on the median $CD$ of a triangle $ABC$. The circle $\mathcal S_1$ passing through $E$ and touching $AB$ at $A$ meets the side $AC$ again at $M$. The circle $S_2$ passing through $E$ and touching $AB$ at $B$ meets the side $BC$ at $N$. Prove that the circumcircle of $\triangle CMN$ is tangent to both $\mathcal S_1$ and $\mathcal S_2$.
PEN O Problems, 18
Let $p$ be an odd prime number. How many $p$-element subsets $A$ of $\{1,2,\ldots \ 2p\}$ are there, the sum of whose elements is divisible by $p$?
PEN A Problems, 54
A natural number $n$ is said to have the property $P$, if whenever $n$ divides $a^{n}-1$ for some integer $a$, $n^2$ also necessarily divides $a^{n}-1$. [list=a] [*] Show that every prime number $n$ has the property $P$. [*] Show that there are infinitely many composite numbers $n$ that possess the property $P$. [/list]
2004 May Olympiad, 3
We have a pool table $8$ meters long and $2$ meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling $29$ meters, it stops at a corner of the table. How many times did the ball hit the edges of the table?
Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.