Found problems: 85335
2009 AMC 10, 21
Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle, In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?
[asy]unitsize(6mm);
defaultpen(linewidth(.8pt));
draw(Circle((0,0),1+sqrt(2)));
draw(Circle((sqrt(2),0),1));
draw(Circle((0,sqrt(2)),1));
draw(Circle((-sqrt(2),0),1));
draw(Circle((0,-sqrt(2)),1));[/asy]$ \textbf{(A)}\ 3\minus{}2\sqrt2 \qquad
\textbf{(B)}\ 2\minus{}\sqrt2 \qquad
\textbf{(C)}\ 4(3\minus{}2\sqrt2) \qquad
\textbf{(D)}\ \frac12(3\minus{}\sqrt2)$
$ \textbf{(E)}\ 2\sqrt2\minus{}2$
2020 LMT Fall, 10
$2020$ magicians are divided into groups of $2$ for the Lexington Magic Tournament. After every $5$ days, which is the duration of one match, teams are rearranged so no $2$ people are ever on the same team. If the longest tournament is $n$ days long, what is the value of $n?$
[i]Proposed by Ephram Chun[/i]
1965 All Russian Mathematical Olympiad, 067
a) A certain committee has gathered $40$ times. There were $10$ members on every meeting. Not a single couple has met on the meetings twice. Prove that there were no less then $60$ members in the committee.
b) Prove that you can not construct more then $30$ subcommittees of $5$ members from the committee of $25$ members, with no couple of subcommittees having more than one common member.
2017 Online Math Open Problems, 14
Let $ABC$ be a triangle, not right-angled, with positive integer angle measures (in degrees) and circumcenter $O$. Say that a triangle $ABC$ is [i]good[/i] if the following three conditions hold:
(a) There exists a point $P\neq A$ on side $AB$ such that the circumcircle of $\triangle POA$ is tangent to $BO$.
(b) There exists a point $Q\neq A$ on side $AC$ such that the circumcircle of $\triangle QOA$ is tangent to $CO$.
(c) The perimeter of $\triangle APQ$ is at least $AB+AC$.
Determine the number of ordered triples $(\angle A, \angle B,\angle C)$ for which $\triangle ABC$ is good.
[i]Proposed by Vincent Huang[/i]
2022 Lusophon Mathematical Olympiad, 3
The positive integers $x$ and $y$ are such that $x^{2022}+x+y^2$ is divisible by $xy$.
a) Give an example of such integers $x$ and $y$, with $x>y$.
b) Prove that $x$ is a perfect square.
2012 South East Mathematical Olympiad, 2
Find the least natural number $n$, such that the following inequality holds:$\sqrt{\dfrac{n-2011}{2012}}-\sqrt{\dfrac{n-2012}{2011}}<\sqrt[3]{\dfrac{n-2013}{2011}}-\sqrt[3]{\dfrac{n-2011}{2013}}$.
2005 Greece JBMO TST, 1
Examine if we can place $9$ convex $6$-angled polygons the one next to the other (with common only one side or part of her) to construct a convex $39$-angled polygon.
2003 National High School Mathematics League, 12
$M_n=\{\overline{0.a_1a_2\cdots a_n}|a_i\in{0,1},i=1,2,\cdots,n,a_n=1\}$. $T_n=|M_n|,S_n=\sum_{x\in M_n}x$, then $\lim_{n\to\infty}\frac{S_n}{T_n}=$________.
1957 Czech and Slovak Olympiad III A, 1
Find all real numbers $p$ such that the equation $$\sqrt{x^2-5p^2}=px-1$$ has a root $x=3$. Then, solve the equation for the determined values of $p$.
2018 Moscow Mathematical Olympiad, 5
On the sides of the convex hexagon $ABCDEF$ into the outer side were built equilateral triangles $ABC_1$, $BCD_1$, $CDE_1$, $DEF_1$, $EFA_1$ and $FAB_1$. The triangle $B_1D_1F_1$ is equilateral too. Prove that, the triangle $A_1C_1E_1$ is also equilateral.
2024 Simon Marais Mathematical Competition, A2
A positive integer $n$ is [i] tripariable [/i] if it is possible to partition the set $\{1, 2, \dots, n\}$ into disjoint pairs such that the sum of two elements in each pair is a power of $3$. For example $6$ is tripariable because $\{1, 2, \dots, n\}=\{1,2\}\cup\{3,6\}\cup\{4,5\}$ and $$1+2=3^1,\quad 3+6 = 3^2\quad\text{and}\quad4+5=3^2$$ are all powers of 3.
How many positive integers less than or equal to 2024 are tripariable?
2019 AIME Problems, 5
Four ambassadors and one advisor for each of them are to be seated at a round table with $12$ chairs numbered in order from $1$ to $12$. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are $N$ ways for the $8$ people to be seated at the table under these conditions. Find the remainder when $N$ is divided by $1000$.
1980 IMO, 4
Given a real number $x>1$, prove that there exists a real number $y >0$ such that
\[\lim_{n \to \infty} \underbrace{\sqrt{y+\sqrt {y + \cdots+\sqrt y}}}_{n \text{ roots}}=x.\]
2016 PUMaC Geometry B, 3
Let $ABCD$ be a square with side length $8$. Let $M$ be the midpoint of $BC$ and let $\omega$ be the circle passing through $M, A$, and $D$. Let $O$ be the center of $\omega, X$ be the intersection point (besides A) of $\omega$ with $AB$, and $Y$ be the intersection point of $OX$ and $AM$. If the length of $OY$ can be written in simplest form as $\frac{m}{n}$ , compute $m + n$.
1978 All Soviet Union Mathematical Olympiad, 256
Given two heaps of checkers. the bigger contains $m$ checkers, the smaller -- $n$ ($m>n$). Two players are taking checkers in turn from the arbitrary heap. The players are allowed to take from the heap a number of checkers (not zero) divisible by the number of checkers in another heap. The player that takes the last checker in any heap wins.
a) Prove that if $m > 2n$, than the first can always win.
b) Find all $x$ such that if $m > xn$, than the first can always win.
2014 IFYM, Sozopol, 8
We will call a rectangular table filled with natural numbers [i]“good”[/i], if for each two rows, there exist a column for which its two cells that are also in these two rows, contain numbers of different parity. Prove that for $\forall$ $n>2$ we can erase a column from a [i]good[/i] $n$ x $n$ table so that the remaining $n$ x $(n-1)$ table is also [i]good[/i].
2001 District Olympiad, 1
A positive integer is called [i]good[/i] if it can be written as a sum of two consecutive positive integers and as a sum of three consecutive positive integers. Prove that:
a)2001 is [i]good[/i], but 3001 isn't [i]good[/i].
b)the product of two [i]good[/i] numbers is a [i]good[/i] number.
c)if the product of two numbers is [i]good[/i], then at least one of the numbers is [i]good[/i].
[i]Bogdan Enescu[/i]
2005 Estonia National Olympiad, 3
How many such four-digit natural numbers divisible by $7$ exist such when changing the first and last number we also get a four-digit divisible by $7$?
2000 Harvard-MIT Mathematics Tournament, 21
How many ways can you color a necklace of $7$ beads with $4$ colors so that no two adjacent beads have the same color?
2019 Latvia Baltic Way TST, 4
Let $P(x)$ be a polynomial with degree $n$ and real coefficients. For all $0 \le y \le 1$ holds $\mid p(y) \mid \le 1$. Prove that $p(-\frac{1}{n}) \le 2^{n+1} -1$
2016 AMC 10, 25
How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600$ and $\text{lcm}(y,z)=900$?
$\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64$
2017 China Team Selection Test, 1
Let $n \geq 4$ be a natural and let $x_1,\ldots,x_n$ be non-negative reals such that $x_1 + \cdots + x_n = 1$. Determine the maximum value of $x_1x_2x_3 + x_2x_3x_4 + \cdots + x_nx_1x_2$.
2014 Contests, 2
Prove that among any $16$ perfect cubes we can always find two cubes whose difference is divisible by $91$.
2017 Math Prize for Girls Problems, 10
Let $C$ be a cube. Let $P$, $Q$, and $R$ be random vertices of $C$, chosen uniformly and independently from the set of vertices of $C$. (Note that $P$, $Q$, and $R$ might be equal.) Compute the probability that some face of $C$ contains $P$, $Q$, and $R$.
Indonesia MO Shortlist - geometry, g3
Given triangle $ABC$. A circle $\Gamma$ is tangent to the circumcircle of triangle $ABC$ at $A$ and tangent to $BC$ at $D$. Let $E$ be the intersection of circle $\Gamma$ and $AC$. Prove that
$$R^2=OE^2+CD^2\left(1- \frac{BC^2}{AB^2+AC^2}\right)$$
where $O$ is the center of the circumcircle of triangle $ABC$, with radius $R$.