Found problems: 85335
2022 Sharygin Geometry Olympiad, 11
Let $ABC$ be a triangle with $\angle A=60^o$ and $T$ be a point such that $\angle ATB=\angle BTC=\angle ATC$. A circle passing through $B,C$ and $T$ meets $AB$ and $AC$ for the second time at points $K$ and $L$.Prove that the distances from $K$ and $L$ to $AT$ are equal.
2025 Euler Olympiad, Round 1, 4
Given any arc $AB$ on a circle and points $C$ and $D$ on segment $AB$, such that $$CD = DB = 2AC.$$ Find the ratio $\frac{CM}{MD}$, where $M$ is a point on arc $AB$, such that $\angle CMD$ is maximized.
[img]https://i.imgur.com/NfjRpgP.png[/img]
[i]
Proposed by Andria Gvaramia, Georgia [/i]
2022 Princeton University Math Competition, A2 / B4
Ten evenly spaced vertical lines in the plane are labeled $\ell_1,\ell_2, \ldots,\ell_{10}$ from left to right. A set $\{a,b,c,d\}$ of four distinct integers $a,b,c,d \in \{1,2,\ldots,10\}$ is [i]squarish[/i] if some square has one vertex on each of the lines $\ell_a,\ell_b,\ell_c,$ and $\ell_d.$ Find the number of squarish sets.
2002 AMC 10, 13
Participation in the local soccer league this year is $10\%$ higher than last year. The number of males increased by $5\%$ and the number of females increased by $20\%$. What fraction of the soccer league is now female?
$\textbf{(A) }\dfrac13\qquad\textbf{(B) }\dfrac4{11}\qquad\textbf{(C) }\dfrac25\qquad\textbf{(D) }\dfrac49\qquad\textbf{(E) }\dfrac12$
2013 Baltic Way, 18
Find all pairs $(x,y)$ of integers such that $y^3-1=x^4+x^2$.
2016-2017 SDML (Middle School), 13
If Scott rolls four fair six-sided dice, what is the probability that he rolls more 2's than 1's?
$\text{(A) }\frac{8}{27}\qquad\text{(B) }\frac{25}{81}\qquad\text{(C) }\frac{103}{324}\qquad\text{(D) }\frac{421}{1296}\qquad\text{(E) }\frac{65}{162}$
III Soros Olympiad 1996 - 97 (Russia), 11.5
All faces of the parallelepiped $ABCDA_1B_1C_1D_1$ are equal rhombuses. Plane angles at vertex $A$ are equal. Points $K$ and $M$ are taken on the edges $A_1B_1$ and $A_1D_1$. It is known that $A_1K = a$, $A_1M = b$, and$ a + b$ is an edge of the parallelepiped. Prove that the plane $AKM$ touches the sphere inscribed in the parallelepiped. Let us denote by $Q$ the touchpoint of this sphere with the plane $AKM $. In what ratio does the straight line $AQ$ divide the segment $KM$?
2004 Bulgaria Team Selection Test, 2
The edges of a graph with $2n$ vertices ($n \ge 4$) are colored in blue and red such that there is no blue triangle and there is no red complete subgraph with $n$ vertices. Find the least possible number of blue edges.
2005 District Olympiad, 3
Prove that if the circumcircles of the faces of a tetrahedron $ABCD$ have equal radii, then $AB=CD$, $AC=BD$ and $AD=BC$.
1981 Polish MO Finals, 6
In a tetrahedron of volume $V$ the sum of the squares of the lengths of its edges equals $S$. Prove that
$$V \le \frac{S\sqrt{S}}{72\sqrt{3}}$$
2007 Today's Calculation Of Integral, 170
Let $a,\ b$ be constant numbers such that $a^{2}\geq b.$
Find the following definite integrals.
(1) $I=\int \frac{dx}{x^{2}+2ax+b}$
(2) $J=\int \frac{dx}{(x^{2}+2ax+b)^{2}}$
2020 Durer Math Competition Finals, 3
Is it possible for the least common multiple of five consecutive positive integers to be a perfect square?
2010 ELMO Shortlist, 2
For a positive integer $n$, let $s(n)$ be the number of ways that $n$ can be written as the sum of strictly increasing perfect $2010^{\text{th}}$ powers. For instance, $s(2) = 0$ and $s(1^{2010} + 2^{2010}) = 1$. Show that for every real number $x$, there exists an integer $N$ such that for all $n > N$,
\[\frac{\max_{1 \leq i \leq n} s(i)}{n} > x.\]
[i]Alex Zhu.[/i]
1975 Canada National Olympiad, 6
(i) 15 chairs are equally placed around a circular table on which are name cards for 15 guests. The guests fail to notice these cards until after they have sat down, and it turns out that no one is sitting in the correct seat. Prove that the table can be rotated so that at least two of the guests are simultaneously correctly seated.
(ii) Give an example of an arrangement in which just one of the 15 guests is correctly seated and for which no rotation correctly places more than one person.
1966 AMC 12/AHSME, 34
Let $r$ be the speed in miles per hour at which a wheel, $11$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\tfrac{1}{4}$ of a second, the speed $r$ is increased by $5$ miles per hour. The $r$ is:
$\text{(A)}\ 9\qquad
\text{(B)}\ 10\qquad
\text{(C)}\ 10\tfrac{1}{2}\qquad
\text{(D)}\ 11\qquad
\text{(E)}\ 12$
2003 Estonia National Olympiad, 4
Call a positive integer [i]lonely [/i] if the sum of reciprocals of its divisors (including $1$ and the integer itself) is not equal to the sum of reciprocals of divisors of any other positive integer. Prove that
a) all primes are lonely,
b) there exist infinitely many non-lonely positive integers.
1964 All Russian Mathematical Olympiad, 054
Find the smallest exact square with last digit not $0$, such that after deleting its last two digits we shall obtain another exact square.
2018 Germany Team Selection Test, 1
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.
2007 China Girls Math Olympiad, 5
Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.
2021 Azerbaijan EGMO TST, 4
Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$.
[i]Demetres Christofides, Cyprus[/i]
2018 MMATHS, 2
Prove that if a triangle has integer side lengths and the area (in square units) equals the perimeter (in units), then the perimeter is not a prime number.
1998 Hong kong National Olympiad, 4
Define a function $f$ on positive real numbers to satisfy
\[f(1)=1 , f(x+1)=xf(x) \textrm{ and } f(x)=10^{g(x)},\]
where $g(x) $ is a function defined on real numbers and for all real numbers $y,z$ and $0\leq t \leq 1$, it satisfies
\[g(ty+(1-t)z) \leq tg(y)+(1-t)g(z).\]
(1) Prove: for any integer $n$ and $0 \leq t \leq 1$, we have
\[t[g(n)-g(n-1)] \leq g(n+t)-g(n) \leq t[g(n+1)-g(n)].\]
(2) Prove that \[\frac{4}{3} \leq f(\frac{1}{2}) \leq \frac{4}{3} \sqrt{2}.\]
2004 AMC 12/AHSME, 3
For how many ordered pairs of positive integers $ (x,y)$ is $ x \plus{} 2y \equal{} 100$?
$ \textbf{(A)}\ 33 \qquad \textbf{(B)}\ 49 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 99 \qquad \textbf{(E)}\ 100$
2022 Stanford Mathematics Tournament, 1
Compute
\[\frac{5+\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{7+\sqrt{12}}{\sqrt{3}+\sqrt{4}}+\dots+\frac{63+\sqrt{992}}{\sqrt{31}+\sqrt{32}}.\]
2016 Dutch IMO TST, 3
Let $k$ be a positive integer, and let $s(n)$ denote the sum of the digits of $n$.
Show that among the positive integers with $k$ digits, there are as many numbers $n$ satisfying $s(n) < s(2n)$ as there are numbers $n$ satisfying $s(n) > s(2n)$.