Found problems: 85335
2014 Contests, 1
Prove that for $\forall$ $a,b,c\in [\frac{1}{3},3]$ the following inequality is true:
$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\geq \frac{7}{5}$.
2012 Princeton University Math Competition, B8
A cyclic quadrilateral $ABCD$ has side lengths $AB = 3, BC = AD = 5$, and $CD = 8$. The radius of its circumcircle can be written in the form $a\sqrt{b}/c$, where $a, b, c$ are positive integers, $a, c$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a + b + c$.
2005 USAMTS Problems, 4
Find, with proof, all irrational numbers $x$ such that both $x^3-6x$ and $x^4-8x^2$ are rational.
2021 Saudi Arabia Training Tests, 18
Let $ABC$ be a triangle with $AB < AC$ and incircle $(I)$ tangent to $BC$ at $D$. Take $K$ on $AD$ such that $CD = CK$. Suppose that $AD$ cuts $(I)$ at $G$ and $BG$ cuts $CK$ at $L$. Prove that K is the midpoint of $CL$.
1993 Polish MO Finals, 3
Find out whether it is possible to determine the volume of a tetrahedron knowing the areas of its faces and its circumradius.
2016 Estonia Team Selection Test, 10
Let $m$ be an integer, $m \ge 2$. Each student in a school is practising $m$ hobbies the most. Among any $m$ students there exist two students who have a common hobby. Find the smallest number of students for which there must exist a hobby which is practised by at least $3$ students .
2015 PAMO, Problem 2
A convex hexagon $ABCDEF$ is such that
$$AB=BC \quad CD=DE \quad EF=FA$$
and
$$\angle ABC=2\angle AEC \quad \angle CDE=2\angle CAE \quad \angle EFA=2\angle ACE$$
Show that $AD$, $CF$ and $EB$ are concurrent.
2005 China Team Selection Test, 3
Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds.
Prove that $\sum_{j=1}^n |a_j| \leq 3$.
2010 National Olympiad First Round, 29
Let $I$ be the incenter of $\triangle ABC$, and $O$ be the excenter corresponding to $B$. If $|BI|=12$, $|IO|=18$, and $|BC|=15$, then what is $|AB|$?
$ \textbf{(A)}\ 16
\qquad\textbf{(B)}\ 18
\qquad\textbf{(C)}\ 20
\qquad\textbf{(D)}\ 22
\qquad\textbf{(E)}\ 24
$
2022 AMC 10, 12
A pair of fair $6$-sided dice is rolled $n$ times. What is the least value of $n$ such that the probability that the sum of the numbers face up on a roll equals $7$ at least once is greater than $\frac{1}{2}$?
$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 6$
1990 AMC 12/AHSME, 2
$\left(\dfrac{1}{4}\right)^{-\frac{1}{4}}=$
$\textbf{(A) }-16\qquad
\textbf{(B) }-\sqrt{2}\qquad
\textbf{(C) }-\dfrac{1}{16}\qquad
\textbf{(D) }-\dfrac{1}{256}\qquad
\textbf{(E) }\sqrt{2}$
1978 Putnam, A6
Let $n$ distinct points in the plane be given. Prove that fewer than $2 n^{3 \slash 2}$ pairs of them are a unit distance apart.
2014 BMT Spring, 10
A plane intersects a sphere of radius $10$ such that the distance from the center of the sphere to the plane is $9$. The plane moves toward the center of the bubble at such a rate that the increase in the area of the intersection of the plane and sphere is constant, and it stops once it reaches the center of the circle. Determine the distance from the center of the sphere to the plane after two-thirds of the time has passed.
2013 Silk Road, 1
Determine all pairs of positive integers $m, n,$ satisfying the equality $(2^{m}+1;2^n+1)=2^{(m;n)}+1$ , where $(a;b)$ is the greatest common divisor
2017 Moldova EGMO TST, 1
Let $a,b,c\geq 0$.
Prove: $$\frac{1+a+a^{2}}{1+b+c^{2}}+\frac{1+b+b^{2}}{1+c+a^{2}}+\frac{1+c+c^{2}}{1+a+b^{2}}\geq 3$$
2020 Online Math Open Problems, 3
Compute the number of ways to write the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 in the cells of a 3 by 3 grid such that
[list]
[*] each cell has exactly one number,
[*] each number goes in exactly one cell,
[*] the numbers in each row are increasing from left to right,
[*] the numbers in each column are increasing from top to bottom, and
[*]the numbers in the diagonal from the upper-right corner cell to the lower-left corner cell are increasing from upper-right to lower-left.
[/list]
[i]Proposed by Ankit Bisain & Luke Robitaille[/i]
2020 BAMO, A
A trapezoid is divided into seven strips of equal width as shown. What fraction of the trapezoid’s area is shaded?
2023 Ukraine National Mathematical Olympiad, 11.3
In the quadrilateral $ABCD$ $\angle ABC = \angle CDA = 90^\circ$. Let $P = AC \cap BD$, $Q = AB\cap CD$, $R = AD \cap BC$. Let $\ell$ be the midline of the triangle $PQR$, parallel to $QR$. Show that the circumcircle of the triangle formed by lines $AB, AD, \ell$ is tangent to the circumcircle of the triangle formed by lines $CD, CB, \ell$.
[i]Proposed by Fedir Yudin[/i]
2017 Purple Comet Problems, 13
Find the number of positive integer divisors of $20^{17}$ that are either perfect squares or perfect cubes.
2019 Saudi Arabia Pre-TST + Training Tests, 4.1
Find the smallest positive integer $n$ with the following property:
After painting black exactly $n$ cells of a $7\times 7$ board there always exists a $2\times 2$ square with at least three black cells.
PEN G Problems, 5
Let $ a, b, c$ be integers, not all equal to $ 0$. Show that
\[ \frac{1}{4a^{2}\plus{}3b^{2}\plus{}2c^{2}}\le\vert\sqrt[3]{4}a\plus{}\sqrt[3]{2}b\plus{}c\vert.\]
1978 All Soviet Union Mathematical Olympiad, 262
The checker is standing on the corner field of a $n\times n$ chess-board. Each of two players moves it in turn to the neighbour (i.e. that has the common side) field. It is forbidden to move to the field, the checker has already visited. That who cannot make a move losts.
a) Prove that for even $n$ the first can always win, and if $n$ is odd, than the second can always win.
b) Who wins if the checker stands initially on the neighbour to the corner field?
1999 Baltic Way, 8
We are given $1999$ coins. No two coins have the same weight. A machine is provided which allows us with one operation to determine, for any three coins, which one has the middle weight. Prove that the coin that is the $1000$th by weight can be determined using no more than $1000000$ operations and that this is the only coin whose position by weight can be determined using this machine.
2024 Spain Mathematical Olympiad, 5
Given two points $p_1=(x_1, y_1)$ and $p_2=(x_2, y_2)$ on the plane, denote by $\mathcal{R}(p_1,p_2)$ the rectangle with sides parallel to the coordinate axes and with $p_1$ and $p_2$ as opposite corners, that is, \[\{(x,y)\in \mathbb{R}^2:\min\{x_1, x_2\}\leq x\leq \max\{x_1, x_2\},\min\{y_1, y_2\}\leq y\leq \max\{y_1, y_2\}\}.\] Find the largest value of $k$ for which the following statement is true: for all sets $\mathcal{S}\subset\mathbb{R}^2$ with $|\mathcal{S}|=2024$, there exist two points $p_1, p_2\in\mathcal{S}$ such that $|\mathcal{S}\cap\mathcal{R}(p_1, p_2)|\geq k$.
1959 AMC 12/AHSME, 30
$A$ can run around a circular track in $40$ seconds. $B$, running in the opposite direction, meets $A$ every $15$ seconds. What is $B$'s time to run around the track, expressed in seconds?
$ \textbf{(A)}\ 12\frac12 \qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 27\frac12\qquad\textbf{(E)}\ 55 $