Found problems: 85335
1995 AMC 12/AHSME, 29
For how many three-element sets of positive integers $\{a,b,c\}$ is it true that $a \times b \times c = 2310$?
$\textbf{(A)}\ 32 \qquad
\textbf{(B)}\ 36 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 43 \qquad
\textbf{(E)}\ 45$
1996 Nordic, 2
Determine all real numbers $x$, such that $x^n+x^{-n}$ is an integer for all integers $n$.
Kyiv City MO Juniors Round2 2010+ geometry, 2018.8.3
In the triangle $ABC$ it is known that $\angle ACB> 90 {} ^ \circ$, $\angle CBA> 45 {} ^ \circ$. On the sides $AC$ and $AB$, respectively, there are points $P$ and $T$ such that $ABC$ and $PT = BC$. The points ${{P} _ {1}}$ and ${{T} _ {1}}$ on the sides $AC$ and $AB$ are such that $AP = C {{P} _ {1}}$ and $AT = B {{T} _ {1}}$. Prove that $\angle CBA- \angle {{P} _ {1}} {{T} _ {1}} A = 45 {} ^ \circ$.
(Anton Trygub)
2009 Stanford Mathematics Tournament, 1
In the future, each country in the world produces its Olympic athletes via cloning and strict training
programs. Therefore, in the fi nals of the 200 m free, there are two indistinguishable athletes from each
of the four countries. How many ways are there to arrange them into eight lanes?
2012 Korea National Olympiad, 4
$a,b,c$ are positive numbers such that $ a^2 + b^2 + c^2 = 2abc + 1 $. Find the maximum value of
\[ (a-2bc)(b-2ca)(c-2ab) \]
PEN A Problems, 16
Determine if there exists a positive integer $n$ such that $n$ has exactly $2000$ prime divisors and $2^{n}+1$ is divisible by $n$.
2010 Princeton University Math Competition, 2
Let $p(x) = x^2 + x + 1$. Find the fourth smallest prime $q$ such that $p(n)$ is divisble by $q$ for some integer $n$.
2014 India Regional Mathematical Olympiad, 6
Let $x_1,x_2,x_3 \ldots x_{2014}$ be positive real numbers such that $\sum_{j=1}^{2014} x_j=1$. Determine with proof the smallest constant $K$ such that
\[K\sum_{j=1}^{2014}\frac{x_j^2}{1-x_j} \ge 1\]
2025 All-Russian Olympiad Regional Round, 10.10
On the graphic of the function $y=x^2$ were selected $1000$ pairwise distinct points, abscissas of which are integer numbers from the segment $[0; 100000]$. Prove that it is possible to choose six different selected points $A$, $B$, $C$, $A'$, $B'$, $C'$ such that areas of triangles $ABC$ and $A'B'C'$ are equals.
[i]A. Tereshin[/i]
2023 239 Open Mathematical Olympiad, 3
In quadrilateral $ABCD$, a circle $\omega$ is inscribed. A point $K$ is chosen on diagonal $AC$. Segment $BK$ intersects $\omega$ at a unique point $X$, and segment $DK$ intersects $\omega$ at a unique point $Y$. It turns out that $XY$ is the diameter of $\omega$. Prove that it is perpendicular to $AC$.
[i]Proposed by Tseren Frantsuzov[/i]
2012-2013 SDML (Middle School), 2
A regular tetrahedron with $5$-inch edges weighs $2.5$ pounds. What is the weight in pounds of a similarly constructed regular tetrahedron that has $6$-inch edges? Express your answer as a decimal rounded to the nearest hundredth.
2002 Korea - Final Round, 3
Let $p_n$ be the $n^{\mbox{th}}$ prime counting from the smallest prime $2$ in increasing order. For example, $p_1=2, p_2=3, p_3 =5, \cdots$
(a) For a given $n \ge 10$, let $r$ be the smallest integer satisfying
\[2\le r \le n-2, \quad n-r+1 < p_r\]
and define $N_s=(sp_1p_2\cdots p_{r-1})-1$ for $s=1,2,\ldots, p_r$. Prove that there exists $j, 1\le j \le p_r$, such that none of $p_1,p_2,\cdots, p_n$ divides $N_j$.
(b) Using the result of (a), find all positive integers $m$ for which
\[p_{m+1}^2 < p_1p_2\cdots p_m\]
2021 Princeton University Math Competition, 1
An evil witch is making a potion to poison the people of PUMAClandia. In order for the potion to work, the number of poison dart frogs cannot exceed $5$, the number of wolves’ teeth must be an even number, and the number of dragon scales has to be a multiple of $6$. She can also put in any number of tiger nails. Given that the stew has exactly $2021$ ingredients, in how many ways can she add ingredients for her potion to work?
2021 Princeton University Math Competition, 8
The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, . . . , 2020$. The students are initially divided up into $20$ groups of $101$, with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these $20$ students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as $\frac{a}{b}$, where $a, b$ are relatively prime integers, determine $a + b$.
1955 AMC 12/AHSME, 31
An equilateral triangle whose side is $ 2$ is divided into a triangle and a trapezoid by a line drawn parallel to one of its sides. If the area of the trapezoid equals one-half of the area of the original triangle, the length of the median of the trapezoid is:
$ \textbf{(A)}\ \frac{\sqrt{6}}{2} \qquad
\textbf{(B)}\ \sqrt{2} \qquad
\textbf{(C)}\ 2\plus{}\sqrt{2} \qquad
\textbf{(D)}\ \frac{2\plus{}\sqrt{2}}{2} \qquad
\textbf{(E)}\ \frac{2\sqrt{3}\minus{}\sqrt{6}}{2}$
2005 Bulgaria Team Selection Test, 3
Let $\mathbb{R}^{*}$ be the set of non-zero real numbers. Find all functions $f : \mathbb{R}^{*} \to \mathbb{R}^{*}$ such that $f(x^{2}+y) = (f(x))^{2} + \frac{f(xy)}{f(x)}$, for all $x,y \in \mathbb{R}^{*}$ and $-x^{2} \not= y$.
2009 IMO Shortlist, 3
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$.
[i]Proposed by Juhan Aru, Estonia[/i]
2022 Kazakhstan National Olympiad, 6
Numbers from $1$ to $49$ are randomly placed in a $35 \times 35$ table such that number $i$ is used exactly $i$ times. Some random cells of the table are removed so that table falls apart into several connected (by sides) polygons. Among them, the one with the largest area is chosen (if there are several of the same largest areas, a random one of them is chosen). What is the largest number of cells that can be removed that guarantees that in the chosen polygon there is a number which occurs at least $15$ times?
1999 Canada National Olympiad, 5
Let $ x$, $ y$, and $ z$ be non-negative real numbers satisfying $ x \plus{} y \plus{} z \equal{} 1$. Show that
\[ x^2 y \plus{} y^2 z \plus{} z^2 x \leq \frac {4}{27}
\]
and find when equality occurs.
2016 Portugal MO, 3
Let $[ABC]$ be an equilateral triangle on the side $1$. Determine the length of the smallest segment $[DE]$, where $D$ and $E$ are on the sides of the triangle, which divides $[ABC]$ into two figures with equal area.
2010 India IMO Training Camp, 6
Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to
\[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]
2023 Baltic Way, 16
Prove that there exist nonconstant polynomials $f, g$ with integer coefficients, such that for infinitely many primes $p$, $p \nmid f(x)-g(y)$ for any integers $x, y$.
2010 South East Mathematical Olympiad, 1
$ABC$ is a triangle with a right angle at $C$. $M_1$ and $M_2$ are two arbitrary points inside $ABC$, and $M$ is the midpoint of $M_1M_2$. The extensions of $BM_1,BM$ and $BM_2$ intersect $AC$ at $N_1,N$ and $N_2$ respectively.
Prove that $\frac{M_1N_1}{BM_1}+\frac{M_2N_2}{BM_2}\geq 2\frac{MN}{BM}$
1999 IMO Shortlist, 6
For $n \geq 3$ and $a_{1} \leq a_{2} \leq \ldots \leq a_{n}$ given real numbers we have the following instructions:
- place out the numbers in some order in a ring;
- delete one of the numbers from the ring;
- if just two numbers are remaining in the ring: let $S$ be the sum of these two numbers. Otherwise, if there are more the two numbers in the ring, replace
Afterwards start again with the step (2). Show that the largest sum $S$ which can result in this way is given by the formula
\[S_{max}= \sum^n_{k=2} \begin{pmatrix} n -2 \\
[\frac{k}{2}] - 1\end{pmatrix}a_{k}.\]
2024 Serbia Team Selection Test, 1
Three coins are placed at the origin of a Cartesian coordinate system. On one move one removes a coin placed at some position $(x, y)$ and places three new coins at $(x+1, y)$, $(x, y+1)$ and $(x+1, y+1)$. Prove that after finitely many moves, there will exist two coins placed at the same point.