Found problems: 85335
1985 IMO Longlists, 90
Factorise $ 5^{1985}\minus{}1$ as a product of three integers, each greater than $ 5^{100}$.
2017 Kazakhstan NMO, Problem 5
Consider all possible sets of natural numbers $(x_1, x_2, ..., x_{100})$ such that $1\leq x_i \leq 2017$ for every $i = 1,2, ..., 100$. We say that the set $(y_1, y_2, ..., y_{100})$ is greater than the set $(z_1, z_2, ..., z_{100})$ if $y_i> z_i$ for every $i = 1,2, ..., 100$. What is the largest number of sets that can be written on the board, so that any set is not more than the other set?
1998 Austrian-Polish Competition, 1
Let $x_1, x_2,y _1,y_2$ be real numbers such that $x_1^2 + x_2^2 \le 1$. Prove the inequality $$(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)$$
2003 AIME Problems, 7
Find the area of rhombus $ABCD$ given that the radii of the circles circumscribed around triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively.
2011 China Second Round Olympiad, 4
Let $A$ be a $3 \times 9$ matrix. All elements of $A$ are positive integers. We call an $m\times n$ submatrix of $A$ "ox" if the sum of its elements is divisible by $10$, and we call an element of $A$ "carboxylic" if it is not an element of any "ox" submatrix. Find the largest possible number of "carboxylic" elements in $A$.
1981 AMC 12/AHSME, 17
The function $f$ is not defined for $x=0$, but, for all non-zero real numbers $x$, $f(x)+2f\left( \frac1x \right)=3x$. The equation $f(x)=f(-x)$ is satisfied by
$\text{(A)} ~\text{exactly one real number}$
$\text{(B)}~\text{exactly two real numbers}$
$\text{(C)} ~\text{no real numbers}$
$\text{(D)} ~\text{infinitely many, but not all, non-zero real numbers}$
$\text{(E)} ~\text{all non-zero real numbers}$
2015 AMC 10, 20
A rectangle has area $A \text{ cm}^2$ and perimeter $P \text{ cm}$, where $A$ and $P$ are positive integers. Which of the following numbers cannot equal $A+P$?
$ \textbf{(A) }100\qquad\textbf{(B) }102\qquad\textbf{(C) }104\qquad\textbf{(D) }106\qquad\textbf{(E) }108 $
2018 Kürschák Competition, 3
In a village (where only dwarfs live) there are $k$ streets, and there are $k(n-1)+1$ clubs each containing $n$ dwarfs. A dwarf can be in more than one clubs, and two dwarfs know each other if they live in the same street or they are in the same club (there is a club they are both in).
Prove that is it possible to choose $n$ different dwarfs from $n$ different clubs (one dwarf from each club), such that the $n$ dwarfs know each other!
2016 Nigerian Senior MO Round 2, Problem 10
Positive numbers $x$ and $y$ satisfy $xy=2^{15}$ and $\log_2{x} \cdot \log_2{y} = 60$. Find $\sqrt[3]{(\log_2{x})^3+(\log_2{y})^3}$
2001 Junior Balkan MO, 4
Let $N$ be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of $N$ which form a triangle of area smaller than 1.
2008 Ukraine Team Selection Test, 4
Two circles $ \omega_1$ and $ \omega_2$ tangents internally in point $ P$. On their common tangent points $ A$, $ B$ are chosen such that $ P$ lies between $ A$ and $ B$. Let $ C$ and $ D$ be the intersection points of tangent from $ A$ to $ \omega_1$, tangent from $ B$ to $ \omega_2$ and tangent from $ A$ to $ \omega_2$, tangent from $ B$ to $ \omega_1$, respectively. Prove that $ CA \plus{} CB \equal{} DA \plus{} DB$.
2020 Harvard-MIT Mathematics Tournament, 3
Let $a=256$. Find the unique real number $x>a^2$ such that
\[\log_a \log_a \log_a x = \log_{a^2} \log_{a^2} \log_{a^2} x.\]
[i]Proposed by James Lin.[/i]
2001 Cuba MO, 1
In each square of a $3 \times 3$ board a real number is written. The element of the $i$ -th row and the $j$ -th column is equal to abso;uteof the difference of the sum of the elements of column $j$ and the sum of the elements of row $i$. Prove that every element of the board is equal to the sum or difference of two other elements on the board.
2004 Nicolae Coculescu, 1
Solve in the real numbers the system:
$$ \left\{ \begin{matrix} x^2+7^x=y^3\\x^2+3=2^y \end{matrix} \right. $$
[i]Eduard Buzdugan[/i]
2002 India IMO Training Camp, 3
Let $X=\{2^m3^n|0 \le m, \ n \le 9 \}$. How many quadratics are there of the form $ax^2+2bx+c$, with equal roots, and such that $a,b,c$ are distinct elements of $X$?
2018 Brazil Team Selection Test, 4
Let $ABC$ be an equilateral triangle. From the vertex $A$ we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle $\alpha$, it leaves with a directed angle $180^{\circ}-\alpha$. After $n$ bounces, the ray returns to $A$ without ever landing on any of the other two vertices. Find all possible values of $n$.
2021 BMT, 6
Consider $27$ unit-cubes assembled into one $3 \times 3 \times 3$ cube. Let $A$ and $B$ be two opposite corners of this large cube. Remove the one unit-cube not visible from the exterior, along with all six unit-cubes in the center of each face. Compute the minimum distance an ant has to walk along the surface of the modified cube to get from $A$ to $B$.
[img]https://cdn.artofproblemsolving.com/attachments/0/5/d3aa802eae40cfe717088445daabd5e7194691.png[/img]
2020 Ukrainian Geometry Olympiad - December, 2
On a circle noted $n$ points. It turned out that among the triangles with vertices in these points exactly half of the acute. Find all values $n$ in which this is possible.
2017 Iran Team Selection Test, 3
Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$
$$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$
$$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$
[i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]
2002 Croatia National Olympiad, Problem 4
Find all natural numbers $n$ for which the equation $\frac1x+\frac1y=\frac1n$ has exactly five solutions $(x,y)$ in the set of natural numbers.
2023 LMT Fall, 4B
In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Let $M$ be the midpoint of side $AB$, $G$ be the centroid of $\triangle ABC$, and $E$ be the foot of the altitude from $A$ to $BC$. Compute the area of quadrilateral $GAME$.
[i]Proposed by Evin Liang[/i]
[hide=Solution][i]Solution[/i]. $\boxed{23}$
Use coordinates with $A = (0,12)$, $B = (5,0)$, and $C = (-9,0)$. Then $M = \left(\dfrac{5}{2},6\right)$ and $E = (0,0)$. By shoelace, the area of $GAME$ is $\boxed{23}$.[/hide]
2019 Iran RMM TST, 2
Let $n >1$ be a natural number and $T_{n}(x)=x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ... + a_1 x^1 + a_0$.\\
Assume that for each nonzero real number $t $ we have $T_{n}(t+\frac {1}{t})=t^n+\frac {1}{t^n} $.\\
Prove that for each $0\le i \le n-1 $
$gcd (a_i,n) >1$.
[i]Proposed by Morteza Saghafian[/i]
1998 Harvard-MIT Mathematics Tournament, 3
Find the sum of all even positive integers less than $233$ not divisible by $10$.
PEN K Problems, 23
Let ${\mathbb Q}^{+}$ be the set of positive rational numbers. Construct a function $f:{\mathbb Q}^{+}\rightarrow{\mathbb Q}^{+}$ such that \[f(xf(y)) = \frac{f(x)}{y}\] for all $x, y \in{\mathbb Q}^{+}$.
2022 Purple Comet Problems, 25
Let $ABCD$ be a parallelogram with diagonal $AC = 10$ such that the distance from $A$ to line $CD$ is $6$ and the distance from $A$ to line $BC$ is $7$. There are two non-congruent configurations of $ABCD$ that satisfy these conditions. The sum of the areas of these two parallelograms is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.