Four congruent spheres are stacked so that each is tangent to the other three. A larger sphere, $R$, contains the four congruent spheres so that all four are internally tangent to $R$. A smaller sphere, $S$, sits in the space between the four congruent spheres so that all four are externally tangent to $S$. The ratio of the surface area of $R$ to the surface area of $S$ can be written $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.

Prove that an arbitrary triangle with area $1$ can be covered by an isosceles triangle with area less than $\sqrt{2}$.

$\boxed{\text{G1}}$ Let $M$ be interior point of the triangle $ABC$ with <BAC=70and <ABC=80 If <ACM=10 and <CBM=20.Prove that $AB=MC$

A positive integer $A_k...A_1A_0$ is called monotonic if $A_k \le ..\le A_1 \le A_0$. Show that for any $n \in N$ there is a monotonic perfect square with $n$ digits.

Into each box of a $ 2012 \times 2012 $ square grid, a real number greater than or equal to $ 0 $ and less than or equal to $ 1 $ is inserted. Consider splitting the grid into $2$ non-empty rectangles consisting of boxes of the grid by drawing a line parallel either to the horizontal or the vertical side of the grid. Suppose that for at least one of the resulting rectangles the sum of the numbers in the boxes within the rectangle is less than or equal to $ 1 $, no matter how the grid is split into $2$ such rectangles. Determine the maximum possible value for the sum of all the $ 2012 \times 2012 $ numbers inserted into the boxes.

In $\triangle ABC$, let $X$ and $Y$ be points on segment $BC$ such that $AX=XB=20$ and $AY=YC=21$. Let $J$ be the $A$-excenter of triangle $\triangle AXY$. Given that $J$ lies on the circumcircle of $\triangle ABC$, the length of $BC$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Andrew Wen[/i]

Consider the statements: $\text{(I)}~~\sqrt{a^2+b^2}=0$ $\text{(II)}~~\sqrt{a^2+b^2}=ab$ $\text{(III)}~~\sqrt{a^2+b^2}=a+b$ $\text{(IV)}~~\sqrt{a^2+b^2}=a-b$, where we allow $a$ and $b$ to be real or complex numbers. Those statements for which there exist solutions other than $a=0$ and $b=0$ are: $\text{(A)} \ \text{(I)},\text{(II)},\text{(III)},\text{(IV)} \qquad \text{(B)} \ \text{(II)},\text{(III)},\text{(IV)} \qquad \text{(C)} \ \text{(I)},\text{(III)},\text{(IV)} \qquad \text{(D)} \ \text{(III)},\text{(IV)} \qquad \text{(E)} \ \text{(I)}$

Let $a_1,a_2,\cdots$ be an infinity sequence of positive integers such that $a_1=2021$ and $$a_{n+1}=(a_1+a_2+\cdots+a_n)^2-1$$ for all positive integers $n$. Prove that for any integer $n\ge 2$, $a_n$ is the product of at least $2n$ (not necessarily distinct) primes.

Do there exist positive integer numbers $a$ and $b$, for which the number $(\sqrt{1+\frac{4}{a}}-1)(\sqrt{1+\frac{4}{b}}-1)$ is rational [i]V. Kamianetski[/i]

Let $p$ be an odd prime.Positive integers $a,b,c,d$ are less than $p$,and satisfy $p|a^2+b^2$ and $p|c^2+d^2$.Prove that exactly one of $ac+bd$ and $ad+bc$ is divisible by $p$

Let $n$ be the answer to this problem. In acute triangle $ABC$, point $D$ is located on side $BC$ so that $\angle BAD = \angle DAC$ and point $E$ is located on $AC$ so that $BE \perp AC$. Segments $BE$ and $AD$ intersect at $X$ such that $\angle BXD = n^o$: Given that $\angle XBA = 16^o$, find the measure of $\angle BCA$.

Let $ABC$ be a triangle with $AB = 2, BC = 5, AC = 4$. Let $M$ be the projection of $C$ onto the external angle bisector at vertex $B$. Similarly, let $N$ be the projection of $B$ onto the external angle bisector at vertex $C$. If the ratio of the area of quadrilateral $BCNM$ to the area of triangle $ABC$ is $a/b$, where $a$ and $b$ are positive integers and $\gcd(a, b) = 1$, find $a + b$.

Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2 n$; that is to say, $p(x)=$ $x^{2 n}+a_{2 n-1} x^{2 n-1}+\cdots+a_1 x+a_0$ for some real coefficients $a_0, \ldots, a_{2 n-1}$. Suppose that $p(1 / k)=k^2$ for all integers $k$ such that $1 \leq|k| \leq n$. Find all other real numbers $x$ for which $p(1 / x)=x^2$.

Let $A$ be set of 20 consecutive positive integers, Which sum and product of elements in $A$ not divisible by 23. Prove that product of elements in $A$ is not perfect square

Find the coefficient of $x$ in the expansion of $(1 + x)(1 - 2x)(1 + 3x)(1 - 4x) ...(1 - 2008x)$.

In the acute triangle $\triangle ABC$ , $H$ is the orthocenter. $S$ is a point on $(AHC)$ st $\angle ASB = 90$. $P$ is on $AC$ and not on the extention of $AC$ from $A$ , st $\angle APS=\angle BAS$.Prove that $CS$ , the circle $(BPC)$ and the circle with diameter $AC$ are concurrent.

Let $T = \{n \in N|$n consists of $2$ digits $\}$ and $$P = \{x|x = n(n + 1)... (n + 7); n,n + 1,..., n + 7 \in T\}.$$ Determine the gcd of the elements of $P$.

Let $n$ be a positive integer not divisible by $3$. A triangular grid of length $n$ is obtained by dissecting a regular triangle with length $n$ into $n^2$ unit regular triangles. There is an orange at each vertex of the grid, which sums up to \[\frac{(n+1)(n+2)}{2}\] oranges. A triple of oranges $A,B,C$ is [i]good[/i] if each $AB,AC$ is some side of some unit regular triangles, and $\angle BAC = 120^{\circ}$. Each time, Yen can take away a good triple of oranges from the grid. Determine the maximum number of oranges Yen can take.

Find all real pairs $(a,b)$ that solve the system of equation \begin{align*} a^2+b^2 &= 25, \\ 3(a+b)-ab &= 15. \end{align*} [i](German MO 2016 - Problem 1)[/i]

Let $p$ be an odd prime and let $Z_{p}$ denote (the field of) integers modulo $p$. How many elements are in the set \[\{x^{2}: x \in Z_{p}\}\cap \{y^{2}+1: y \in Z_{p}\}?\]

Vertices $A, B, C$ of a equilateral triangle of side $1$ are in the surface of a sphere with radius $1$ and center $O$. Let $D$ be the orthogonal projection of $A$ on the plane $\alpha$ determined by points $B, C, O$. Let $N$ be one of the intersections of the line perpendicular to $\alpha$ passing through $O$ with the sphere. Find the angle $\angle DNO$.

Let $n$ be a positive integer. There is a pawn in one of the cells of an $n\times n$ table. The pawn moves from an arbitrary cell of the $k$th column, $k \in \{1,2, \cdots, n \}$, to an arbitrary cell in the $k$th row. Prove that there exists a sequence of $n^{2}$ moves such that the pawn goes through every cell of the table and finishes in the starting cell.

Anis, Banu, and Cholis are going to play a game. They are given an $n\times n$ board consisting of $n^2$ unit squares, where $n$ is an integer and $n > 5$. In the beginning of the game, the number $n$ is written on each unit square. Then Anis, Banu, and Cholis take turns playing the game, repeatedly in that order, according to the following procedure: On every turn, an arrangement of $n$ squares on the same row or column is chosen, and every number from the chosen squares is subtracted by $1$. The turn cannot be done if it results in a negative number, that is, no arrangement of $n$ unit squares on the same column or row in which all of its unit squares contain a positive number can be found. The last person to get a turn wins. Determine which player will win the game.

Let $a,b,c\in\mathbb{R^+}$ If $3=a+b+c\le 3abc$ , prove that $$\frac{1}{\sqrt{2a+1}}+ \frac{1}{\sqrt{2b+1}}+\frac{1}{\sqrt{2c+1}}\le \left( \frac32\right)^{3/2}$$ [i](Real Matrik)[/i]

Let \( S \) be a set consisting of \( 2024 \) points on a plane, such that no three points in \( S \) are collinear. A line \( \ell \) passing through two points in \( S \) is called a "weakly balanced line" if it satisfies the following condition: (Condition) The line \( \ell \) divides the plane into two regions, one containing exactly \( 1010 \) points of \( S \), and the other containing exactly \( 1012 \) points of \( S \) (where each region contains no points lying on \( \ell \)). Let \( \omega(S) \) denote the number of weakly balanced lines among the lines passing through two points in \( S \). Find the smallest possible value of \( \omega(S) \).