A sequence of real numbers $x_1,x_2,\ldots ,x_n$ is given such that $x_{i+1}=x_i+\frac{1}{30000}\sqrt{1-x_i^2},\ i=1,2,\ldots ,$ and $x_1=0$. Can $n$ be equal to $50000$ if $x_n<1$?

A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle? [asy] size(200); defaultpen(linewidth(0.6pt)+fontsize(10pt)); real y = sqrt(3); pair A,B,C,D,E,F,G,H; A = (0,0); B = (0,y); C = (y,y); D = (y,0); E = ((y + 1)/2,y); F = (y, (y - 1)/2); G = ((y - 1)/2, 0); H = (0,(y + 1)/2); fill(H--B--E--cycle, gray); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); [/asy] $\textbf{(A) }\frac15\qquad\textbf{(B) }\frac14\qquad\textbf{(C) }2-\sqrt3\qquad\textbf{(D) }\sqrt3-\sqrt2\qquad\textbf{(E) }\sqrt2-1$

Positive integers $a,b$ are such that $137$ divides $a+139b$ and $139$ divides $a+137b$. Find the minimal posible value of $a+b$.

Let $f_1(x)=\sqrt{1-x}$, and for integers $n \ge 2$, let $f_n(x)=f_{n-1}(\sqrt{n^2-x})$. If $N$ is the largest value of $n$ for which the domain of $f_n$ is nonempty, the domain of $f_N$ is ${c}$. What is $N+c$? $ \textbf{(A)}\ -226 \qquad \textbf{(B)}\ -144 \qquad \textbf{(C)}\ -20 \qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ 144$

Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^3$

What mass of the compound $\ce{CrO3}$ $\text{(M = 100.0)}$ contains $4.5\times10^{23}$ oxygen atoms? $ \textbf{(A) }\text{2.25 g}\qquad\textbf{(B) }\text{12.0 g}\qquad\textbf{(C) }\text{25.0 g}\qquad\textbf{(D) }\text{75.0 g}\qquad$

In quadrilateral $ABCD$ sides $AD$ and $BC$ aren't parallel. Diagonals $AC$ and $BD$ intersect in $E$. $F$ and $G$ are points on sides $AB$ and $DC$ such $\frac{AF}{FB}=\frac{DG}{GC}=\frac{AD}{BC}$ Prove that if $E, F, G$ are collinear then $ABCD$ is cyclic.

Find $n$ such that\[\frac{1!\cdot2!\cdot3!\cdots10!}{(1!)^2(3!)^2(5!)^2(7!)^2(9!)^2}=15\cdot2^n.\]

Define sequence $ (a_n)$ by $ \sum_{d|n} a_d \equal{} 2^n.$ Show that $ n|a_n.$

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

A walker and a jogger travel along the same straight line in the same direction. The walker walks at one meter per second, while the jogger runs at two meters per second. The jogger starts one meter in front of the walker. A dog starts with the walker, and then runs back and forth between the walker and the jogger with constant speed of three meters per second. Let $f(n)$ meters denote the total distance travelled by the dog when it has returned to the walker for the nth time (so $f(0)=0$). Find a formula for $f(n)$.

Nico wants to write the $100$ integers from $1$ to $100$ around a circle in some order and without repetition, such that they have the following property: when moving around the circle clockwise, the sum of the $100$ distances between each number and its next number is equal to $198$. Determine in how many ways the $100$ numbers can be ordered so that Nico achieves his goal. [b]Note:[/b] The distance between two numbers $a$ and $b$ is equal to the absolute value of their difference: $|a - b|$.

Let $n > 1$ be a positive integer and $S$ be a collection of $\frac{1}{2}\binom{2n}{n}$ distinct $n$-element subsets of $\{1, 2, \dotsc, 2n\}$. Show that there exists $A, B\in S$ such that $|A\cap B|\leq 1$. [i]Proposed by Michael Ren.[/i]

On the base $BC$ of the isosceles triangle $ABC$ chose a point $D$ and in each of the triangles $ABD$ and $ACD$ inscribe a circle. Then everything was wiped, leaving only two circles. It is known from which side of their line of centers the apex $A$ is located . Use a compass and ruler to restore the triangle $ABC$ , if we know that : a) $AD$ is angle bisector, b) $AD$ is median.

For what values of $ n$ does there exist an $ n \times n$ array of entries -1, 0 or 1 such that the $ 2 \cdot n$ sums obtained by summing the elements of the rows and the columns are all different?

Given three numbers $x, y, z$, and set $x_1 = |x - y|, y_1 = | y -z|, z_1 = |z- x|$. From $x_1, y_1, z_1$, form in the same fashion the numbers $x_2, y_2, z_2$, and so on. It is known that $x_n = x, y_n = y, z_n = z$ for some $n$. Find all possible values of $(x, y, z)$.

In a school there are 1200 students. Each student must join exactly $k$ clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of $k$.

The circle $C_{1}$ with radius $6$ and the circle $C_{2}$ with radius $8$ are externally tangent to each other at $A$. The circle $C_3$ which is externally tangent to $C_{1}$ and $C_{2}$ has a radius with length $21$. The common tangent of $C_{1}$ and $C_{2}$ which passes through $A$ meets $C_{3}$ at $B$ and $C$. What is $|BC|$? $ \textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 14\sqrt{3} \qquad\textbf{(D)}\ 24\sqrt{3} \qquad\textbf{(E)}\ 25\sqrt{3} $

We have a closed path on a vertices of a $ n$×$ n$ square which pass from each vertice exactly once . prove that we have two adjacent vertices such that if we cut the path from these points then length of each pieces is not less than quarter of total path .

Let $n$ be an integer greater than $6$.Show that if $n+1$ is a prime number,than $\left\lceil \frac{(n-1)!}{n(n+1)}\right \rceil$ is $ODD.$

In a convex hexagon $AC'BA'CB'$, every two opposite sides are equal. Let $A_1$ denote the point of intersection of $BC$ with the perpendicular bisector of $AA'$. Define $B_1$ and $C_1$ similarly. Prove that $A_1$, $B_1$, and $C_1$ are collinear.

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

Let $n \geq 2, n \in \mathbb{N}$ and $A_0 = (a_{01},a_{02}, \ldots, a_{0n})$ be any $n-$tuple of natural numbers, such that $0 \leq a_{0i} \leq i-1,$ for $i = 1, \ldots, n.$ $n-$tuples $A_1= (a_{11},a_{12}, \ldots, a_{1n}), A_2 = (a_{21},a_{22}, \ldots, a_{2n}), \ldots$ are defined by: $a_{i+1,j} = Card \{a_{i,l}| 1 \leq l \leq j-1, a_{i,l} \geq a_{i,j}\},$ for $i \in \mathbb{N}$ and $j = 1, \ldots, n.$ Prove that there exists $k \in \mathbb{N},$ such that $A_{k+2} = A_{k}.$

[i]The Game.[/i] Eric and Greg are watching their new favorite TV show, [i]The Price is Right[/i]. Bob Barker recently raised the intellectual level of his program, and he begins the latest installment with bidding on following question: How many Carmichael numbers are there less than $100,000$? Each team is to list one nonnegative integer not greater than $100,000$. Let $X$ denote the answer to Bob’s question. The teams listing $N$, a maximal bid (of those submitted) not greater than $X$, will receive $N$ points, and all other teams will neither receive nor lose points. (A Carmichael number is an odd composite integer $n$ such that $n$ divides $a^{n-1}-1$ for all integers $a$ relatively prime to $n$ with $1<a<n$.)

Find all prime numbers $p$ such that $16p + 1$ is a perfect cube.