Let $x$ and $y$ be real numbers satisfying \[(x^2+x-1)(x^2-x+1)=2(y^3-2\sqrt{5}-1)\] and \[(y^2+y-1)(y^2-y+1)=2(x^3+2\sqrt{5}-1)\] Find $8x^2+4y^3$.

Does there exist a $6$-digit number $A$ such that none of its $500 000$ multiples $A$, $2A$, $3A$, ..., $500 000A$ ends in $6$ identical digits? (S Tokarev)

In a convex quadrilateral $ABCD$, let $AB + CD = BC + AD$. Prove that the circle inscribed in $ABC$ is tangent to the circle inscribed in $ACD$.

We call a monic polynomial $P(x) \in \mathbb{Z}[x]$ [i]square-free mod n[/i] if there [u]dose not[/u] exist polynomials $Q(x),R(x) \in \mathbb{Z}[x]$ with $Q$ being non-constant and $P(x) \equiv Q(x)^2 R(x) \mod n$. Given a prime $p$ and integer $m \geq 2$. Find the number of monic [i]square-free mod p[/i] $P(x)$ with degree $m$ and coeeficients in $\{0,1,2,3,...,p-1\}$. [i]Proposed by Masud Shafaie[/i]

The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length $5{.}$ The numbers in positions $(5, 5), \,(2,4),\,(4,3),$ and $(3, 1)$ are $0, 48, 16,$ and $12{,}$ respectively. What number is in position $(1, 2)?$ \[ \begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}\] $\textbf{(A) } 19 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 29 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 39$

In making Euclidean constructions in geometry it is permitted to use a ruler and a pair of compasses. In the constructions considered in this question no compasses are permitted, but the ruler is assumed to have two parallel edges, which can be used for constructing two parallel lines through two given points whose distance is at least equal to the breadth of the rule. Then the distance between the parallel lines is equal to the breadth of the ruler. Carry through the following constructions with such a ruler. Construct: [b]a)[/b] The bisector of a given angle. [b]b)[/b] The midpoint of a given rectilinear line segment. [b]c)[/b] The center of a circle through three given non-collinear points. [b]d)[/b] A line through a given point parallel to a given line.

There are $n$ students $A_1,A_2,...,A_n$ and some of them shaked hands with each other. ($A_i$ and $A-j$ can shake hands more than one time.) Let the student $A_i$ shaked hands $d_i$ times. Suppose $d_1 + d_2 +... + d_n > 0$. Prove that there exist $1 \le i < j \le n$ satisfying the following conditions: (a) Two students $A_i$ and $A_j$ shaked hands each other. (b) $\frac{(d_1 + d_2 +... + d_n)^2}{n^2}\le d_id_j$

Let $n \geq 2$ be an integer. Define $x_i =1$ or $-1$ for every $i=1,2,3,\cdots, n$. Call an operation [i]adhesion[/i], if it changes the string $(x_1,x_2,\cdots,x_n)$ to $(x_1x_2, x_2x_3, \cdots ,x_{n-1}x_n, x_nx_1)$ . Find all integers $n \geq 2$ such that the string $(x_1,x_2,\cdots, x_n)$ changes to $(1,1,\cdots,1)$ after finitely [i]adhesion[/i] operations.

Define $f$ on the positive integers by $f(n) = k^2 + k + 1$, where $n=2^k(2l+1)$ for some $k,l$ nonnegative integers. Find the smallest $n$ such that $f(1) + f(2) + ... + f(n) \geq 123456$.

Evaluate the infinite sum $ \sum_{n\equal{}1}^{\infty}\frac{n}{n^4\plus{}4}$.

Find the greatest and smallest value of $f(x, y) = y-2x$, if x, y are distinct non-negative real numbers with $\frac{x^2+y^2}{x+y}\leq 4$.

Denote by $S(k)$ the sum of the digits in the decimal representation of $k$. Prove that there are infinitely many $n\in \mathbb{Z_{+}}$ such that: ${S(2^{n}+n})<S(2^{n})$.

Let $p_n$ be the probability that $c+d$ is a perfect square when the integers $c$ and $d$ are selected independently at random from the set $\{1,2,\ldots,n\}$. Show that $\lim_{n\to\infty}p_n\sqrt n$ exists and express this limit in the form $r(\sqrt s-t)$, where $s$ and $t$ are integers and $r$ is a rational number.

There are $2^n$ airports, numbered with binary strings of length $n{}$. Any two stations whose numbers differ in exactly one digit are connected by a flight that has a price (which is the same in both directions). The sum of the prices of all $n{}$ flights leaving any station does not exceed 1. Prove that one can travel between any two airports by paying no more than 1.

Find a countable family of natural solutions to $ \frac{1}{a} +\frac{1}{b} +\frac{1}{ab}=\frac{1}{c} . $

Denote by $M(a, b, c, . . . , k)$ the least common multiple and by $D(a, b, c, . . . , k)$ the greatest common divisor of $a, b, c, . . . , k$. Prove that: a) $M(a, b)D(a, b) = ab$, b) $\frac{M(a, b, c)D(a, b)D(b, c)D(a, c)}{D(a, b, c)}= abc$.

An electric field varies according the the relationship, \[ \textbf{E} = \left( 0.57 \, \dfrac{\text{N}}{\text{C}} \right) \cdot \sin \left[ \left( 1720 \, \text{s}^{-1} \right) \cdot t \right]. \]Find the maximum displacement current through a $ 1.0 \, \text{m}^2 $ area perpendicular to $\vec{\mathbf{E}}$. Assume the permittivity of free space to be $ 8.85 \times 10^{-12} \, \text{F}/\text{m} $. Round to two significant figures. [i]Problem proposed by Kimberly Geddes[/i]

Let triangle $ABC$ have side lengths $AB=16, BC=20, AC=26.$ Let $ACDE, ABFG,$ and $BCHI$ be squares that are entirely outside of triangle $ABC$. Let $J$ be the midpoint of $EH$, $K$ be the midpoint of $DG$, and $L$ be the midpoint of $AC$. Find the area of triangle $JKL$.

If $x$, $y$, $z$ are nonnegative integers satisfying the equation below, then compute $x+y+z$. \[\left(\frac{16}{3}\right)^x\times \left(\frac{27}{25}\right)^y\times \left(\frac{5}{4}\right)^z=256.\] [i]Proposed by Jeffrey Shi[/i]

[asy] size(200); dotfactor=3; pair A=(0,0),B=(1,0),C=(2,0),D=(3,0),X=(1.2,0.7); draw(A--D); dot(A);dot(B);dot(C);dot(D); draw(arc((0.4,0.4),0.4,180,110),arrow = Arrow(TeXHead)); draw(arc((2.6,0.4),0.4,0,70),arrow = Arrow(TeXHead)); draw(B--X,dotted); draw(C--X,dotted); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,S); label("$D$",D,S); label("x",X,fontsize(5pt)); //Credit to TheMaskedMagician for the diagram [/asy] Points $A , B, C$, and $D$ are distinct and lie, in the given order, on a straight line. Line segments $AB, AC$, and $AD$ have lengths $x, y$, and $z$ , respectively. If line segments $AB$ and $CD$ may be rotated about points $B$ and $C$, respectively, so that points $A$ and $D$ coincide, to form a triangle with positive area, then which of the following three inequalities must be satisfied? $\textbf{I. }x<\frac{z}{2}\qquad\textbf{II. }y<x+\frac{z}{2}\qquad\textbf{III. }y<\frac{z}{2}$ $\textbf{(A) }\textbf{I. }\text{only}\qquad\textbf{(B) }\textbf{II. }\text{only}\qquad$ $\textbf{(C) }\textbf{I. }\text{and }\textbf{II. }\text{only}\qquad\textbf{(D) }\textbf{II. }\text{and }\textbf{III. }\text{only}\qquad\textbf{(E) }\textbf{I. },\textbf{II. },\text{and }\textbf{III. }$

Prove that $$\lim_{n\to\infty}n^2\left(\int^1_0\sqrt[n]{1+x^n}\text dx-1\right)=\frac{\pi^2}{12}.$$

Let $a_1,a_2,...,a_n$ be an arithmetic progression of positive real numbers. Prove that $\tfrac {1}{\sqrt a_1+\sqrt a_2}+\tfrac {1}{\sqrt a_2+\sqrt a_3}+...+\tfrac {1}{\sqrt a_{n-1}+\sqrt a_n}=\tfrac{n-1}{\sqrt {a_1}+\sqrt{a_n}}$.

The natural numbers $x, y > 1$, are such that $x^2 + xy -y$ is the square of a natural number. Prove that $x + y + 1$ is a composite number.

Let $S=\{a_1,a_2,\ldots,a_n\}$ of nenul vectors in a plane. Show that $S{}$ can be partitioned in nenul subsets $B_1, B_2,\ldots, B_m$ with the properties: 1) each vector from $S{}$ is part of only on subset; 2) if $a_i\in B_j$ then the angle between vectors $a_i$ and $c_j$, which is the sum of all vectors from $B_j$ is not greater than $\frac{\pi}{2}$; 3) if $i\neq j$ then the angle between vectors $c_i$ and $c_j$, which is the sum of all vectors from $B_i$ and $B_j$, respectively, is greater than $\frac{\pi}{2}$. What are the possible values of $m$?

In triangle $ ABC$, points $ M,N$ lie on line $ AC$ such that $ MA\equal{}AB$ and $ NB\equal{}NC$. Also $ K,L$ lie on line $ BC$ such that $ KA\equal{}KB$ and $ LA\equal{}LC$. It is know that $ KL\equal{}\frac12{BC}$ and $ MN\equal{}AC$. Find angles of triangle $ ABC$.