Consider $x_1,\ldots,x_n>0$. Show that there exists $a_1,a_2,\ldots,a_n\in \{-1,1\}$ such that \[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\geqslant (a_1x_1+a_2x_2+\ldots +a_nx_n)^2.\] [i]Laurențiu Panaitopol[/i]

In an isosceles trapezoid $ABCD$ with bases $AD$ and $BC$, diagonals intersect at point $P$, and lines $AB$ and $CD$ intersect at point $Q$. $O_1$ and $O_2$ are the centers of the circles circumscribed around the triangles $ABP$ and $CDP$, $r$ is the radius of these circles. Construct the trapezoid ABCD given the segments $O_1O_2$, $PQ$ and radius $r$.

Find all integers $x,y,z$ such that \[x^3+y^3+z^3=x+y+z=8\]

Two circles $O_1$ and $O_2$ intersect at $A,B$. Diameter $AC$ of $\odot O_1$ intersects $\odot O_2$ at $E$, Diameter $AD$ of $\odot O_2$ intersects $\odot O_1$ at $F$. $CF$ intersects $O_2$ at $H$, $DE$ intersects $O_1$ at $G,H$. $GH\cap O_1=P$. Prove that $PH=PK$.

Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.

Figure 1 shows a road-map connecting 14 cities. Is there a path passing through each city exactly once?

Let be a point $ P $ on the diagonal $ BD $ (excluding its endpoints) of a quadrilateral $ ABCD, $ and $ Q $ be a point in the interior of $ ABD. $ The projections of $ P $ on $ AB,AD $ are $ P_1,P_2, $ respectively, and the projections of $ Q $ on $ AB,AD $ are $ Q_1,Q_2, $ respectively, and verify the equations $ AQ_1=\frac{1}{4}AB $ and $ AQ_2=\frac{1}{4}AD. $ Show that the point $ Q $ is not in the interior of $ AP_1P_2. $

Find all triplets of real numbers $(x,y,z)$ such that the following equations are satisfied simultaneously: \begin{align*} x^3+y=z^2 \\ y^3+z=x^2 \\ z^3+x =y^2 \end{align*}

We are given a $4\times 4$ table. a) Place $7$ stars in the cells in such a way that the erasing of any two rows and two columns will leave at least one of the stars. b) Prove that if there are less than $7$ stars, you can always find two columns and two rows such that erasing them, no star remains in the table.

For positive integer $n$ let $z_n=\sqrt{\frac{3}{n}}+i$, where $i=\sqrt{-1}$. Find $|z_1 \cdot z_2 \cdot z_3 \cdots z_{47}|$.

Let $n\ge1$ and $r\ge2$ be positive integers. Prove that there is no integer $m$ such that $n(n+1)(n+2)=m^r$.

Compute the number of ways to rearrange nine white cubes and eighteen black cubes into a $3\times 3\times 3$ cube such that each $1\times1\times3$ row or column contains exactly one white cube. Note that rotations are considered $\textit{distinct}$.

Lizzie writes a list of fractions as follows. First, she writes $\frac11$, the only fraction whose numerator and denominator add to $2$. Then she writes the two fractions whose numerator and denominator add to $3$, in increasing order of denominator. Then she writes the three fractions whose numerator and denominator sum to $4$ in increasing order of denominator. She continues in this way until she has written all the fractions whose numerator and denominator sum to at most $1000$. So Lizzie's list looks like: $$\frac11, \frac21, \frac12 , \frac31 , \frac22, \frac13, \frac41, \frac32, \frac23, \frac14, ..., \frac{1}{999}.$$ Let $p_k$ be the product of the first $k$ fractions in Lizzie's list. Find, with proof, the value of $p_1 + p_2 + ...+ p_{499500}$.

Let $ k$ be a positive integer. Set $ A \subseteq \mathbb{Z}$ is called a $ \textbf{k \minus{} set}$ if there exists $ x_1, x_2, \cdots, x_k \in \mathbb{Z}$ such that for any $ i \neq j$, $ (x_i \plus{} A) \cap (x_j \plus{} A) \equal{} \emptyset$, where $ x \plus{} A \equal{} \{ x \plus{} a \mid a \in A \}$. Prove that if $ A_i$ is $ \textbf{k}_i\textbf{ \minus{} set}$($ i \equal{} 1,2, \cdots, t$), and $ A_1 \cup A_2 \cup \cdots \cup A_t \equal{} \mathbb{Z}$, then $ \displaystyle \frac {1}{k_1} \plus{} \frac {1}{k_2} \plus{} \cdots \plus{} \frac {1}{k_t} \geq 1$.

Which positive numbers $ x$ satisfy the equation $ (\log_3x)(\log_x5)\equal{}\log_35$? $ \textbf{(A)}\ 3 \text{ and } 5 \text{ only} \qquad \textbf{(B)}\ 3, 5, \text{ and } 15 \text{ only} \qquad$ $ \textbf{(C)}\ \text{only numbers of the form } 5^n \cdot 3^m, \text{ where } n \text{ and } m \text{ are }$ $ \text{positive integers} \qquad$ $ \textbf{(D)}\ \text{all positive } x \neq 1 \qquad \textbf{(E)}\ \text{none of these}$

Let $n$ be a constant positive integer. Show that for only non-negative integers $k$, the Diophantine equation $\sum_{i=1 }^{n}{ x_i ^3}=y^{3k+2}$ has infinitely many solutions in the positive integers $x_i, y$.

In a triangle $\vartriangle ABC$, $\angle B = 2\angle C$. Let $P$ and $Q$ be points on the perpendicular bisector of segment $BC$ such that rays $AP$ and $AQ$ trisect $\angle A$. Prove that $PQ$ is smaller than $AB$ if and only if $\angle B$ is obtuse.

Let $\phi=\tfrac{1+\sqrt 5}{2}$. Find \[\left(4+\phi^{\frac12}\right)\left(4-\phi^{\frac12}\right)\left(4+i\phi^{-\frac12}\right)\left(4-i\phi^{-\frac12}\right).\]

On the extension of the largest side $AC$ of the triangle $ABC$ set aside the segment $CM$ such that $CM = BC$. Prove that the angle $ABM$ is obtuse or right.

Solve system of equation $$8(x^3+y^3+z^3)=73$$ $$2(x^2+y^2+z^2)=3(xy+yz+zx)$$ $$xyz=1$$ in set $\mathbb{R}^3$

Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y$. What is the value of \[(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?\] $ \textbf{(A)}\ -2 \qquad \textbf{(B)}\ -1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

Let $A$ be an $n\times n$ matrix of real numbers for some $n\ge 1.$ For each positive integer $k,$ let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th power. Show that if $A^k=A^{[k]}$ for $k=1,2,\cdots,n+1,$ then $A^k=A^{[k]}$ for all $k\ge 1.$

Jim has two fair $6$-sided dice, one whose faces are labelled from $1$ to $6$, and the second whose faces are labelled from $3$ to $8$. Twice, he randomly picks one of the dice (each die equally likely) and rolls it. Given the sum of the resulting two rolls is $9$, if $\frac{m}{n}$ is the probability he rolled the same die twice where $m, n$ are relatively prime positive integers, then what is $m + n$?

Function $f(n), n \in \mathbb N$, is defined as follows: Let $\frac{(2n)!}{n!(n+1000)!} = \frac{A(n)}{B(n)}$ , where $A(n), B(n)$ are coprime positive integers; if $B(n) = 1$, then $f(n) = 1$; if $B(n) \neq 1$, then $f(n)$ is the largest prime factor of $B(n)$. Prove that the values of $f(n)$ are finite, and find the maximum value of $f(n).$

Five bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes at most $36$ hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone? [i]Author: Ray Li[/i]