Let $n\in\mathbb{N}^*$ and $v_1,v_2,\ldots ,v_n$ be vectors in the plane with lengths less than or equal to $1$. Prove that there exists $\xi_1,\xi_2,\ldots ,\xi_n\in\{-1,1\}$ such that \[ | \xi_1v_1+\xi_2v_2+\ldots +\xi_nv_n|\le\sqrt{2}\]

Let $a, b, c,$ be nonnegative reals with $ a+b+c=3 $, find the largest positive real $ k $ so that for all $a,b,c,$ we have $$ a^2+b^2+c^2+k(abc-1)\ge 3 $$

Show that, exist infinite triples $(a, b, c)$ where $a, b, c$ are natural numbers, such that: $2a^2 + 3b^2 - 5c^2 = 1997$

$ A_1 , A_2 , \cdots , A_n $ are given subsets. Let $ S = \left\{ 1, 2, \cdots , n \right\} $. For any $ X \subset S $, let \[ N(X)= \left\{ i \in S-X \ | \ \forall j \in X, \ A_i \cap A_j \ne \emptyset \right\} \] Let $ m $ be an integer such that $ 3 \le m \le n-2 $. Prove that there exist $ X \subset S $ such that $ |X|=m $ and $ |N(X)| \ne 1 $.

Let $n \geq 2$ be a fixed positive integer and let $a_{0},a_{1},...,a_{n-1}$ be real numbers. Assume that all of the roots of the polynomial $P(x) = x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_{1}x+a_{0}$ are strictly positive real numbers. Determine the smallest possible value of $\frac{a_{n-1}^{2}}{a_{n-2}}$ over all such polynomials. [i]Proposed by Nikola Velov[/i]

Let $\alpha(n)$ be the number of digits equal to one in the dyadic representation of a positive integer $n$. Prove that [list=a] [*] the inequality $\alpha(n^2 ) \le \frac{1}{2} \alpha(n) (1+\alpha(n))$ holds, [*] equality is attained for infinitely $n\in\mathbb{N}$, [*] there exists a sequence $\{n_i\}$ such that $\lim_{i \to \infty} \frac{ \alpha({n_{i}}^2 )}{ \alpha(n_{i}) } = 0$.[/list]

(My problem. :D) Call the number of times that the digits of a number change from increasing to decreasing, or vice versa, from the left to right while ignoring consecutive digits that are equal the [i]flux[/i] of the number. For example, the flux of 123 is 0 (since the digits are always increasing from left to right) and the flux of 12333332 is 1, while the flux of 9182736450 is 8. What is the average value of the flux of the positive integers from 1 to 999, inclusive?

The expression $ (81)^{ \minus{} 2^{ \minus{} 2}}$ has the same value as: $ \textbf{(A)}\ \frac {1}{81} \qquad \textbf{(B)}\ \frac {1}{3} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 81 \qquad \textbf{(E)}\ 81^4$

Point $C$ is chosen on the arc of a semicircle with diameter $AB$. The two circles with diameters of $AC$ and $BC$ intersect again at point $D$. If $DA=20$ and $DB=16$, compute the length of $DC$.

A convex pentagon has the property that each of its diagonals cuts off a triangle of unit area. Find the area of the pentagon.

Prove that for every integer $n > 10000$ there exists an integer $m$ such that it can be written as the sum of two squares, and $0<m-n<3\sqrt[4]n$.

A number of runners competed in a race. When Ammar finished, there were half as many runners who had finished before him compared to the number who finished behind him. Julia was the 10th runner to finish behind Ammar. There were twice as many runners who had finished before Julia compared to the number who finished behind her. How many runners were there in the race?

For a given integer $ n\geq 2,$ determine the necessary and sufficient conditions that real numbers $ a_{1},a_{2},\cdots, a_{n},$ not all zero satisfy such that there exist integers $ 0<x_{1}<x_{2}<\cdots<x_{n},$ satisfying $ a_{1}x_{1}\plus{}a_{2}x_{2}\plus{}\cdots\plus{}a_{n}x_{n}\geq 0.$

[b]T[/b]wo ordered pairs $(a,b)$ and $(c,d)$, where $a,b,c,d$ are real numbers, form a basis of the coordinate plane if $ad \neq bc$. Determine the number of ordered quadruples $(a,b,c)$ of integers between $1$ and $3$ inclusive for which $(a,b)$ and $(c,d)$ form a basis for the coordinate plane.

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(0) \in \mathbb Q$ and \[f(x+f(y)^2 ) = {f(x+y)}^2.\] (25 points)

Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m.$

Denote $\mathbb{Z}_{>0}=\{1,2,3,...\}$ the set of all positive integers. Determine all functions $f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$ such that, for each positive integer $n$, $\hspace{1cm}i) \sum_{k=1}^{n}f(k)$ is a perfect square, and $\vspace{0.1cm}$ $\hspace{1cm}ii) f(n)$ divides $n^3$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

A pentagon $ABCDE$ is such that $ABCD$ is cyclic, $BE\parallel CD$, and $DB=DE$. Let us fix the points $B,C,D,E$ and vary $A$ on the circumcircle of $BCD$. Let $P=AC\cap BE$, and $Q=BC\cap DE$. Prove that the second intersection of circles $(ABE)$ and $(PQE)$ lie on a fixed circle.

Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors.

A long time ago, there existed Martians with $3$ different colours: red, green and blue. As Mars was devastated by an intergalactic war, only $2$ Martians of each colours survived. In order to reconstruct the Martian population, they decided to use a machine that transforms two Martians of distinct colours into four Martians of colour different to the two initial ones. For example, if a red Martian and a blue Martian use the machine, they'll be transformed into four green Martians. a) Is it possible that, after using that machine finitely many times, we have $2022$ red Martians, $2022$ green Martians and $2022$ blue Martians? b) Is it possible that, after using that machine finitely many times, we have $2021$ red Martians, $2022$ green Martians and $2023$ blue Martians?

For a positive integer $n$, denote $p(n)$ to be the number of nonnegative integer tuples $(x,y,z,w)$ such that $x+2y+2z+3w=n$. Also, denote $q(n)$ to be the number of nonnegative integer tuples $(a,b,c,d)$ such that (i) $a+b+c+d=n$ (ii) $a \ge b \ge d$ (iii) $a \ge c \ge d$ Prove that for all $n$, $p(n) = q(n)$.

Alice and Bob are playing "factoring game." On the paper, $270000(=2^43^35^4)$ is written and each person picks one number from the paper(call it $N$) and erase $N$ and writes integer $X,Y$ such that $N=XY$ and $\text{gcd}(X,Y)\ne1.$ Alice goes first and the person who can no longer make this factoring loses. If two people use optimal strategy, prove that Alice always win.

Points $A,B,C$ lie on a circle $\omega$ such that $BC$ is a diameter. $AB$ is extended past $B$ to point $B'$ and $AC$ is extended past $C$ to point $C'$ such that line $B'C'$ is parallel to $BC$ and tangent to $\omega$ at point $D$. If $B'D = 4$ and $C'D = 6$, compute $BC$.

A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring? [asy] size(200); defaultpen(linewidth(3)); real[] inrad = {40,34,28,21}; real[] outrad = {55,49,37,30}; real[] center; path[][] quad = new path[4][4]; center[0] = 0; for(int i=0;i<=3;i=i+1) { if(i != 0) { center[i] = center[i-1] - inrad[i-1] - inrad[i]+3.5; } quad[0][i] = arc((0,center[i]),inrad[i],0,90)--arc((0,center[i]),outrad[i],90,0)--cycle; quad[1][i] = arc((0,center[i]),inrad[i],90,180)--arc((0,center[i]),outrad[i],180,90)--cycle; quad[2][i] = arc((0,center[i]),inrad[i],180,270)--arc((0,center[i]),outrad[i],270,180)--cycle; quad[3][i] = arc((0,center[i]),inrad[i],270,360)--arc((0,center[i]),outrad[i],360,270)--cycle; draw(circle((0,center[i]),inrad[i])^^circle((0,center[i]),outrad[i])); } void fillring(int i,int j) { if ((j % 2) == 0) { fill(quad[i][j],white); } else { filldraw(quad[i][j],black); } } for(int i=0;i<=3;i=i+1) { for(int j=0;j<=3;j=j+1) { fillring(((2-i) % 4),j); } } for(int k=0;k<=2;k=k+1) { filldraw(circle((0,-228 - 25 * k),3),black); } real r = 130, s = -90; draw((0,57)--(r,57)^^(0,-57)--(r,-57),linewidth(0.7)); draw((2*r/3,56)--(2*r/3,-56),linewidth(0.7),Arrows(size=3)); label("$20$",(2*r/3,-10),E); draw((0,39)--(s,39)^^(0,-39)--(s,-39),linewidth(0.7)); draw((9*s/10,38)--(9*s/10,-38),linewidth(0.7),Arrows(size=3)); label("$18$",(9*s/10,0),W); [/asy] $ \textbf{(A) } 171\qquad \textbf{(B) } 173\qquad \textbf{(C) } 182\qquad \textbf{(D) } 188\qquad \textbf{(E) } 210$

We say that distinct positive integers $n, m$ are $friends$ if $\vert n-m \vert$ is a divisor of both ${}n$ and $m$. Prove that, for any positive integer $k{}$, there exist $k{}$ distinct positive integers such that any two of these integers are friends.