Fix positive integers $n$ and $k$, and $2n$ positive (not neccesarily distinct) real numbers $a_1,\cdots, a_n$, $b_1, \cdots, b_n$. An equation is written on a whiteboard: $$t=*\times*\times\cdots\times*$$ where $t$ is a fixed positive real number, with exactly $k$ asterisks. Ebi fills each asterisk with a number from $a_1, a_2,\cdots, a_n$, while Rubi fills each asterisk with a number from $b_1, b_2,\cdots, b_n$, so that the equation on the whiteboard is correct. Suppose for every positive real number $t$, the number of ways for Ebi and Rubi to do so are equal. Prove that the sequences $a_1,\cdots, a_n$ and $b_1, \cdots, b_n$ are permutations of each other. [i](Note: $t=a_1a_2a_3$ and $t=a_2a_3a_1$ are considered different ways to fill the asterisks, and the chosen terms need not be distinct, for example $t=a_1a_1a_2$.)[/i] [i]Proposed by Wong Jer Ren[/i]

There are $3$ sticks of each color between blue, red and green, such that we can make a triangle $T$ with sides sticks with all different colors. Dana makes $2$ two arrangements, she starts with $T$ and uses the other six sticks to extend the sides of $T$, as shown in the figure. This leads to two hexagons with vertex the ends of these six sticks. Prove that the area of the both hexagons it´s the same. [asy]size(300); pair A, B, C, D, M, N, P, Q, R, S, T, U, V, W, X, Y, Z, K; A = (0, 0); B = (1, 0); C=(-0.5,2); D=(-1.1063,4.4254); M=(-1.7369,3.6492); N=(3.5,0); P=(-2.0616,0); Q=(0.2425,-0.9701); R=(1.6,-0.8); S=(7.5164,0.8552); T=(8.5064,0.8552); U=(7.0214,2.8352); V=(8.1167,-1.546); W=(9.731,-0.7776); X=(10.5474,0.8552); Y=(6.7813,3.7956); Z=(6.4274,3.6272); K=(5.0414,0.8552); draw(A--B, blue); label("$b$", (A + B) / 2, dir(270), fontsize(10)); label("$g$", (B+C) / 2, dir(10), fontsize(10)); label("$r$", (A+C) / 2, dir(230), fontsize(10)); draw(B--C,green); draw(D--C,green); label("$g$", (C + D) / 2, dir(10), fontsize(10)); draw(C--A,red); label("$r$", (C + M) / 2, dir(200), fontsize(10)); draw(B--N,green); label("$g$", (B + N) / 2, dir(70), fontsize(10)); draw(A--P,red); label("$r$", (A+P) / 2, dir(70), fontsize(10)); draw(A--Q,blue); label("$b$", (A+Q) / 2, dir(540), fontsize(10)); draw(B--R,blue); draw(C--M,red); label("$b$", (B+R) / 2, dir(600), fontsize(10)); draw(Q--R--N--D--M--P--Q, dashed); draw(Y--Z--K--V--W--X--Y, dashed); draw(S--T,blue); draw(U--T,green); draw(U--S,red); draw(T--W,red); draw(T--X,red); draw(S--K,green); draw(S--V,green); draw(Y--U,blue); draw(U--Z,blue); label("$b$", (Y+U) / 2, dir(0), fontsize(10)); label("$b$", (U+Z) / 2, dir(200), fontsize(10)); label("$b$", (S+T) / 2, dir(100), fontsize(10)); label("$r$", (S+U) / 2, dir(200), fontsize(10)); label("$r$", (T+W) / 2, dir(70), fontsize(10)); label("$r$", (T+X) / 2, dir(70), fontsize(10)); label("$g$", (U+T) / 2, dir(70), fontsize(10)); label("$g$", (S+K) / 2, dir(70), fontsize(10)); label("$g$", (V+S) / 2, dir(30), fontsize(10)); [/asy]

For positive $x, y$, and $z$ that satisfy the condition $xyz = 1$, prove the inequality $$\sqrt[3]{\frac{x+y}{2z}}+\sqrt[3]{\frac{y+z}{2x}}+\sqrt[3]{\frac{z+x}{2y}}\le \frac{5(x+y+z)+9}{8}$$

Blondes, brunettes, redheads and brown-haired women participate in the Olympiad. There are twice as many redheads as brown-haired. Blondes and redheads make up a quarter of the total number of participants, and Brown-haired and Blondes one fifth part. Prove that the number of Brunettes is divisible by $7$.

Solve the system of equations \[x^2 + y^2 + z^2 + t^2 = 50;\] \[x^2 - y^2 + z^2 - t^2 = -24;\] \[xy = zt;\] \[x - y + z - t = 0.\]

[center][u]Guts Round[/u] / [u]Set 7[/u][/center] [b]p19.[/b] Let $N_{21}$ be the answer to question 21. Suppose a jar has $3N_{21}$ colored balls in it: $N_{21}$ red, $N_{21}$ green, and $N_{21}$ blue balls. Jonathan takes one ball at a time out of the jar uniformly at random without replacement until all the balls left in the jar are the same color. Compute the expected number of balls left in the jar after all balls are the same color. [b]p20.[/b] Let $N_{19}$ be the answer to question 19. For every non-negative integer $k$, define $$f_k(x) = x(x - 1) + (x + 1)(x - 2) + ...+ (x + k)(x - k - 1),$$ and let $r_k$ and $s_k$ be the two roots of $f_k(x)$. Compute the smallest positive integer $m$ such that $|r_m - s_m| > 10N_{19}$. [b]p21.[/b] Let $N_{20}$ be the answer to question 20. In isosceles trapezoid $ABCD$ (where $\overline{BC}$ and $\overline{AD}$ are parallel to each other), the angle bisectors of $A$ and $D$ intersect at $F$, and the angle bisectors of points $B$ and $C$ intersect at $H$. Let $\overline{BH}$ and $\overline{AF}$ intersect at $E$, and let $\overline{CH}$ and $\overline{DF}$ intersect at $G$. If $CG = 3$, $AE = 15$, and $EG = N_{20}$, compute the area of the quadrilateral formed by the four tangency points of the largest circle that can fit inside quadrilateral $EFGH$.

Prove that: $ \sum_{i\equal{}1}^{n^2} \lfloor \frac{i}{3} \rfloor\equal{} \frac{n^2(n^2\minus{}1)}{6}$ For all $ n \in N$.

Given $f_1(x)=2x-2$ and, for $k\ge2$, defined $f_k(x)=f(f_{k-1}(x))$ to be a real-valued function of $x$. Find the remainder when $f_{2013}(2012)$ is divided by the prime $2011$.

The game involves two players $A$ and $B$. Player $A$ sets the value of one of the coefficients $a, b$ or $c$ of the polynomial $$x^3 + ax^2 + bx + c.$$ Player $B$ indicates the value of any of the two remaining coefficients . Player $A$ then sets the value of the last coefficients. Is there a strategy for player A such that no matter how player $B$ plays, the equation $$x^3 + ax^2 + bx + c = 0$$ to have three different (real) solutions?

The factors of the expression $ x^2\plus{}y^2$ are: $ \textbf{(A)}\ (x\plus{}y)(x\minus{}y) \qquad\textbf{(B)}\ (x\plus{}y)^2 \qquad\textbf{(C)}\ (x^{\frac{2}{3}}\plus{}y^{\frac{2}{3}})(x^{\frac{4}{3}}\plus{}y^{\frac{4}{3}}) \\ \textbf{(D)}\ (x\plus{}iy)(x\minus{}iy) \qquad\textbf{(E)}\ \text{none of these}$

Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that $$ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd. $$ Show that $$ \frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12 $$ and that $-12$ is the maximum.

Find all values of the parameter $a$ for which there are exactly $1998$ integers $x$ satisfying the inequality $$x^2 -\pi x +a < 0.$$

Find all values of $A$ such that $0^o < A < 360^o$ and also $\frac{\sin A}{\cos A - 1} \ge 1$ and $\frac{3\cos A - 1}{\sin A} \ge 1.$

The remainder $R$ obtained by dividing $x^{100}$ by $x^2-3x+2$ is a polynomial of degree less than $2$. Then $R$ may be written as: $\textbf{(A) }2^{100}-1\qquad \textbf{(B) }2^{100}(x-1)-(x-2)\qquad \textbf{(C) }2^{100}(x-3)\qquad$ $\textbf{(D) }x(2^{100}-1)+2(2^{99}-1)\qquad \textbf{(E) }2^{100}(x+1)-(x+2)$

We call a function $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ [i]good[/i] if for all $x,y \in \mathbb{Q}^+$ we have: $$f(x)+f(y)\geq 4f(x+y).$$ a) Prove that for all good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ $$f(x)+f(y)+f(z) \geq 8f(x+y+z)$$ b) Does there exists a good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ such that $$f(x)+f(y)+f(z) < 9f(x+y+z) ?$$

There are $11$ quadratic equations on the board, where each coefficient is replaced by a star. Initially, each of them looks like this $$\star x^2 + \star x + \star= 0.$$ Two players are playing a game making alternating moves. In one move each ofthem replaces one star with a real nonzero number. The first player tries to make as many equations as possible without roots and the second player tries to make the number of equations without roots as small as possible. What is the maximal number of equations without roots that the first player can achieve if the second player plays to her best? Describe the strategies of both players.

Find all $x$ for which, for all $n,$ \[\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.\]

What is the largest possible length of an arithmetic progression of positive integers $ a_{1}, a_{2},\cdots , a_{n}$ with difference $ 2$, such that $ {a_{k}}^{2}\plus{}1$ is prime for $ k \equal{} 1, 2, . . . , n$?

Find all pairs $(a, b)$ of real numbers such that $a \cdot \lfloor b \cdot n\rfloor = b \cdot \lfloor a \cdot n \rfloor$ applies to all positive integers$ n$. (For a real number $x, \lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.)

Euler is selling Mathematician cards to Gauss. Three Fermat cards plus $5$ Newton cards costs $95$ Euros, while $5$ Fermat cards plus $2$ Newton cards also costs $95$ Euros. How many Euroes does one Fermat card cost? $$ \mathrm a. ~ 10\qquad \mathrm b.~15\qquad \mathrm c. ~20 \qquad \mathrm d. ~30 \qquad \mathrm e. ~35 $$

Let $a_1,a_2,\cdots,a_n$ be integers such that $1=a_1\le a_2\le \cdots\le a_{2019}=99$. Find the minimum $f_0$ of the expression $$f=(a_1^2+a_2^2+\cdots+a_{2019}^2)-(a_1a_3+a_2a_4+\cdots+a_{2017}a_{2019}),$$ and determine the number of sequences $(a_1,a_2,\cdots,a_n)$ such that $f=f_0$.

Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called [i]good[/i], if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have \[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \] Find the number of [i]good[/i] functions.

There are $n$ coins lying in a circle. Each coin has two sides, $+$ and $-$. A $flop$ means to flip every coin that has two different neighbors simultaneously, while leaving the others alone. For instance, $++-+$, after one $flop$, becomes $+---$. For $n$ coins, let us define $M$ to be a $perfect$ $number$ if for any initial arrangement of the coins, the arrangement of the coins after $m$ $flops$ is exactly the same as the initial one. (a) When $n=1024$, find a perfect number $M$. (b) Find all $n$ for which a perfect number $M$ exist.

Let $a_1,a_2,\cdots ,a_9$ be $9$ positive integers (not necessarily distinct) satisfying: for all $1\le i<j<k\le 9$, there exists $l (1\le l\le 9)$ distinct from $i,j$ and $j$ such that $a_i+a_j+a_k+a_l=100$. Find the number of $9$-tuples $(a_1,a_2,\cdots ,a_9)$ satisfying the above conditions.

Consider a polynomial with coefficients of real numbers $ \phi(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ with three positive real roots. Assume that $ \phi(0)<0$, prove that \[ 2b^3\plus{}9a^2d\minus{}7abc \le 0.\] [i]Hery Susanto, Malang[/i]