Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying (i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$. (ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.

Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0. \] Determine when equality occurs.

A parallelogram $ABCD$ with $|AD| =|BD|$ has been given. A point $E$ lies on line segment $|BD|$ in such a way that $|AE| = |DE|$. The (extended) line $AE$ intersects line segment $BC$ in $F$. Line $DF$ is the angle bisector of angle $CDE$. Determine the size of angle $ABD$. [asy] unitsize (3 cm); pair A, B, C, D, E, F; D = (0,0); A = dir(250); B = dir(290); C = B + D - A; E = extension((A + D)/2, (A + D)/2 + rotate(90)*(A - D), B, D); F = extension(A, E, B, C); draw(A--B--C--D--cycle); draw(A--F--D--B); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, NE); dot("$D$", D, NW); dot("$E$", E, S); dot("$F$", F, SE); [/asy]

The [i]rank[/i] of a rational number $q$ is the unique $k$ for which $q=\frac{1}{a_1}+\cdots+\frac{1}{a_k}$, where each $a_i$ is the smallest positive integer $q$ such that $q\geq \frac{1}{a_1}+\cdots+\frac{1}{a_i}$. Let $q$ be the largest rational number less than $\frac{1}{4}$ with rank $3$, and suppose the expression for $q$ is $\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}$. Find the ordered triple $(a_1,a_2,a_3)$.

Triangle $ABC$ is given. On its sides $AB$, $BC$ and $CA$, respectively, points $X, Y, Z$ are chosen so that $$AX : XB =BY : YC = CZ : ZA = 2:1.$$ It turned out that the triangle $XYZ$ is equilateral. Prove that the original triangle $ABC$ is also equilateral.

Let $a$ be any integer. Define the sequence $x_0,x_1,\ldots$ by $x_0=a$, $x_1=3$, and for all $n>1$ \[x_n=2x_{n-1}-4x_{n-2}+3.\] Determine the largest integer $k_a$ for which there exists a prime $p$ such that $p^{k_a}$ divides $x_{2011}-1$.

For real numbers $a$, $b$, $c$ and $d$ holds: $$ a+b+c+d=0$$ $$a^3+b^3+c^3+d^3=0$$ Prove that sum of some two numbers $a$, $b$, $c$ and $d$ is equal to zero

Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units? $\textbf{(A) } 644 \qquad \textbf{(B) } 658 \qquad \textbf{(C) } 664 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 749$

Source: 2018 Canadian Open Math Challenge Part B Problem 3 ----- The [i]doubling sum[/i] function is defined by \[D(a,n)=\overbrace{a+2a+4a+8a+...}^{\text{n terms}}.\] For example, we have \[D(5,3)=5+10+20=35\] and \[D(11,5)=11+22+44+88+176=341.\] Determine the smallest positive integer $n$ such that for every integer $i$ between $1$ and $6$, inclusive, there exists a positive integer $a_i$ such that $D(a_i,i)=n.$

On the diagonal $AC$ of the convex quadrilateral $ABCD$ points $P$,$Q$ are chosen such that triangles $ABD$,$PCD$ and $QBD$ are similar to each other in this order. Prove that $AQ=PC$ [i]M. Zorka[/i]

Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all functions $f:\mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that for all $x,y\in \mathbb{R}_{>0}$ we have \[f(x) = f(f(f(x)) + y) + f(xf(y)) f(x+y).\]

Find the number of pairs $(m, n)$ of positive integers, both of which are $\le 1000$, such that $\frac{m}{n+1}< \sqrt2 < \frac{m+1}{n}$ (recalling that $ \sqrt2 = 1.414213..$.). (D. Fomin, Leningrad)

Let be two $ 2\times 2 $ real matrices $A,B$ having the property that all their natural powers are not real multiples of the identity. Prove that if some natural power of $ A $ is equal to some natural power of $ B, $ then, $ A,B $ commute. Is the converse statement true? [i]Cosmin Nitu[/i]

a) Let $x$ be a positive real number. Show that $x + 1 / x\ge 2$. b) Let $n\ge 2$ be a positive integer and let $x _1,y_1,x_2,y_2,...,x_n,y_n$ be positive real numbers such that $x _1+x _2+...+x _n \ge x _1y_1+x _2y_2+...+x _ny_n$. Show that $x _1+x _2+...+x _n \le \frac{x _1}{y_1}+\frac{x _2}{y_2}+...+\frac{x _n}{y_n}$

Given a regular triangle $ABC$, whose area is $1$, and the point $P$ on its circumscribed circle. Lines $AP, BP, CP$ intersect, respectively, lines $BC, CA, AB$ at points $A', B', C'$. Find the area of the triangle $A'B'C'$.

On the squares $a1, a2,... , a8$ of a chessboard there are respectively $2^0, 2^1, ..., 2^7$ grains of oat, on the squares $b8, b7,..., b1$ respectively $2^8, 2^9, ..., 2^{15}$ grains of oat, on the squares $c1, c2,..., c8$ respectively $2^{16}, 2^{17}, ..., 2^{23}$ grains of oat etc. (so there are $2^{63}$ grains of oat on the square $h1$). A knight starts moving from some square and eats after each move all the grains of oat on the square to which it had jumped, but immediately after the knight leaves the square the same number of grains of oat reappear. With the last move the knight arrives to the same square from which it started moving. Prove that the number of grains of oat eaten by the knight is divisible by $3$.