A function $ f:\mathbb{Q}_{>0}\longrightarrow\mathbb{Q} $ has the following property: $$ f(xy)=f(x)+f(y),\quad x,y\in\mathbb{Q}_{>0} $$ [b]a)[/b] Demonstrate that there are no injective functions with this property. [b]b)[/b] Do exist surjective functions having this property?

$ 2000(2000^{2000}) \equal{}$ $ \textbf{(A)}\ 2000^{2001} \qquad \textbf{(B)}\ 4000^{2000} \qquad \textbf{(C)}\ 2000^{4000}\qquad \textbf{(D)}\ 4,000,000^{2000} \qquad \textbf{(E)}\ 2000^{4,000,000}$

Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$? $\textbf{(A)}\ 9\pi \qquad\textbf{(B)}\ 12\pi \qquad\textbf{(C)}\ 18\pi \qquad\textbf{(D)}\ 24\pi \qquad\textbf{(E)}\ 27\pi$

An $m\times n$ checkerboard is colored randomly: each square is independently assigned red or black with probability $\frac12.$ we say that two squares, $p$ and $q$, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starting at $p$ and ending at $q,$ in which successive squares in the sequence share a common side. Show that the expected number of connected monochromatic regions is greater than $\frac{mn}8.$

[b]Problem 6.[/b] Let $p>2$ be prime. Find the number of the subsets $B$ of the set $A=\{1,2,\ldots,p-1\}$ such that, the sum of the elements of $B$ is divisible by $p.$ [i] Ivan Landgev[/i]

We call a sequence of $n$ digits one or zero a code. Subsequence of a code is a palindrome if it is the same after we reverse the order of its digits. A palindrome is called nice if its digits occur consecutively in the code. (Code $(1101)$ contains $10$ palindromes, of which $6$ are nice.) a) What is the least number of palindromes in a code? b) What is the least number of nice palindromes in a code?

Prove that $ 5^x+6^x\le 4^x+8^x, $ for any nonegative real numbers $ x. $

Find all pairs of integers $(a, b)$ that satisfy the equation $$(a + 1)(b- 1) = a^2b^2.$$

Let be given two concentric circles of radii $R$ and $R_1 > R$. Let quadrilateral $ABCD$ is inscribed in the smaller circle and let the rays $CD, DA, AB, BC$ meet the larger circle at $A_1, B_1, C_1, D_1$ respectively. Prove that $$ \frac{\sigma(A_1B_1C_1D_1)}{\sigma(ABCD)} \geq \frac{R_1^2}{R^2}$$ where $\sigma(P)$ denotes the area of a polygon $P.$

Let us assume that each of the equations $x^7 + x^2 + 1= 0$ and $x^5- x^4 + x^2- x + 1.001 = 0$ has a single root. Which of these roots is larger?

Let $\Gamma$ be the circumcircle of an acute triangle $ABC.$ The perpendicular to $AB$ from $C$ meets $AB$ at $D$ and $\Gamma$ again at $E.$ The bisector of angle $C$ meets $AB$ at $F$ and $\Gamma$ again at $G.$ The line $GD$ meets $\Gamma$ again at $H$ and the line $HF$ meets $\Gamma$ again at $I.$ Prove that $AI = EB.$

In the plane there is a worm of length 1. Prove that it can be always covered by means of half of a circular disk of diameter 1. [i]Note.[/i] Under a "worm", we understand a continuous curve. The "half of a circular disk" is a semicircle including its boundary.

Define the Fibonacci numbers via $F_0=0$, $f_1=1$, and $F_{n-1}+F_{n-2}$. Olivia flips two fair coins at the same time, repeatedly, until she has flipped a tails on both, not necessarily on the same throw. She records the number of pairs of flips $c$ until this happens (not including the last pair, so if on the last flip both coins turned up tails $c$ would be $0$). What is the expected value $F_c$?

Determine all pairs of integers $(x, y)$ that satisfy equation $(y - 2) x^2 + (y^2 - 6y + 8) x = y^2 - 5y + 62$.

The area of the figure enclosed by the $x$-axis, $y$-axis, and line $7x + 8y = 15$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $a,b,c$ be rational numbers such that $a+b+c$ and $a^2+b^2+c^2$ are [b]equal[/b] integers. Prove that the number $abc$ can be written as the ratio of a perfect cube and a perfect square which are relatively prime.

Prove that, for every integer $n > 2$, the number $$\left[\left( \sqrt[3]{n}+\sqrt[3]{n+2}\right)^3\right]+1$$ is divisible by $8$.

Let $f (x)$ be a polynomial with integer coefficients. Prove that if $f (x)= 1979$ for four different integer values of $x$, then $f (x)$ cannot be equal to $2\times 1979$ for any integral value of $x$.

An integer $c$ is [i]square-friendly[/i] if it has the following property: For every integer $m$, the number $m^2+18m+c$ is a perfect square. (A perfect square is a number of the form $n^2$, where $n$ is an integer. For example, $49 = 7^2$ is a perfect square while $46$ is not a perfect square. Further, as an example, $6$ is not [i]square-friendly[/i] because for $m = 2$, we have $(2)2 +(18)(2)+6 = 46$, and $46$ is not a perfect square.) In fact, exactly one square-friendly integer exists. Show that this is the case by doing the following: (a) Find a square-friendly integer, and prove that it is square-friendly. (b) Prove that there cannot be two different square-friendly integers.

A triangle $ ABC$ can be divided into two isosceles triangles by a line through $ A$. Given that one of the angles of the triangles is $ 30^{\circ}$, find all possible values of the other two angles.

Let $n$ be an integer greater than $1.$ $n$ pupils are seated around a round table, each having a certain number of candies (it is possible that some pupils don't have a candy) such that the sum of all the candies they possess is a multiple of $n.$ They exchange their candies as follows: For each student's candies at first, there is at least a student who has more candies than the student sitting to his/her right side, in which case, the student on the right side is given a candy by that student. After a round of exchanging, if there is at least a student who has candies greater than the right side student, then he/she will give a candy to the next student sitting to his/her right side. Prove that after the exchange of candies is completed (ie, when it reaches equilibrium), all students have the same number of candies.

A total fixed amount of $N$ thousand rupees is given to $A,B,C$ every year, each being given an amount proportional to her age. In the first year, A got half the total amount. When the sixth payment was made, A got six-seventh of the amount that she had in the first year; B got 1000 Rs less than that she had in the first year, and C got twice of that she had in the first year. Find N.

A truck travels $\frac{b}{6}$ feet every $t$ seconds. There are $3$ feet in a yard. How many yards does the truck travel in $3$ minutes? $ \textbf {(A) } \frac{b}{1080t} \qquad \textbf {(B) } \frac{30t}{b} \qquad \textbf {(C) } \frac{30b}{t}\qquad \textbf {(D) } \frac{10t}{b} \qquad \textbf {(E) } \frac{10b}{t}$

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

Let $E$ denote the set of all natural numbers $n$ such that $3 < n < 100$ and the set $\{ 1, 2, 3, \ldots , n\}$ can be partitioned in to $3$ subsets with equal sums. Find the number of elements of $E$.