Find all differentiable functions $f, g:[0,\infty) \to \mathbb{R}$ and the real constant $k\geq 0$ such that \begin{align*} f(x) &=k+ \int_0^x \frac{g(t)}{f(t)}dt \\ g(x) &= -k-\int_0^x f(t)g(t) dt \end{align*} and $f(0)=k, f'(0)=-k^2/3$ and also $f(x)\neq 0$ for all $x\geq 0$.\\ \\ [i] (Nora Gavrea)[/i]

Let $a,b,c$ be positive reals such that $a^2+b^2+c^2=3abc$. Prove that \[\frac{a}{b^2c^2}+\frac{b}{c^2a^2}+\frac{c}{a^2b^2} \geq \frac{9}{a+b+c}\]

Given are the real numbers $x_1,x_2,...,x_n$ and $t_1,t_2,...,t_n$ for which holds: $\sum_{i=1}^n x_i = 0$. Prove that $$\sum_{i=1}^n \left( \sum_{j=1}^n (t_i-t_j)^2x_ix_j \right)\le 0.$$

Two towns, $A$ and $B$, are connected by a straight road, $15$ miles long. Traveling from town $A$ to town $B$, the speed limit changes every $5$ miles: from $25$ to $40$ to $20$ miles per hour (mph). Two cars, one at town $A$ and one at town $B$, start moving toward each other at the same time. They drive exactly the speed limit in each portion of the road. How far from town $A$, in miles, will the two cars meet? $\textbf{(A) }7.75 \qquad\textbf{(B) }8 \qquad\textbf{(C) }8.25\qquad\textbf{(D) }8.5 \qquad\textbf{(E) }8.75$

a) If positive integer $N$ is a perfect cube and is not divisible by $10$, then $N=(10m+n)^2$ where $m,n \in N$ with $1\le n\le 9$ b) Find all the positive integers $N$ which are perfect cubes, are not divisible by $10$, such that the number obtained by erasing the last three digits to be also also a perfect cube.

The positive real numbers $a, b, c$ satisfy the equality $a + b + c = 1$. For every natural number $n$ find the minimal possible value of the expression $$E=\frac{a^{-n}+b}{1-a}+\frac{b^{-n}+c}{1-b}+\frac{c^{-n}+a}{1-c}$$

In a video game, there is a board divided into squares, with $27$ rows and $27$ columns. The squares are painted alternately in black, gray and white as follows: $\bullet$ in the first row, the first square is black, the next is gray, the next is white, the next is black, and so on; $\bullet$ in the second row, the first is white, the next is black, the next is gray, the next is white, and so on; $\bullet$ in the third row, the order would be gray-white-black-gray and so on; $\bullet$ the fourth row is painted the same as the first, the fifth the same as the second, $\bullet$ the sixth the same as the third, and so on. In the box in row $i$ and column $j$, there are $ij$ coins. For example, in the box in row $15$ and column $20$ there are $(15) (20) = 300$ coins. Verify that in total there are, in the black squares, $9^2 (13^2 + 14^2 + 15^2)$ coins.

Compute the number of positive integers $n \le 1890$ such that n leaves an odd remainder when divided by all of $2, 3, 5$, and $7$.

For all positive real numbers $a,b,c$, prove that \[\sqrt{\frac{a^4 + 2b^2c^2}{a^2+2bc}} + \sqrt{\frac{b^4+2c^2a^2}{b^2+2ca}} + \sqrt{\frac{c^4 + 2a^2b^2}{c^2 + 2ab}} \geq a + b + c.\] [i]In-Sung Na.[/i]

Suppose that $ABC$ is a triangle with $AB = 6, BC = 12$, and $\angle B = 90^{\circ}$. Point $D$ lies on side $BC$, and point $E$ is constructed on $AC$ such that $\angle ADE = 90^{\circ}$. Given that $DE = EC = \frac{a\sqrt{b}}{c}$ for positive integers $a, b,$ and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, find $a+ b+c$. [i]Proposed by Andrew Wu[/i]

Title is self explanatory. Pick two points on the unit sphere. What is the expected distance between them?

Bucket $A$ contains 3 litres of syrup. Bucket $B$ contains $n$ litres of water. Bucket $C$ is empty. We can perform any combination of the following operations: - Pour away the entire amount in bucket $X$, - Pour the entire amount in bucket $X$ into bucket $Y$, - Pour from bucket $X$ into bucket $Y$ until buckets $Y$ and $Z$ contain the same amount. [b](a)[/b] How can we obtain 10 litres of 30% syrup if $n = 20$? [b](b)[/b] Determine all possible values of $n$ for which the task in (a) is possible.

Prove that the set $\{1,2,…,12001\}$ can be partitioned into 5 groups so that none of them contains an arithmetic progression with length 11.

Fix a sequence $ a_1,a_2,a_3,... $ of integers satisfying the following condition:for all prime numbers $ p $ and all positive integers $ k $, we have $ a_{pk+1}=pa_k-3a_p+13 $.Determine all possible values of $ a_{2013} $.

A $25$ foot ladder is placed against a vertical wall of a building. The foot of the ladder is $7$ feet from the base of the building. If the top of the ladder slips $4$ feet, then the foot of the ladder will slide: $\textbf{(A)}\ 9\text{ ft} \qquad \textbf{(B)}\ 15\text{ ft} \qquad \textbf{(C)}\ 5\text{ ft} \qquad \textbf{(D)}\ 8\text{ ft} \qquad \textbf{(E)}\ 4\text{ ft}$

$ \int \frac{1+2x^3}{1+x^2-2x^3+x^6} dx $ [i]Ion Nedelcu[/i] and [i]Lucian Tutescu[/i]

A set of $n$ points in space is given, no three of which are collinear and no four of which are co-planar (on a single plane), and each pair of points is connected by a line segment. Initially, all the line segments are colorless. A positive integer $b$ is given and Alice and Bob play the following game. In each turn Alice colors one segment red and then Bob colors up to $b$ segments blue. This is repeated until there are no more colorless segments left. If Alice colors a red triangle, Alice wins. If there are no more colorless segments and Alice hasn’t succeeded in coloring a red triangle, Bob wins. Neither player is allowed to color over an already colored line segment. 1. Prove that if $b < \sqrt{2n - 2} -\frac32$ , then Alice has a winning strategy. 2. Prove that if $b \ge 2\sqrt{n}$, then Bob has a winning strategy.

Find all positive integers $x$ and $y$ such that $${x \choose y} = 1432$$

A table consisting of $5$ columns and $32$ rows, which are filled with zero and one numbers, are "varied", if no two lines are filled in the same way.\\ On the exterior of a cylinder, a table with $32$ rows and $16$ columns is constructed. Is it possible to fill the numbers cells of the table with numbers zero and one, such that any five consecutive columns, table $32\times5$ created by these columns, is a varied one? [i]Proposed by Morteza Saghafian[/i]

Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Numbers from$ 1$ to $8$ were written at the vertices of the cube, and on each edge the absolute value of the difference between the numbers at its ends.. What is the smallest number of different numbers that can be written on the edges?

Find all triples $(a,b,c)$ of positive integers such that (i) $a \leq b \leq c$; (ii) $\text{gcd}(a,b,c)=1$; and (iii) $a^3+b^3+c^3$ is divisible by each of the numbers $a^2b, b^2c, c^2a$.

Let $a,\ b,\ c$ be positive real numbers such that $b^2>ac.$ Evaluate \[\int_0^{\infty} \frac{dx}{ax^4+2bx^2+c}.\] [i]1981 Tokyo University, Master Course[/i]

For some constant $b$, the graph of $y = x^2 + b^2 + 2bx - b + 2$ has only one $x$ intercept. What is the value of $b$? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }10$