Let function pair $f,g : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies \[ f(g(x)y + f(x)) = (y+2015)f(x) \] for every $x,y \in \mathbb{R^+} $ a. Prove that $f(x) = 2015g(x)$ for every $x \in \mathbb{R^+}$ b. Give an example of function pair $(f,g)$ that satisfies the statement above and $f(x), g(x) \geq 1$ for every $x \in \mathbb{R^+}$

In a circle with center $O$ is inscribed a polygon, which is triangulated. Show that the sum of the squares of the distances from $O$ to the incenters of the formed triangles is independent of the triangulation.

We have $n\ge3$ points on the plane such that no three are collinear. Prove that it's possible to name them $P_1,P_2,\ldots,P_n$ such that for all $1<i<n$, the angle $\angle P_{i-1}P_iP_{i+1}$ is acute.

Let $(a_n)_{n\ge0}$ be a sequence of positive real numbers such that \[\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0.\] Prove that $(a_n)_{n\ge0}$ is a geometric sequence. [i]Lucian Dragomir[/i]

Let $S$ be the set of natural numbers whose digits are different and belong to the set $\{1, 3, 5, 7\}$. Calculate the sum of the elements of $S$.

Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.

Suppose that $ \left(u_n\right)$ is a sequence of real numbers satisfying $ u_{n \plus{} 2} \equal{} 2u_{n \plus{} 1} \plus{} u_{n}$, and that $ u_3 \equal{} 9$ and $ u_6 \equal{} 128$. What is $ u_5$? $ \textbf{(A)}\ 40 \qquad \textbf{(B)}\ 53 \qquad \textbf{(C)}\ 68 \qquad \textbf{(D)}\ 88 \qquad \textbf{(E)}\ 104$

We have a $6 \times 6$ square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw?

Let $A_1, A_2, ..., A_m$ and $B_2 , B_3,..., B_n$ be the points on a circle such that $A_1A_2... A_n$ is a regular $m$-gon and $A_1B_2...B_n$ is a regular $n$-gon whereby $n > m$ and the point $B_2$ lies between $A_1$ and $A_2$. Find $\angle B_2A_1A_2$.

Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle's area is numerically equal to $6$ times its perimeter.

Let $x_0=a, x_1= b, x_2 = c$ be given real numbers and let $x_{n+2} = \frac{x_n + x_{n-1}}{2}$ for all $n\geq 1$. Show that the sequence $(x_n)_{n\geq 0}$ converges and find its limit.

For all $x,y,z\in \mathbb{R}\backslash \{1\}$, such that $xyz=1$, prove that \[ \frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1 \]

For a positive integer $A = \overline{a_n ...a_1a_0}$ with nonzero digits which are not all the same ($n \ge 0$), the numbers $A_k = \overline{a_{n-k}...a_1a_0a_n ...a_{n-k+1}}$ are obtained for $k = 1,2,...,n$ by cyclic permutations of its digits. Find all $A$ for which each of the $A_k$ is divisible by $A$.

Positive integers are put into the following table. \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & & \\ \hline 2 & 5 & 9 & 14 & 20 & 27 & 35 & 44 & & \\ \hline 4 & 8 & 13 & 19 & 26 & 34 & 43 & 53 & & \\ \hline 7 & 12 & 18 & 25 & 33 & 42 & & & & \\ \hline 11 & 17 & 24 & 32 & 41 & & & & & \\ \hline 16 & 23 & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline \end{tabular} Find the number of the line and column where the number $2015$ stays.

The sum of squares of two consecutive integers can be a square, as in $3^2+4^2 =5^2$. Prove that the sum of squares of $m$ consecutive integers cannot be a square for $m = 3$ or $6$ and find an example of $11$ consecutive integers the sum of whose squares is a square.

Viorel claims that for any natural number $n$ greater than $2024$, the number $2024^n + 1$ is prime. Is Viorel's statement true?

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$. Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set. [i]Dan Schwarz, Romania[/i]

Determine all polynomials $P$ with integer coefficients for which there exists an integer $a_n$ such that $P(a_n)=n^n$ for all positive integers $n$.

On a table with $25$ columns and $300$ rows, Kostya painted all its cells in three colors. Then, Lesha, looking at the table, for each row names one of the three colors and marks in that row all cells of that color (if there are no cells of that color in that row, he does nothing). After that, all columns that have at least a marked square will be deleted. Kostya wants to be left as few as possible columns in the table, and Lesha wants there to be as many as possible columns in the table. What is the largest number of columns Lesha can guarantee to leave?

The circles $\mathcal C_1(O_1, r_1)$ and $\mathcal C_2(O_2, r_2)$, $r_2 > r_1$, intersect at $A$ and $B$ such that $\angle O_1AO_2 = 90^\circ$. The line $O_1O_2$ meets $\mathcal C_1$ at $C$ and $D$, and $\mathcal C_2$ at $E$ and $F$ (in the order $C$, $E$, $D$, $F$). The line $BE$ meets $\mathcal C_1$ at $K$ and $AC$ at $M$, and the line $BD$ meets $\mathcal C_2$ at $L$ and $AF$ at $N$. Prove that \[ \frac{ r_2}{r_1} = \frac{KE}{KM} \cdot \frac{LN}{LD} . \] [i]Greece[/i]

Compute the sum of all real numbers x which satisfy the following equation $$\frac {8^x - 19 \cdot 4^x}{16 - 25 \cdot 2^x}= 2$$

a. Prove that there are infinitely many positive integers $t$ such that both $2012t+1$ and $2013t+1$ are perfect squares. b. Suppose that $m,n$ are positive integers such that both $mn+1$ and $mn+n+1$ are perfect squares. Prove that $8(2m+1)$ divides $n$.

Let us choose arbitrarily $n$ vertices of a regular $2n$-gon and color them red. The remaining vertices are colored blue. We arrange all red-red distances into a non-decreasing sequence and do the same with the blue-blue distances. Prove that the sequences are equal.

In the $5\times6$ grid shown, fill in all of the grid cells with the digits $0\textendash9$ so that the following conditions are satisfied: [list=1][*] Each digit gets used exactly $3$ times. [*] No digit is greater than the digit directly above it. [*] In any four cells that form a $2\times2$ subgrid, the sum of the four digits must be a multiple of $3$.[/list] You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that works. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) \[\begin{Large}\begin{array}{|c|c|c|c|c|c|}\hline&&&&\,7\,&\\ \hline&\,8\,&&&&\,6\,\\\hline&&\,2\,&\,4\,&&\\ \hline\,5\,&&&&1&\\ \hline&3&&&&\\ \hline\end{array}\end{Large}\]