Two real number sequences are guiven, one arithmetic $\left(a_n\right)_{n\in \mathbb {N}}$ and another geometric sequence $\left(g_n\right)_{n\in \mathbb {N}}$ none of them constant. Those sequences verifies $a_1=g_1\neq 0$, $a_2=g_2$ and $a_{10}=g_3$. Find with proof that, for every positive integer $p$, there is a positive integer $m$, such that $g_p=a_m$.

Let $A,B$ be two matrices with positive integer entries such that sum of entries of a row in $A$ is equal to sum of entries of the same row in $B$ and sum of entries of a column in $A$ is equal to sum of entries of the same column in $B$. Show that there exists a sequence of matrices $A_1,A_2,A_3,\cdots , A_n$ such that all entries of the matrix $A_i$ are positive integers and in the sequence \[A=A_0,A_1,A_2,A_3,\cdots , A_n=B,\] for each index $i$, there exist indexes $k,j,m,n$ such that \[\begin{array}{*{20}{c}} \\ {{A_{i + 1}} - {A_{i}} = } \end{array}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} \quad \quad \ \ j& \ \ \ {k} \end{array}} \\ {\begin{array}{*{20}{c}} m \\ n \end{array}\left( {\begin{array}{*{20}{c}} { + 1}&{ - 1} \\ { - 1}&{ + 1} \end{array}} \right)} \end{array} \ \text{or} \ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} \quad \quad \ \ j& \ \ \ {k} \end{array}} \\ {\begin{array}{*{20}{c}} m \\ n \end{array}\left( {\begin{array}{*{20}{c}} { - 1}&{ + 1} \\ { + 1}&{ - 1} \end{array}} \right)} \end{array}.\] That is, all indices of ${A_{i + 1}} - {A_{i}}$ are zero, except the indices $(m,j), (m,k), (n,j)$, and $(n,k)$.

Let $ABC$ be an isosceles triangle with $\measuredangle C=\measuredangle B=36$. The point $M$ is in interior of $ ABC$ such that $\measuredangle MBC=24^{\circ} , \measuredangle BCM=30^{\circ}$ $N = AM \cap BC.$. Find $\measuredangle MCB$ .

Solve in none-negative integers ${x^3} + 7{x^2} + 35x + 27 = {y^3}$.

Given are two points $B,C$. Consider point $A$ not lying on the line $BC$ and draw the circles $C_1(K_1,R_1)$ (with center $K_1$ and radius $R_1$) and $C_2(K_2,R_2)$ with chord $AB, AC$ respectively such that their centers lie on the interior of the triangle $ABC$ and also $R_1 \cdot AC= R_2 \cdot AB$. Let $T$ be the intersection point of the two circles, different from $A$, and M be a random pointof line $AT$, prove that $TC \cdot S_{(MBT)}=TB \cdot S_{(MCT)}$

Each $1 \times 1$ square of an $n \times n$ table contains a different number. The smallest number in each row is marked, and these marked numbers are in different columns. Then the smallest number in each column is marked, and these marked numbers are in different rows. Prove that the two sets of marked numbers are identical. (V Klepcyn)

There are $ n \ge 3 $ puddings in a room. If a pudding $ A $ hates a pudding $ B $, then $ B $ hates $ A $ as well. Suppose the following two conditions holds: 1. Given any four puddings, there are two puddings who like each other. 2. For any positive integer $ m $, if there are $ m $ puddings who like each other, then there exists $ 3 $ puddings (from the other $ n-m $ puddings) that hate each other. Find the smallest possible value of $ n $.

Let be $ABC$ an acute-angled triangle. Consider point $P$ lying on the opposite ray to the ray $BC$ such that $|AB|=|BP|$. Similarly, consider point $Q$ on the opposite ray to the ray $CB$ such that $|AC|=|CQ|$. Denote $J$ the excenter of $ABC$ with respect to $A$ and $D,E$ tangent points of this excircle with the lines $AB$ and $AC$, respectively. Suppose that the opposite rays to $DP$ and $EQ$ intersect in $F\neq J$. Prove that $AF\perp FJ$.

Prove that $46^{2n+1} + 296 \cdot 13^{2n+1}$ is divisible by $1947$.

Compute $\displaystyle\int_0^{\pi/3}x\tan^2(x)dx$.

Let $ABC$ be an acute angled triangle with orthocentre $H$. Let $E = BH \cap AC$ and $F= CH \cap AB$. Let $D, M, N$ denote the midpoints of segments $AH, BD, CD$ respectively, and $T = FM \cap EN$. Suppose $D, E, T, F$ are concylic. Prove that $DT$ passes through the circumcentre of $ABC$. [i]Proposed by Pranjal Srivastava[/i]

The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? $\textbf{(A) }19\qquad \textbf{(B) }31\qquad \textbf{(C) }271\qquad \textbf{(D) }319\qquad \textbf{(E) }511\qquad$

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

Let $a\%b$ denote the remainder when $a$ is divided by $b$. Find $\Sigma_{i=1}^{100}(100\%i)$.

Solve the following equation over the integers $$ 25x^2y^2+10x^2y+25xy^2+x^2+30xy+2y^2+5x+7y+6= 0.$$

[b]1)[/b] Prove that there are finite sequences, of any length, of nonegative integers having the property that the arithmetic mean of any choice of its elements is natural. [b]2)[/b] Study if there is an increasing infinite sequence of nonegative integers having the property that the arithmetic mean of any finite choice of its elements is natural.

Given a circle, point $C$ on it and point $A$ outside the circle. The equilateral triangle $ACP$ is constructed on the segment $AC$. Point $C$ moves along the circle. What trajectory will the point $P$ describe?

Let $G$ be a simple graph with $3n^2$ vertices ($n\geq 2$). It is known that the degree of each vertex of $G$ is not greater than $4n$, there exists at least a vertex of degree one, and between any two vertices, there is a path of length $\leq 3$. Prove that the minimum number of edges that $G$ might have is equal to $\frac{(7n^2- 3n)}{2}$.

Find all pairs of real numbers $(x,y)$ for which \[ \begin{aligned} x^2+y^2+xy&=133 \\ x+y+\sqrt{xy}&=19 \end{aligned} \]

A cell phone plan costs $\$20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay? $ \textbf{(A)}\ \$ 24.00 \qquad \textbf{(B)}\ \$ 24.50\qquad \textbf{(C)}\ \$ 25.50\qquad \textbf{(D)}\ \$ 28.00\qquad \textbf{(E)}\ \$ 30.00$

Let $a_i, b_i$ ($1 \le i \le n$, $n \ge 2$) be positive real numbers such that $\sum_{i=1}^n a_i = \sum_{i=1}^n b_i$. Prove that $\sum_{i=1}^n \frac{(a_{i+1}+b_{i+1})^2}{n(a_i-b_i)^2+4(n-1)\sum_{j=1}^n a_jb_j} \ge \frac{1}{n-1}$

Given a polynomial with integer coefficents$$P(x)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0,n\ge 1$$satisfying these conditions: i) $|a_0|$ is not a perfect square. ii) $P(x)$ is irreducible in $\mathbb{Q}[x].$ Prove that: $P(x^2)$ is irreducible in $\mathbb{Q}[x].$

Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing $6$ cans of cat food. On what day of the week did the cat finish eating all the cat food in the box? ${ \textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$

Let $a \ge 2$ be an integer. Find all polynomials $f$ with real coefficients such that $$A = \{a^{n^2} | n \ge 1, n \in Z\} \subset \{f(n) | n \ge 1, n \in Z\} = B.$$

Let $n$ and $k$ be positive integers with $n\geq k\geq 2$. For $i=1,\dots,n$, let $S_i$ be a nonempty set of consecutive integers such that among any $k$ of them, there are two with nonempty intersection. Prove that there is a set $X$ of $k-1$ integers such that each $S_i$, $i=1,\dots,n$ contains at least one integer in $X$.