Define the two sequences $a_0, a_1, a_2, \cdots$ and $b_0, b_1, b_2, \cdots$ by $a_0 = 3$ and $b_0 = 1$ with the recurrence relations $a_{n+1} = 3a_n + b_n$ and $b_{n+1} = 3b_n - a_n$ for all nonnegative integers $n.$ Let $r$ and $s$ be the remainders when $a_{32}$ and $b_{32}$ are divided by $31,$ respectively. Compute $100r + s.$

Three bells begin to ring simultaneously. The intervals between strikes for these bells are, respectively, $\frac43$ seconds, $\frac53$ second and $2$ seconds. Impacts that coincide in time are perceived as one. How many beats will be heard in $1$ minute? (Include first and last.)

Show that if we choose any $n + 2$ distinct numbers from the set $\{1, 2, 3, . . . , 3n\}$ there will be two whose difference is greater than $n$ and smaller than $2n$.

A point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, is chosen at random, with all such points having an equal probability of being chosen. What is the probability that the distance from the point to the origin is at most two units? $\textbf{(A) }\frac{13}{81}\qquad\textbf{(B) }\frac{15}{81}\qquad\textbf{(C) }\frac{13}{64}\qquad\textbf{(D) }\frac{\pi}{16}\qquad \textbf{(E) }\text{the square of a rational number}$

Let $\triangle ABC$ be a triangle with circumcenter $O$ satisfying $AB=13$, $BC = 15$, and $AC = 14$. Suppose there is a point $P$ such that $PB \perp BC$ and $PA \perp AB$. Let $X$ be a point on $AC$ such that $BX \perp OP$. What is the ratio $AX/XC$? [i]Proposed by Thomas Lam[/i]

Let $ABC$ an acute triangle and $D$ and $E$ the feet of heights by $A$ and $B$, respectively, and let $M$ be the midpoint of $AC$. The circle that passes through $D$ and $B$ and is tangent to $BE$ in $B$ intersects the line $BM$ in $F, F\neq B$. Show that $FM$ is the angle bisector of $\angle AFD$.

In an $ h$-meter race, Sunny is exactly $ d$ meters ahead of Windy when Sunny finishes the race. The next time they race, Sunny sportingly starts $ d$ meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. How many meters ahead is Sunny when Sunny finishes the second race? $ \textbf{(A)}\ \frac {d}{h} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ \frac {d^2}{h} \qquad \textbf{(D)}\ \frac {h^2}{d} \qquad \textbf{(E)}\ \frac {d^2}{h \minus{} d}$

Given positive real numbers $a,b,c$ with $ab+bc+ca\ge a+b+c$ , prove that $$(a + b + c)(ab + bc+ca) + 3abc \ge 4(ab + bc + ca).$$ (I. Gorodnin)

For $n = 1,2,3,...$. $a_n$ is defined by: $$a_n =\frac{1 \cdot 4 \cdot 7 \cdot ... (3n-2)}{2 \cdot 5 \cdot 8 \cdot ... (3n-1)}$$ Prove that for every $n$ holds that $$\frac{1}{\sqrt{3n+1}}\le a_n \le \frac{1}{\sqrt[3]{3n+1}}$$

In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.

In $\triangle ABC$, max $\{\angle A, \angle B \} = \angle C + 30^{\circ}$ and $\frac{R}{r} = \sqrt{3} + 1$, where $R$ is the radius of the circumcircle and $r$ is the radius of the incircle. Find $\angle C$ in degrees.

There are $2014$ airports in the faraway land of Artinia. Each pair of airports is connected by a nonstop flight in one or both directions. Show that there is some airport from which it is possible to reach every other airport in at most two flights.

Determine the locus of points, for whom the ratio of the distances to two given points has a constant value.

Suppose that $\alpha$ is a real number and $a_1<a_2<.....$ is a strictly increasing sequence of natural numbers such that for each natural number $n$ we have $a_n\le n^{\alpha}$. We call the prime number $q$ golden if there exists a natural number $m$ such that $q|a_m$. Suppose that $q_1<q_2<q_3<.....$ are all the golden prime numbers of the sequence $\{a_n\}$. [b]a)[/b] Prove that if $\alpha=1.5$, then $q_n\le 1390^n$. Can you find a better bound for $q_n$? [b]b)[/b] Prove that if $\alpha=2.4$, then $q_n\le 1390^{2n}$. Can you find a better bound for $q_n$? [i]part [b]a[/b] proposed by mahyar sefidgaran by an idea of this question that the $n$th prime number is less than $2^{2n-2}$ part [b]b[/b] proposed by mostafa einollah zade[/i]

3. Consider all 100-digit positive integers such that each decimal digit of these equals $2,3,4,5,6$, or 7 . How many of these integers are divisible by $2^{100}$ ? Pavel Kozhevnikov

In a convex pentagon $ABCDE$, the area of the triangles $ABC, ABD, ACD$ and $ADE$ are equal and have the value $F$. What is the area of the triangle $BCE$ ?

Given a real number $\alpha$ satisfying $\alpha^3 = \alpha + 1$. Determine all $4$-tuples of rational numbers $(a, b, c, d)$ satisfying: $a\alpha^2 + b\alpha+ c = \sqrt{d}.$

Let $n\geq2$ be an integer. $n$ is a prime if it is only divisible by $1$ and $n$. Prove that there are infinitely many prime numbers.

In Cartesian space, let $\Omega = \{(a, b, c) : a, b, c$ are integers between $1$ and $30\}$. A point of $\Omega$ is said to be [i]visible [/i] from the origin if the segment that joins said point with the origin does not contain any other elements of $\Omega$. Find the number of points of $\Omega$ that are [i]visible [/i] from the origin.

Denote $X,Y$ two convex polygons, such that $X$ is contained inside $Y$. Denote $S (X)$, $P (X)$, $S (Y)$, $P (Y)$ the area and perimeter of the first and second polygons, respectively. Prove that $$ \frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.$$

Decompose the polynomial $$x^8 + x^4 +1$$ to factors of at most second degree.

Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true." (1) It is prime. (2) It is even. (3) It is divisible by 7. (4) One of its digits is 9. This information allows Malcolm to determine Isabella's house number. What is its units digit? $\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Count the sum of the four-digit positive integers containing only odd digits in their decimal representation.

Convex quadrilateral $ABCD$ is described on a circle $\omega$, and is not a trapezius inscribed in a circle. Let the tangency points of $\omega$ and $AB, BC, CD, DA$ be $K, L, M, N$ respectively. A circle with a center $I_K$, different from $\omega$ is tangent to the segement $AB$ and lines $AD, BC$. A circle with center $I_L$, different from $\omega$ is tangent to segment $BC$ and lines $AB, CD$. A circle with center $I_M$, different from $\omega$ is tangent to segment $CD$ and lines $AD, BC$. A circle with center $I_N$, different from $\omega$ is tangent to segment $AD$ and lines $AB, CD$. Prove that the lines $I_KK, I_LL, I_MM, I_NN$ are concurrent.

Let $p_{1}, p_{2}, \cdots, p_{n}$ be distinct primes greater than $3$. Show that \[2^{p_{1}p_{2}\cdots p_{n}}+1\] has at least $4^{n}$ divisors.