Let $\mathbb{Q}^+$ be the set of all positive rational numbers. Find all functions $f:\mathbb{Q}^+\rightarrow \mathbb{Q}^+$ satisfying $f(1)=1$ and \[ f(x+n)=f(x)+nf(\frac{1}{x}) \forall n\in\mathbb{N},x\in\mathbb{Q}^+\]

Find the sum of the values of $x$ such that $\cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x$, where $x$ is measured in degrees and $100< x< 200$.

In fields of a $19 \times 19$ table are written integers so that any two lying on neighboring fields differ at most by $2$ (two fields are neighboring if they share a side). Find the greatest possible number of mutually different integers in such a table.

Let $P(x)$ be a polynomial with integer coefficients. Prove that if there is an integer $k$ such that none of the integers $P(1),P(2), ..., P(k)$ is divisible by $k$, then $P(x)$ does not have integer roots.

There are 100 boxers, each of them having different strengths, who participate in a tournament. Any of them fights each other only once. Several boxers form a plot. In one of their matches, they hide in their glove a horse shoe. If in a fight, only one of the boxers has a horse shoe hidden, he wins the fight; otherwise, the stronger boxer wins. It is known that there are three boxers who obtained (strictly) more wins than the strongest three boxers. What is the minimum number of plotters ? [i]Proposed by N. Kalinin[/i]

Twelve places have been arranged at a round table for members of the Jury, with a name tag at each place . Professor K. being absent-minded instead of occupying his place, sits down at the next place (clockwise) . Each of the other Jury members in turn either occupies the place assigned to this member or, if it has been already occupied, sits down at the first free place in the clockwise order. The resulting seating arrangement depends on the order in which the Jury members come to the table. How many different seating arrangements of this kind are possible? (A Shapovalov)

Let $f(x)=x^4-4x^3-3x^2-4x+1$. Compute the sum of the real roots of $f(x)$.

In triangle $ABC$, $D$ is the middle point of side $BC$ and $M$ is a point on segment $AD$ such that $AM=3MD$. The barycenter of $ABC$ and $M$ are on the inscribed circumference of $ABC$. Prove that $AB+AC>3BC$.

A birdbath is designed to overflow so that it will be self-cleaning. Water flows in at the rate of 20 milliliters per minute and drains at the rate of 18 milliliters per minute. One of these graphs shows the volume of water in the birdbath during the filling time and continuing into the overflow time. Which one is it? [asy] size(450); defaultpen(linewidth(0.8)); path[] p={origin--(8,8)--(14,8), (0,10)--(4,10)--(14,0), origin--(14,14), (0,14)--(14,14), origin--(7,7)--(14,0)}; int i; for(i=0; i<5; i=i+1) { draw(shift(21i,0)*((0,16)--origin--(14,0))); draw(shift(21i,0)*(p[i])); label("Time", (7+21i,0), S); label(rotate(90)*"Volume", (21i,8), W); } label("$A$", (0*21 + 7,-5), S); label("$B$", (1*21 + 7,-5), S); label("$C$", (2*21 + 7,-5), S); label("$D$", (3*21 + 7,-5), S); label("$E$", (4*21 + 7,-5), S); [/asy] $\text{(A)}\ \text{A} \qquad \text{(B)}\ \text{B} \qquad \text{(C)}\ \text{C} \qquad \text{(D)}\ \text{D} \qquad \text{(E)}\ \text{E}$

There is located real number $f(A)$ in any point A on the plane. It's known that if $M$ will be centroid of triangle $ABC$ then $f(M)=f(A)+f(B)+f(C)$. Prove that $f(A)=0$ for all points A.

Let $\mathcal{A}_n$ be the set of $n$-tuples $x = (x_1, ..., x_n)$ with $x_i \in \{0, 1, 2\}$. A triple $x, y, z$ of distinct elements of $\mathcal{A}_n$ is called [i]good[/i] if there is some $i$ such that $\{x_i, y_i, z_i\} = \{0, 1, 2\}$. A subset $A$ of $\mathcal{A}_n$ is called [i]good[/i] if every three distinct elements of $A$ form a good triple. Prove that every good subset of $\mathcal{A}_n$ has at most $2(\frac{3}{2})^n$ elements.

Find the real numbers $ x>1 $ having the property that $ \sqrt[n]{\lfloor x^n \rfloor } $ is an integer for any natural number $ n\ge 2. $ [i]Mihai Piticari[/i] and [i]Dan Popescu[/i]

Let $n(n\geq2)$ be a natural number and $a_1,a_2,...,a_n$ natural positive real numbers. Determine the least possible value of the expression $$E_n=\frac{(1+a_1)\cdot(a_1+a_2)\cdot(a_2+a_3)\cdot...\cdot(a_{n-1}+a_n)\cdot(a_n+3^{n+1})} {a_1\cdot a_2\cdot a_3\cdot...\cdot a_n}$$

A positive integer $n$ is [i]magical[/i] if $\lfloor \sqrt{\lceil \sqrt{n} \rceil} \rfloor=\lceil \sqrt{\lfloor \sqrt{n} \rfloor} \rceil$. Find the number of magical integers between $1$ and $10,000$ inclusive.

Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$, $k \geq 7$, and for which the following equalities hold: $$d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1$$ [i]Proposed by Mykyta Kharin[/i]

In quadrilateral $ABCD$, $AB=7, BC=24, CD=15, DA=20,$ and $AC=25$. Let segments $AC$ and $BD$ intersect at $E$. What is the length of $EC$? [i]Proposed by James Lin[/i]

We wish to write $n$ distinct real numbers $(n\geq3)$ on the circumference of a circle in such a way that each number is equal to the product of its immediate neighbors to the left and right. Determine all of the values of $n$ such that this is possible.

Is it possible to put distinct positive integers less than $1991$ in the cells of a $9\times 9$ table so that the products of all the numbers in every column and every row are equal to each other? (N.B. Vasiliev, Moscow)

Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$

A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T = aL + bG,$ where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimate it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet? $\textbf{(A) } 240 \qquad \textbf{(B) } 246 \qquad \textbf{(C) } 252 \qquad \textbf{(D) } 258 \qquad \textbf{(E) } 264$

The first step in finding the product $ (3x \plus{} 2)(x \minus{} 5)$ by use of the distributive property in the form $ a(b \plus{} c) \equal{} ab \plus{} ac$ is: $ \textbf{(A)}\ 3x^2 \minus{} 13x \minus{} 10 \qquad \textbf{(B)}\ 3x(x \minus{} 5) \plus{} 2(x \minus{} 5)\qquad \\\textbf{(C)}\ (3x \plus{} 2)x \plus{} (3x \plus{} 2)( \minus{} 5)\qquad \textbf{(D)}\ 3x^2 \minus{} 17x \minus{} 10\qquad \textbf{(E)}\ 3x^2 \plus{} 2x \minus{} 15x \minus{} 10$

Let $ABCD$ be a convex quadrilateral with $AB=CD$. From a random point $P$ of it's diagonal $BD$, we draw a line parallel to $AB$ that intersects $AD$ at point $M$ and a line parallel to $CD$ that intersects $BC$ at point $N$. Prove that: a) The sum $PM+PN$ is constant, independent of the position of $P$ on the diagonal $BD$. b) $MN\le BD$. When the equality holds?

For positive real numbers $x$ and $y$ define $x*y=\frac{x\cdot y}{x+y}$; then $ \textbf{(A)}\ \text{"*" is commutative but not associative} \\ \qquad\textbf{(B)}\ \text{"*" is associative but not commutative} \\ \qquad\textbf{(C)}\ \text{"*" is neither commutative nor associative} \\ \qquad\textbf{(D)}\ \text{"*" is commutative and associative} \\ \qquad\textbf{(E)}\ \text{none of these} $

Anselmo and Bonifacio start a game where they alternatively substitute a number written on a board. In each turn, a player can substitute the written number by either the number of divisors of the written number or by the difference between the written number and the number of divisors it has. Anselmo is the first player to play, and whichever player is the first player to write the number $0$ is the winner. Given that the initial number is $1036$, determine which player has a winning strategy and describe that strategy. Note: For example, the number of divisors of $14$ is $4$, since its divisors are $1$, $2$, $7$, and $14$.

A unit sphere is centered at $(0, 0, 1)$. There is a point light source located at $(1, 0, 4)$ that sends out light uniformly in every direction but is blocked by the sphere. What is the area of the sphere’s shadow on the $x-y$ plane? (A point $(a, b, c)$ denotes the point in three dimensions with $x$-coordinate $a$, $y$-coordinate $b$, and $z$-coordinate $c$)