$f: \mathbb{R} \rightarrow \mathbb{R}$ is a function such that for $\forall x,y\in \mathbb{R}$ the equation $f(xy+x+y)=f(xy)+f(x)+f(y)$ is true. Prove that $f(x+y)=f(x)+f(y)$ for $\forall$ $x,y\in \mathbb{R}$.

$(2\times 3\times 4)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right) =$ $\text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 26$

If $x +\frac{1}{x} = 5$, what is the value of $x^3 +\frac{1}{x^3} $ ?

Consider an odd prime $p$ and a positive integer $N < 50p$. Let $a_1, a_2, \ldots , a_N$ be a list of positive integers less than $p$ such that any specific value occurs at most $\frac{51}{100}N$ times and $a_1 + a_2 + \cdots· + a_N$ is not divisible by $p$. Prove that there exists a permutation $b_1, b_2, \ldots , b_N$ of the $a_i$ such that, for all $k = 1, 2, \ldots , N$, the sum $b_1 + b_2 + \cdots + b_k$ is not divisible by $p$. [i]Will Steinberg, United Kingdom[/i]

What is the graph of $y^4+1=x^4+2y^2$ in the coordinate plane? $ \textbf{(A)}\ \textbf{Two intersecting parabolas} \qquad \textbf{(B)}\ \textbf{Two nonintersecting parabolas} \qquad \textbf{(C)}\ \textbf{Two intersecting circles} \qquad \textbf{(D)}\ \textbf{A circle and a hyperbola} \qquad \textbf{(E)}\ \textbf{A circle and two parabolas}$

For every natural $n$ let $\phi (n)$ be the number of coprime numbers $k \in \{1,2,...,n\}$. (Example: $\phi (12) = 4$, because among the numbers $1, 2, ..., 12$ there are only the$ 4$ numbers, $1, 5, 7$ and $11$ coprime to$12.$) If $k$ is a natural number, then one defines $\phi^k (n)=\underbrace{\strut \phi (\phi ...(\phi (n)) ...)}_{(k \, times \phi)}$ (Example: $\phi^3 (n)=\phi (\phi (\phi (n))) $) For every whole $n> 2$ let $c(n)$ be the smallest natural number $k$ with $\phi^k (n)= 2$. Prove that $c (ab) = c (a) + c (b)$ for odd integers $a$ and $b$, both of which are greater than $2$, .

During the class interval, $n$ children sit in a circle and play the game described below. The teacher goes around the children clockwisely and hands out candies to them according to the following regulations: Select a child, give him a candy; and give the child next to the first child a candy too; then skip over one child and give next child a candy; then skip over two children; give the next child a candy; then skip over three children; give the next child a candy;... Find the value of $n$ for which the teacher can ensure that every child get at least one candy eventually (maybe after many circles).

Let $n$ be a given positive integer. A restaurant offers a choice of $n$ starters, $n$ main dishes, $n$ desserts and $n$ wines. A merry company dines at the restaurant, with each guest choosing a starter, a main dish, a dessert and a wine. No two people place exactly the same order. It turns out that there is no collection of $n$ guests such that their orders coincide in three of these aspects, but in the fourth one they all differ. (For example, there are no $n$ people that order exactly the same three courses of food, but $n$ different wines.) What is the maximal number of guests?

Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$. [i]Alex Zhu.[/i]

The circle is inscribed in a triangle, inscribed in a semicircle. Find the marked angle $a$. [img]https://cdn.artofproblemsolving.com/attachments/8/e/334c8662377155086e9211da3589145f460b52.png[/img]

a) A square table with $49$ small squares is filled with numbers $1$ to $7$ so that in each row and in each column all numbers from $1$ to $7$ are present. Let the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , 7$ are present. b) A square table with $n^2$ small squares is filled with numbers $1$ to $n$ so that in each row and in each column all numbers from $1$ to $n$ are present. Let $n$ be odd and the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , n$ are present.

In the triangle $ABC$ points $M$ and $N$ are the midpoints of sides $AC$ and $AB$ respectively. $I$ is the incenter of the triangle. It is known that the angle $MIC$ is a right angle. Find the angle $NIB$.

A Pythagorean triple is a solution of the equation $x^2 + y^2 = z^2$ in positive integers such that $x < y$. Given any non-negative integer $n$ , show that some positive integer appears in precisely $n$ distinct Pythagorean triples.

For each positive integer $n>1$, if $n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$($p_i$ are pairwise different prime numbers and $\alpha_i$ are positive integers), define $f(n)$ as $\alpha_1+\alpha_2+\cdots+\alpha_k$. For $n=1$, let $f(1)=0$. Find all pairs of integer polynomials $P(x)$ and $Q(x)$ such that for any positive integer $m$, $f(P(m))=Q(f(m))$ holds.

Prove that there exists a set $S$ of lines in the three dimensional space satisfying the following conditions: $i)$ For each point $P$ in the space, there exist a unique line of $S$ containing $P$. $ii)$ There are no two lines of $S$ which are parallel.

Find all positive integers $n$ for which there exist positive integers $a, b,$ and $c$ that satisfy the following three conditions: $\bullet$ $a+b+c=n$ $\bullet$ $a$ is a divisor of $b$ and $b$ is a divisor of $c$ $\bullet$ $a < b < c$

A snowman is built on a level plane by placing a ball radius $6$ on top of a ball radius $8$ on top of a ball radius $10$ as shown. If the average height above the plane of a point in the snowman is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m + n$. [asy] size(150); draw(circle((0,0),24)); draw(ellipse((0,0),24,9)); draw(circle((0,-56),32)); draw(ellipse((0,-56),32,12)); draw(circle((0,-128),40)); draw(ellipse((0,-128),40,15)); [/asy]

Compute the area of the smallest triangle which can contain six congruent, non-overlapping unit circles.

Compute \[\dfrac{2^3-1}{2^3+1}\cdot\dfrac{3^3-1}{3^3+1}\cdot\dfrac{4^3-1}{4^3+1}\cdot\dfrac{5^3-1}{5^3+1}\cdot\dfrac{6^3-1}{6^3+1}.\]

In a circle, $15$ equally spaced points are drawn and arbitrary triangles are formed connecting $3$ of these points. How many non-congruent triangles can be drawn?

Dexter is running a pyramid scheme. In Dexter's scheme, he hires ambassadors for his company, Lie Ultimate. Any ambassador for his company can recruit up to two more ambassadors (who are not already ambassadors), who can in turn recruit up to two more ambassadors each, and so on (Dexter is a special ambassador that can recruit as many ambassadors as he would like). An ambassador's downline consists of the people they recruited directly as well as the downlines of those people. An ambassador earns executive status if they recruit two new people and each of those people has at least $70$ people in their downline (Dexter is not considered an executive). If there are $2020$ ambassadors (including Dexter) at Lie Ultimate, what is the maximum number of ambassadors with executive status?

Solve for integers $x,y,z$: \[ \{ \begin{array}{ccc} x + y &=& 1 - z \\ x^3 + y^3 &=& 1 - z^2 . \end{array} \]

Kate rode her bicycle for $ 30$ minutes at a speed of $ 16$ mph, then walked for $ 90$ minutes at a speed of $ 4$ mph. What was her overall average speed in miles per hour? $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14$

You have two blackboards $A$ and $B$. You have to write on them some of the integers greater than or equal to $2$ and less than or equal to $20$ in such a way that each number on blackboard $A$ is co-prime with each number on blackboard $B.$ Determine the maximum possible value of multiplying the number of numbers written in $A$ by the number of numbers written in $B$.

Let $p_1<p_2<p_3<p_4$ and $q_1<q_2<q_3<q_4$ be two sets of prime numbers, such that $p_4 - p_1 = 8$ and $q_4 - q_1= 8$. Suppose $p_1 > 5$ and $q_1>5$. Prove that $30$ divides $p_1 - q_1$.