Let $ S$ be a set of all positive integers which can be represented as $ a^2 \plus{} 5b^2$ for some integers $ a,b$ such that $ a\bot b$. Let $ p$ be a prime number such that $ p \equal{} 4n \plus{} 3$ for some integer $ n$. Show that if for some positive integer $ k$ the number $ kp$ is in $ S$, then $ 2p$ is in $ S$ as well. Here, the notation $ a\bot b$ means that the integers $ a$ and $ b$ are coprime.

Let $a,b,c$ be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: $ax^2+bx+c, bx^2+cx+a,$ and $cx^2+ax+b $.

A set of points $P_i$ [i]covers[/i] a polygon if for every point in the polygon, a line can be drawn inside the polygon to at least one $P_i$. Points $A_1, A_2, \cdots, A_n$ in the plane form a $2007$-gon, not necessarily convex. Find the minimum value of $n$ such that for any such polygon, we can pick $n$ points inside it that cover the polygon.

Let $a, b, c, d$ be positive real numbers satisfying $a+b+c+d =1$. Show that \[\frac{2}{(a+b)(c+d)} \leq \frac{1}{\sqrt{ab}}+ \frac{1}{\sqrt{cd}}\mbox{.}\] [i](Swiss Mathematical Olympiad 2011, Final round, problem 6)[/i]

Estimate the number of ordered pairs $(p,q)$ of positive integers at most $2020$ such that the cubic equation $x^3-px-q=0$ has three distinct real roots. If your estimate is $E$ and the answer is $A$, your score for this problem will be \[\Big\lfloor15\min\Big(\frac{A}{E},\frac{E}{A}\Big)\Big\rfloor.\] [i]Proposed by Alex Li[/i]

How many ways are there to paint an $m \times n$ board, so that each square is painted blue, white, brown or gold, and in each $2 \times 2$ square there is one square of each color?

Given a set of boys and girls, we call a pair $(A,B)$ amicable if $A$ and $B$ are friends. The friendship relation is symmetric. A set of people is affectionate if it satisfy the following conditions: i) The set has the same number of boys and girls. ii) For every four different people $A,B,C,D$ if the pairs $(A,B),(B,C),(C,D)$ and $(D,A)$ are all amicable, then at least one of the pairs $(A,C)$ and $(B,D)$ is also amicable. iii) At least $\frac{1}{2013}$ of all boy-girl pairs are amicable. Let $m$ be a positive integer. Prove that there exists an integer $N(m)$ such that if a affectionate set has al least $N(m)$ people, then there exists $m$ boys that are pairwise friends or $m$ girls that are pairwise friends.

Construct a parallelogram $ABCD$, if three points are marked on the plane: the midpoints of its altitudes $BH$ and $BP$ and the midpoint of the side $AD$.

For which $k$ does the system $x^2 - y^2 = 0, (x-k)^2 + y^2 = 1$ have exactly: (i) two, (ii) three real solutions?

If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is $\text{(A)}\ 16 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 44 \qquad \text{(E)}\ 52$

Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$

How many numbers of $ n $ digits formed only with $ 1,9,8 $ and $ 6 $ divide themselves by $ 3 $ ?

Each day Walter gets $\$3$ for doing his chores or $\$5$ for doing them exceptionally well. After 10 days of doing his chores daily, Walter has received a total of $\$36$. On how many days did Walter do them exceptionally well? $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

The [i]spikiness[/i] of a sequence $a_1, a_2, \ldots, a_n$ of at least two real numbers is the sum $\textstyle\sum_{i=1}^{n-1} |a_{i+1}-a_i|.$ Suppose $x_1, x_2, \ldots, x_9$ are chosen uniformly at random from the set $[0, 1].$ Let $M$ be the largest possible value of the spikiness of a permutation of $x_1, x_2, \ldots, x_9.$ Compute the expected value of $M.$

Given triangle $ABC$ and point $P$. Points $A', B', C'$ are the projections of $P$ to $BC, CA, AB$. A line passing through $P$ and parallel to $AB$ meets the circumcircle of triangle $PA'B'$ for the second time in point $C_{1}$. Points $A_{1}, B_{1}$ are defined similarly. Prove that a) lines $AA_{1}, BB_{1}, CC_{1}$ concur; b) triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.

Pictures are taken of $100$ adults and $100$ children, with one adult and one child in each, the adult being the taller of the two. Each picture is reduced to $\frac 1k$ of its original size, where $k$ is a positive integer which may vary from picture to picture. Prove that it is possible to have the reduced image of each adult taller than the reduced image of every child.

A bus and a cyclist left town $A$ at $10$ o'clock in the same direction, and a motorcyclist left town $B$ to meet them $15$ minutes later. The bus drove past the pedestrian at $10$ o'clock $30$ minutes, met the motorcyclist at $11$ o'clock and arrived in the city of $B$ at $12$ o'clock. The motorcyclist met the cyclist $15$ minutes after meeting the bus and another $15$ minutes later caught up with the pedestrian. At what time did the cyclist and the pedestrian meet? (The speeds and directions of movement of all participants were equal, the pedestrian and the motorcyclist were moving in the direction of city $A$.)

To [i]dissect [/i] a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals: [img]https://cdn.artofproblemsolving.com/attachments/0/a/378e477bcbcec26fc90412c3eada855ae52b45.png[/img] An [i]integer-ratio[/i] right triangle is a right triangle whose side lengths are in an integer ratio. For example, a triangle with sides $3,4,5$ is an[i] integer-ratio[/i] right triangle, and so is a triangle with sides $\frac52 \sqrt3 ,6\sqrt3, \frac{13}{2} \sqrt3$. On the other hand, the right triangle with sides$ \sqrt2 ,\sqrt5, \sqrt7$ is not an [i]integer-ratio[/i] right triangle. Determine, with proof, all integers $n$ for which it is possible to completely [i]dissect [/i] a regular $n$-sided polygon into [i]integer-ratio[/i] right triangles.

Let $n\ge 3$ be an integer. In a country there are $n$ airports and $n$ airlines operating two-way flights. For each airline, there is an odd integer $m\ge 3$, and $m$ distinct airports $c_1, \dots, c_m$, where the flights offered by the airline are exactly those between the following pairs of airports: $c_1$ and $c_2$; $c_2$ and $c_3$; $\dots$ ; $c_{m-1}$ and $c_m$; $c_m$ and $c_1$. Prove that there is a closed route consisting of an odd number of flights where no two flights are operated by the same airline.

Compute $$\prod^{12}_{k=1} \left(\prod^{10}_{j=1} \left(e^{2\pi ji/11} - e^{2\pi ki/13}\right) \right) .$$ (The notation $\prod^{b}_{k=a}f(k)$means the product $f(a)f(a + 1)... f(b)$.)

Show that $19^{93} - 13^{99}$ is a positive integer divisible by $162$.

The digits $ 1$, $ 2$, $ 3$, $ 4$, $ 5$, $ 6$, $ 7$, and $ 9$ are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four primes? $ \text{(A)}\ 150 \qquad \text{(B)}\ 160 \qquad \text{(C)}\ 170 \qquad \text{(D)}\ 180 \qquad \text{(E)}\ 190$

Consider all binary sequences (sequences consisting of 0’s and 1’s). In such a sequence the following four types of operation are allowed: (a) $010 \rightarrow 1$, (b) $1 \rightarrow 010$, (c) $110 \rightarrow 0$, and (d) $0 \rightarrow 110$. Determine if it is possible to obtain the sequence $100...0$ (with $2003$ zeroes) from the sequence $0...01$ (with $2003$ zeroes).

A set $A_1 , A_2 , A_3 , A_4$ of 4 points in the plane is said to be [i]Athenian[/i] set if there is a point $P$ of the plane satsifying (*) $P$ does not lie on any of the lines $A_i A_j$ for $1 \leq i < j \leq 4$; (**) the line joining $P$ to the mid-point of the line $A_i A_j$ is perpendicular to the line joining $P$ to the mid-point of $A_k A_l$, $i,j,k,l$ being distinct. (a) Find all [i]Athenian[/i] sets in the plane. (b) For a given [i]Athenian[/i] set, find the set of all points $P$ in the plane satisfying (*) and (**)

A $9\times 9$ array is filled with integers from 1 to 81. Prove that there exists $k\in\{1,2,3,\ldots, 9\}$ such that the product of the elements in the row $k$ is different from the product of the elements in the column $k$ of the array.