Let $n$ be a positive integer. How many integer solutions $(i, j, k, l) , \ 1 \leq i, j, k, l \leq n$, does the following system of inequalities have: \[1 \leq -j + k + l \leq n\]\[1 \leq i - k + l \leq n\]\[1 \leq i - j + l \leq n\]\[1 \leq i + j - k \leq n \ ?\]

There are $n$ students. Denoted the number of the selections to select two students (with their weights are $a$ and $b$) such that $\left| {a - b} \right| \le 1$ (kg) and $\left| {a - b} \right| \le 2$ (kg) by $A_1$ and $A_2$, respectively. Prove that $A_2<3A_1+n$.

A rectangular field is half as wide as it is long and is completely enclosed by $ x$ yards of fencing. The area in terms of $ x$ is: $ \textbf{(A)}\ \frac {x^2}{2} \qquad\textbf{(B)}\ 2x^2 \qquad\textbf{(C)}\ \frac {2x^2}{9} \qquad\textbf{(D)}\ \frac {x^2}{18} \qquad\textbf{(E)}\ \frac {x^2}{72}$

Quadrilateral $ABCD$ is inscribed in a circle with $\angle ABC=60^\circ$ and $BC=CD$. Prove that $AB=AD+DC$.

Let $n$ be a natural number and $f(x)$ be a polynomial with real coefficients having $n$ different positive real roots. Is it possible the polynomial: $$x(x+1)(x+2)(x+4)f(x)+a$$ to be presented as the $k$-th power of a polynomial with real coefficients, for some natural $k\geq 2$ and real $a$?

[u]Round 1[/u] [b]p1.[/b] In a chemical lab there are three vials: one that can hold $1$ oz of fluid, another that can hold $2$ oz, and a third that can hold $3$ oz. The first is filled with grape juice, the second with sulfuric acid, and the third with water. There are also $3$ empty vials in the cupboard, also of sizes $1$ oz, $2$ oz, and $3$ oz. In order to save the world with grape-flavored acid, James Bond must make three full bottles, one of each size, filled with a mixture of all three liquids so that each bottle has the same ratio of juice to acid to water. How can he do this, considering he was silly enough not to bring any equipment? [b]p2.[/b] Twelve people, some are knights and some are knaves, are sitting around a table. Knaves always lie and knights always tell the truth. At some point they start up a conversation. The first person says, “There are no knights around this table.” The second says, “There is at most one knight at this table.” The third – “There are at most two knights at the table.” And so on until the $12$th says, “There are at most eleven knights at the table.” How many knights are at the table? Justify your answer. [b]p3.[/b] Aquaman has a barrel divided up into six sections, and he has placed a red herring in each. Aquaman can command any fish of his choice to either ‘jump counterclockwise to the next sector’ or ‘jump clockwise to the next sector.’ Using a sequence of exactly $30$ of these commands, can he relocate all the red herrings to one sector? If yes, show how. If no, explain why not. [img]https://cdn.artofproblemsolving.com/attachments/0/f/956f64e346bae82dee5cbd1326b0d1789100f3.png[/img] [b]p4.[/b] Is it possible to place $13$ integers around a circle so that the sum of any $3$ adjacent numbers is exactly $13$? [b]p5.[/b] Two girls are playing a game. The first player writes the letters $A$ or $B$ in a row, left to right, adding one letter on her turn. The second player switches any two letters after each move by the first player (the letters do not have to be adjacent), or does nothing, which also counts as a move. The game is over when each player has made $2011$ moves. Can the second player plan her moves so that the resulting letters form a palindrome? (A palindrome is a sequence that reads the same forward and backwards, e.g. $AABABAA$.) [u]Round 2[/u] [b]p6.[/b] Eight students participated in a math competition. There were eight problems to solve. Each problem was solved by exactly five people. Show that there are two students who solved all eight problems between them. [b]p7.[/b] There are $3n$ checkers of three different colors: $n$ red, $n$ green and $n$ blue. They were used to randomly fill a board with $3$ rows and $n$ columns so that each square of the board has one checker on it. Prove that it is possible to reshuffle the checkers within each row so that in each column there are checkers of all three colors. Moving checkers to a different row is not allowed. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the sums of the form $\sum_{k=1}^{n} \epsilon_k k^3,$ where $\epsilon_k \in \{-1, 1\}.$ Is any of these sums equal to $0$ if [b](a)[/b] $n=2000;$ [b](b)[/b] $n=2001 \ ?$

Let $ P(x) \in \mathbb{Z}[x]$ be a polynomial of degree $ \text{deg} P \equal{} n > 1$. Determine the largest number of consecutive integers to be found in $ P(\mathbb{Z})$. [i]B. Berceanu[/i]

Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that for all $x$ with $0<|x|\le c$ and for all $\alpha$ with $0<\alpha<1$, we have: $$a^\alpha b^{1-\alpha}\le\frac{a\sinh\alpha x}{\sinh x}+\frac{b\sinh x(1-\alpha)}{\sinh x}.$$

10. Let $n \ge 5$ be an odd number and let $r$ be an integer such that $1\le r \le (n-1)/2$. IN a sports tournament, $n$ players take part in a series of contests. In each contest, $2r+1$ players participate, and the scores obtained by the players are the numbers $$-r, -(r-1),\cdots, -1, 0, 1 \cdots, r-1, r$$ in some order. Each possible subset of $2r+1$ players takes part together in exactly one contest. let the final score of player $i$ be $S_i$, for each $i=1, 2,\cdots,n$. Define $N$ to be the smallest difference between the final scores of two players, i.e., $$N = \min_{i<j}|S_i - S_j|.$$ Determine, with proof, the maximum possible value of $N$.

Find all triples $(k, m, n)$ of positive integers such that $m$ is a prime and: (1) $kn$ is a perfect square; (2) $\frac{k(k-1)}{2}+n$ is a fourth power of a prime; (3) $k-m^2=p$ where $p$ is a prime; (4) $\frac{n+2}{m^2}=p^4$.

The real factors of $ x^2\plus{}4$ are: $\textbf{(A)}\ (x^2+2)(x^2+2) \qquad \textbf{(B)}\ (x^2+2)(x^2-2) \qquad \textbf{(C)}\ x^2(x^2+4) \qquad\\ \textbf{(D)}\ (x^2-2x+2)(x^2+2x+2) \qquad \textbf{(E)}\ \text{Non-existent}$

Two circles $A$ and $B$, both with radius $1$, touch each other externally. Four circles $P, Q, R$ and $S$, all four with the same radius $r$, lie such that $P$ externally touches on $A, B, Q$ and $S$, $Q$ externally touches on $P, B$ and $R$, $R$ externally touches on $A, B, Q$ and $S$, $S$ externally touches on $P, A$ and $R$. Calculate the length of $r.$ [asy] unitsize(0.3 cm); pair A, B, P, Q, R, S; real r = (3 + sqrt(17))/2; A = (-1,0); B = (1,0); P = intersectionpoint(arc(A,r + 1,0,180), arc(B,r + 1,0,180)); R = -P; Q = (r + 2,0); S = (-r - 2,0); draw(Circle(A,1)); draw(Circle(B,1)); draw(Circle(P,r)); draw(Circle(Q,r)); draw(Circle(R,r)); draw(Circle(S,r)); label("$A$", A); label("$B$", B); label("$P$", P); label("$Q$", Q); label("$R$", R); label("$S$", S); [/asy]

Let $m,k$ be positive integers, $k<m$ and $M$ a set with $m$ elements. Prove that the maximal number of subsets $A_1,A_2,\ldots ,A_p$ of $M$ for which $A_i\cap A_j$ has at most $k$ elements, for every $1\le i<j\le p$, equals \[ p_{max}=\binom{m}{0}+\binom{m}{1}+\binom{m}{2}+\ldots+\binom{m}{k+1}\]

Non-negative integers are placed on the vertices of a $100$-gon, the sum of the numbers is $99$. Every minute at one of the vertices that is equal to $0$ will be replaced by $2$ and both its neighboring numbers are subtracted by $1$. Prove that after a while a negative number will appear on the board.

We have to divide a cube onto $k$ non-overlapping tetrahedrons. For what smallest $k$ is it possible?

Is there a polyhedron whose area ratio of any two faces is at least $2$ ?

Compute the sum of the positive integers $n \le 100$ for which the polynomial $x^n + x + 1$ can be written as the product of at least $2$ polynomials of positive degree with integer coefficients.

Let $ABC$ be an equilateral triangle, and let $D,E$ and $F$ be points on $BC,BA$ and $AB$ respectively. Let $\angle BAD= \alpha, \angle CBE=\beta$ and $\angle ACF =\gamma$. Prove that if $\alpha+\beta+\gamma \geq 120^\circ$, then the union of the triangular regions $BAD,CBE,ACF$ covers the triangle $ABC$.

A box contains 36 pink, 18 blue, 9 green, 6 red, and 3 purple cubes that are identical in size. If a cube is selected at random, what is the probability that it is green? $\textbf{(A)}\ \dfrac{1}{9} \qquad \textbf{(B)}\ \dfrac{1}{8} \qquad \textbf{(C)}\ \dfrac{1}{5} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{9}{70}$

A Princeton slot machine has $100$ pictures, each equally likely to occur. One is a picture of a tiger. Alice and Bob independently use the slot machine, and each repeatedly makes independent plays. Alice keeps playing until she sees a tiger, at which point she stops. Similarly, Bob keeps playing until he sees a tiger. Given that Bob plays twice as much as Alice, let the expected number of plays for Alice be $\tfrac{a}{b}$ with $a, b$ relatively prime positive integers. Find the remainder when $a + b$ is divided by $1000$.

In $\bigtriangleup ABC$, $AB=BC=29$, and $AC=42$. What is the area of $\bigtriangleup ABC$? $\textbf{(A) }100\qquad\textbf{(B) }420\qquad\textbf{(C) }500\qquad\textbf{(D) }609\qquad \textbf{(E) }701$

Let $ABCD$ is a trapezoid with $\angle A = \angle B = 90^o$ and let $E$ is a point lying on side $CD$. Let the circle $\omega$ is inscribed to triangle $ABE$ and tangents sides $AB, AE$ and $BE$ at points $P, F$ and $K$ respectively. Let $KF$ intersects segments $BC$ and $AD$ at points $M$ and $N$ respectively, as well as $PM$ and $PN$ intersect $\omega$ at points $H$ and $T$ respectively. Prove that $PH = PT$.

Given a triangle $ABC$, $A',B',C'$ are the midpoints of $\overline{BC},\overline{AC},\overline{AB}$, respectively. $B^*,C^*$ lie in $\overline{AC},\overline{AB}$, respectively, such that $\overline{BB^*},\overline{CC^*}$ are the altitudes of the triangle $ABC$. Let $B^{\#},C^{\#}$ be the midpoints of $\overline{BB^*},\overline{CC^*}$, respectively. $\overline{B'B^{\#}}$ and $\overline{C'C^{\#}}$ meet at $K$, and $\overline{AK}$ and $\overline{BC}$ meet at $L$. Prove that $\angle{BAL}=\angle{CAA'}$

$4026$ points were noted on the plane, not three of which lie on a straight line. The $2013$ points are the vertices of a convex polygon, and the other $2013$ vertices are inside this polygon. It is allowed to paint each point in one of two colors. Coloring will be good if some pairs of dots can be combined segments with the following conditions: $\bullet$ Each segment connects dots of the same color. $\bullet$ No two drawn segments intersect at their inner points. $\bullet$ For an arbitrary pair of dots of the same color, there is a path along the lines from one point to another. Please note that the sides of the convex $2013$ rectangle are not automatically drawn segments, although some (or all) can be drawn as needed. Prove that the total number of good colors does not depend on the specific locations of the points and find that number.