Let $a, b, c,$ and $d$ be real numbers that satisfy the system of equations \begin{align*} a+b&=-3\\ ab+bc+ca&= -4\\ abc+bcd+cda+dab&=14\\ abcd&=30. \end{align*} There exist relatively prime positive integers $m$ and $n$ such that $$a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.$$ Find $m + n$.

In trapezoid $ ABCD$ with bases $ AB$ and $ CD$, we have $ AB\equal{}52$, $ BC\equal{}12$, $ CD\equal{}39$, and $ DA\equal{}5$. The area of $ ABCD$ is [asy] pair A,B,C,D; A=(0,0); B=(52,0); C=(38,20); D=(5,20); dot(A); dot(B); dot(C); dot(D); draw(A--B--C--D--cycle); label("$A$",A,S); label("$B$",B,S); label("$C$",C,N); label("$D$",D,N); label("52",(A+B)/2,S); label("39",(C+D)/2,N); label("12",(B+C)/2,E); label("5",(D+A)/2,W);[/asy] $ \text{(A)}\ 182 \qquad \text{(B)}\ 195 \qquad \text{(C)}\ 210 \qquad \text{(D)}\ 234 \qquad \text{(E)}\ 260$

Positive numbers $x,y$ satisfy $x^2+y^3 \ge x^3+y^4$. Prove that $x^3+y^3 \le 2$.

A trapezoid $ABCD$ and a line $\ell$ perpendicular to its bases $AD$ and $BC$ are given. A point $X$ moves along $\ell$. The perpendiculars from $A$ to $BX$ and from $D$ to $CX$ meet at point $Y$ . Find the locus of $Y$ . by D.Prokopenko

Let $ a_1,\ldots,a_n$ be positive reals so that $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le\frac n2$. Find the minimal value of $ \sqrt{a_1^2\plus{}\frac1{a_2^2}}\plus{}\sqrt{a_2^2\plus{}\frac1{a_3^2}}\plus{}\ldots\plus{}\sqrt{a_n^2\plus{}\frac1{a_1^2}}$.

Let $c \geq 4$ be an even integer. In some football league, each team has a home uniform and anaway uniform. Every home uniform is coloured in two different colours, and every away uniformis coloured in one colour. A team’s away uniform cannot be coloured in one of the colours fromthe home uniform. There are at most $c$ distinct colours on all of the uniforms. If two teams havethe same two colours on their home uniforms, then they have different colours on their away uniforms. We say a pair of uniforms is clashing if some colour appears on both of them. Suppose that for every team $X$ in the league, there is no team $Y$ in the league such that the home uniform of $X$ is clashing with both uniforms of $Y$. Determine the maximum possible number of teams in the league.

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

Square $ABCD$ with sides of length $4$ is a base of a cuboid $ABCDA'B'C'D'$. Side edges $AA'$, $BB'$, $CC'$, $DD'$ of this cuboid have length $7$. Points $K, L, M$ lie respectively on line segments $AA'$, $BB'$, $CC'$, and $AK = 3$, $BL = 2$, $CM = 5$. Plane passing through points $K, L, M$ cuts cuboid on two blocks. Calculate volumes of these blocks.

Given positive integers $n,k$, $n \ge 2$. Find the minimum constant $c$ satisfies the following assertion: For any positive integer $m$ and a $kn$-regular graph $G$ with $m$ vertices, one could color the vertices of $G$ with $n$ different colors, such that the number of monochrome edges is at most $cm$.

In the triangle $ABC$ with circumcircle $\Gamma$, the incircle $\omega$ touches sides $BC, CA$, and $AB$ at $D, E$, and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $K\neq D$. Line $AK$ meets $\Gamma$ at $L\neq A$. Rays $KI$ and $IL$ meet the circumcircle of triangle $BIC$ at $Q\neq I$ and $P\neq I$, respectively. The circumcircles of triangles $KFB$ and $KEC$ meet $EF$ at $R\neq F$ and $S\neq E$, respectively. Prove that $PQRS$ is cyclic. [i]India, Anant Mugdal[/i]

The degree-$6$ polynomial $f$ satisfies $f(7) - f(1) = 1, f(8) - f(2) = 16, f(9) - f(3) = 81, f(10) - f(4) = 256$ and $f(11) - f(5) = 625.$ Compute $f(15) - f(-3).$

Which of the following sets of whole numbers has the largest average? \[ \textbf{(A)}\ \text{multiples of 2 between 1 and 101} \qquad \textbf{(B)}\ \text{multiples of 3 between 1 and 101} \\ \textbf{(C)}\ \text{multiples of 4 between 1 and 101} \qquad \textbf{(D)}\ \text{multiples of 5 between 1 and 101} \\ \textbf{(E)}\ \text{multiples of 6 between 1 and 101} \]

Let $n\geq 3$ be a positive integer. Consider an $n\times n$ square. In each cell of the square, one of the numbers from the set $M=\{1,2,\ldots,2n-1\}$ is to be written. One such filling is called [i]good[/i] if, for every index $1\leq i\leq n,$ row no. $i$ and column no. $i,$ together, contain all the elements of $M$. [list=a] [*]Prove that there exists $n\geq 3$ for which a good filling exists. [*]Prove that for $n=2017$ there is no good filling of the $n\times n$ square. [/list]

There are $4$ piles of stones with the following quantities: $1004$, $1005$, $2009$ and $2010$. A legitimate move is to remove a stone from each from $3$ different piles. Two players $A$ and $B$ play in turns. $A$ begins the game . The player who, on his turn, cannot make a legitimate move, loses. Determine which of the players has a winning strategy and give a strategy for that player.

Given: $ x > 0, y > 0, x > y$ and $ z\not \equal{} 0$. The inequality which is not always correct is: $ \textbf{(A)}\ x \plus{} z > y \plus{} z \qquad\textbf{(B)}\ x \minus{} z > y \minus{} z \qquad\textbf{(C)}\ xz > yz$ $ \textbf{(D)}\ \frac {x}{z^2} > \frac {y}{z^2} \qquad\textbf{(E)}\ xz^2 > yz^2$

$(MON 5)$ Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation $x^2 - ax + b = 0$.