Michael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tournament. Each pair of competitors plays exactly one game, in which each competitor has an equal chance of winning (and there are no ties). The probability that none of the five players wins all of his/her games is $\tfrac{m}{n}$ for relatively prime positive integers $m$, $n$. Compute $100m + n$. [i]Proposed by Evan Chen[/i]

Four congruent and pairwise externally tangent circles are inscribed in square $ABCD$ as shown. The ray through $A$ passing through the center of the square hits the opposing circle at point $E$, shown in the diagram below. Given $AE= 5 + 2\sqrt{2}$, the area of the square can be expressed as $a + b\sqrt{c}$, where $a$, $b$, $c$ are positive integers and $c$ is square-free. Find $a+b+c.$ [asy] size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-18.99425911800572,xmax=23.81538435842469,ymin=-15.51769962526155,ymax=6.464951807764648; pen zzttqq=rgb(0.6,0.2,0.); pair A,B,C,D,M,E; A=(0,0); B=(0,4); C=(4,4); D=(4,0); M=(2,2); E=(2.29289,2.29289); draw(A--B--C--D--cycle); draw(A--E); draw(circle((1,1),1));draw(circle((1,3),1));draw(circle((3,1),1));draw(circle((3,3),1)); dot(A);dot(B);dot(C);dot(D);dot(E);label("$A$",A,SW);label("$B$",B,NW);label("$C$",C,NE);label("$D$",D,SE);label("$E$",E,NE); [/asy] [i]Lightning 1.4[/i]

Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\triangle KJM$ is an isosceles triangle.

On the edges of a convex polyhedra we draw arrows such that from each vertex at least an arrow is pointing in and at least one is pointing out. Prove that there exists a face of the polyhedra such that the arrows on its edges form a circuit. [i]Dan Schwartz[/i]

In the figure, a circle is located inside a trapezoid with two right angles so that a point of tangency of the circle is the midpoint of the side perpendicular to the two bases. The circle also has points of tangency on each base of the trapezoid. The diameter of the circle is $\frac{2}{3}$ the length of $EF$. If the area of the circle is $9\pi$ square units, what is the area of the trapezoid? [asy] draw((0,0) -- (11, 0) -- (7,6) -- (0,6) -- cycle); draw((0,3) -- (9,3)); draw(circle((3,3), 3)); draw(rightanglemark((1,0),(0,0),(0,1),12)); draw(rightanglemark((0,0),(0,6),(6,6), 12)); label("E", (0,3), W); label("F", (9,3), E); [/asy]

Let $f, g$ be bijections on $\{1, 2, 3, \dots, 2016\}$. Determine the value of $$\sum_{i=1}^{2016}\sum_{j=1}^{2016}[f(i)-g(j)]^{2559}.$$

For how many integers $x$ is $|15x^2-32x-28|$ a prime number? $ \textbf{a)}\ 0 \qquad\textbf{b)}\ 1 \qquad\textbf{c)}\ 2 \qquad\textbf{d)}\ 4 \qquad\textbf{e)}\ \text{None of above} $

[b]10.[/b] Given a straight line $g$ in the plane and a point $O$ on $g$. Construct, without making use of the Parallel Axiom, the half-line perpendicular to $g$ at the point $O$ and lying in one of the half-planes defined by $g$, under the following restrictions: The construction must be effected by use of a ruler and of a length standard (i.e. an etalon-segment) only; moreover, all lines and points of the construction must lie in the chosen half-plane. [b](G. 20)[/b]

Let $S$ be a set of $10$ positive integers. Prove that one can find two disjoint subsets $A =\{a_1, ..., a_k\}$ and $B = \{b_1, ... , b_k\}$ of $S$ with $|A| = |B|$ such that the sums $x =\frac{1}{a_1}+ ... +\frac{1}{a_k}$ and $y =\frac{1}{b_1}+ ... +\frac{1}{b_k}$ differ by less than $0.01$, i.e., $|x - y| < 1/100$.

A positive integer is called a "double number" if its decimal representation consists of a block of digits, not commencing with $0$, followed immediately by an identical block. So, for instance, $360360$ is a double number, but $36036$ is not. Show that there are infinitely many double numbers which are perfect squares.

Let $A$ be a set $A=\{1,2,3,...,2017\}$. Subset $S$ of set $A$ is [i]good [/i] if for all $x\in A$ sum of remaining elements of set $S$ has same last digit as $x$. Prove that [i]good[/i] subset with $405$ elements is not possible.

The vertex C of the triangle ABC is allowed to vary along a line parallel to AB. Find the locus of the orthocenter.

\[\int_1^e \frac{\log(x^{2024})}{x} \mathrm dx\] [i]Proposed by Connor Gordon[/i]

Find all ordered pairs of integers $(a,b)$ such that $3^a + 7^b$ is a perfect square.

For each integer $C>1$ decide whether there exist pairwise distinct positive integers $a_1,a_2,a_3,...$ such that for every $k\ge 1$, $a_{k+1}^k$ divides $C^ka_1a_2...a_k$. [i]Proposed by Daniel Liu

A "Hishgad" lottery ticket contains the numbers $1$ to $mn$, arranged in some order in a table with $n$ rows and $m$ columns. It is known that the numbers in each row increase from left to right and the numbers in each column increase from top to bottom. An example for $n=3$ and $m=4$: [asy] size(3cm); Label[][] numbers = {{"$1$", "$2$", "$3$", "$9$"}, {"$4$", "$6$", "$7$", "$10$"}, {"$5$", "$8$", "$11$", "$12$"}}; for (int i=0; i<5;++i) { draw((i,0)--(i,3)); } for (int i=0; i<4;++i) { draw((0,i)--(4,i)); } for (int i=0; i<4;++i){ for (int j=0; j<3;++j){ label(numbers[2-j][i], (i+0.5, j+0.5)); }} [/asy] When the ticket is bought the numbers are hidden, and one must "scratch" the ticket to reveal them. How many cells does it always suffice to reveal in order to determine the whole table with certainty?

The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet? $\textbf{(A) } 41 \qquad\textbf{(B) } 47 \qquad\textbf{(C) } 59 \qquad\textbf{(D) } 61 \qquad\textbf{(E) } 66 $

The points $P$ and $Q$ are placed in the interior of the triangle $\Delta ABC$ such that $m(\angle PAB)=m(\angle QAC)<\frac{1}{2}m(\angle BAC)$ and similarly for the other $2$ vertices($P$ and $Q$ are isogonal conjugates). Let $P_{A}$ and $Q_{A}$ be the intersection points of $AP$ and $AQ$ with the circumcircle of $CPB$, respectively $CQB$. Similarly the pairs of points $(P_{B},Q_{B})$ and $(P_{C},Q_{C})$ are defined. Let $PQ_{A}\cap QP_{A}=\{M_{A}\}$, $PQ_{B}\cap QP_{B}=\{M_{B}\}$, $PQ_{C}\cap QP_{C}=\{M_{C}\}$. Prove the following statements: $1.$ Lines $AM_{A}$, $BM_{B}$, $CM_{C}$ concur. $2. $ $M_{A}\in BC$, $M_{B}\in CA$, $M_{C}\in AB$

The sum of the distances from one vertex of a square with sides of length two to the midpoints of each of the sides of the square is $\textbf{(A) }2\sqrt{5}\qquad\textbf{(B) }2+\sqrt{3}\qquad\textbf{(C) }2+2\sqrt{3}\qquad\textbf{(D) }2+\sqrt{5}\qquad \textbf{(E) }2+2\sqrt{5}$

Two subsets of the set $ S\equal{}\{a,b,c,d,e\}$ are to be chosen so that their union is $ S$ and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter? $ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 60 \qquad \textbf{(D)}\ 160 \qquad \textbf{(E)}\ 320$

In a triangle $ABC, AM$ is a median, $BK$ is a bisectrix, $L=AM\cap BK$. It is known that $BC=a, AB=c, a>c$. Given that the circumcenter of triangle $ABC$ lies on the line $CL$, find $AC$ I. Voronovich

Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres? $ \textbf{(A)}\ \sqrt 2\qquad \textbf{(B)}\ \sqrt 3\qquad \textbf{(C)}\ 1 \plus{} \sqrt 2\qquad \textbf{(D)}\ 1 \plus{} \sqrt 3\qquad \textbf{(E)}\ 3$

Each of the $39$ students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and $26$ students have a cat. How many students have both a dog and a cat? $\textbf{(A)}\ 7\qquad \textbf{(B)}\ 13\qquad \textbf{(C)}\ 19\qquad \textbf{(D)}\ 39\qquad \textbf{(E)}\ 46$

Let us denote the midpoint of $AB$ with $O$. The point $C$, different from $A$ and $B$ is on the circle $\Omega$ with center $O$ and radius $OA$ and the point $D$ is the foot of the perpendicular from $C$ to $AB$. The circle with center $C$ and radius $CD$ and $\omega$ intersect at $M$, $N$. Prove that $MN$ cuts $CD$ in two equal segments.

A rectangle with a diagonal of length $ x$ is twice as long as it is wide. What is the area of the rectangle? $ \textbf{(A)}\ \frac14x^2 \qquad \textbf{(B)}\ \frac25x^2 \qquad \textbf{(C)}\ \frac12x^2 \qquad \textbf{(D)}\ x^2 \qquad \textbf{(E)}\ \frac32x^2$