Each of 100 students sends messages to 50 different students. What is the least number of pairs of students who send messages to each other? $\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 75 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 25 \qquad\textbf{(E)}\ \text{None}$

You have $2$ columns of $11$ squares in the middle, in the right and in the left you have columns of $9$ squares (centered on the ones of $11$ squares), then columns of $7,5,3,1$ squares. (This is the way it was explained in the original thread, http://www.artofproblemsolving.com/Forum/viewtopic.php?t=44430 ; anyway, i think you can understand how it looks) Several rooks stand on the table and beat all the squares ( a rook beats the square it stands in, too). Prove that one can remove several rooks such that not more than $11$ rooks are left and still beat all the table. [i]Proposed by D. Rostovsky, based on folklore[/i]

A prism is called [i]binary [/i] if it can be assigned to each of its vertices a number from the set $\{-1, 1\}$, such that the product of the numbers assigned to the vertices of each face is equal to $-1$. a) Prove that the number of vertices of the binary prisms is divisible for $8$. b) Prove that a prism with $2000$ vertices is binary.

Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively. a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$. b) Does the converse implication also always hold?

Let $ABC$ be an acute triangle and $D, E, F$ are the midpoints of $BC, CA, AB$, respectively. The circle with diameter $AD$ intersects the lines $AB$ and $AC$ at points $P$ and $Q$ , respectively. The lines through $P$ and $Q$ parallel to $BC$ intersect $DE$ at point $R$ and $DF$ at point $S$, respectively. The circumcircle of $DPR$ intersects $AB$ at $X$, the circumcircle of $DQS$ intersects $AC$ in $Y$, and these two circles intersect again point $Z$. Prove that $Z$ is the midpoint of $XY$.

Two equal rectangles are intersecting in $8$ points. Prove that the common part area is greater than the half of the rectangle's area.

Let $H$ be the orthocenter of the acute triangle $ABC.$ In the plane of the triangle $ABC$ we consider a point $X$ such that the triangle $XAH$ is right and isosceles, having the hypotenuse $AH,$ and $B$ and $X$ are on each part of the line $AH.$ Prove that $\overrightarrow{XA}+\overrightarrow{XC}+\overrightarrow{XH}=\overrightarrow{XB}$ if and only if $ \angle BAC=45^{\circ}.$

Find all prime numbers $p, q, r$ such that \[15p+7pq+qr=pqr.\]

Let $ ABCDE$ be an arbitrary convex pentagon. Suppose that $ BD\cap CE \equal{} A'$, $ CE \cap DA \equal{} B'$, $ DA\cap EB \equal{} C'$, $ EB\cap AC \equal{} D'$ and $ AC \cap BD \equal{} E'$. Suppose also that $ (ABD')\cap (AC'E) \equal{} A''$, $ (BCE')\cap (BD'A) \equal{} B''$, $ (CDA')\cap (CE'B) \equal{} C''$, $ (DEB')\cap DA'C \equal{} D''$ and $ (EAC')\cap (EB'D) \equal{} E''$. Prove that $ AA''$, $ BB''$, $ CC''$, $ DD''$ and $ EE''$ are concurrent.

In an isosceles trapezoid, the diagonals are perpendicular. Find the distance from the center of the circle described around the trapezoid to the point of intersection of its diagonals, if the lengths of the bases are equal to $a$ and $b$.

What is the greatest number of parts into which the plane can be cut by the edges of $ n $ squares?

Let $ABC$ be a triangle with $AB \ne AC$. Let the angle bisector of $\angle BAC$ intersects $BC$ at $P$ and intersects the perpendicular bisector of segment $BC$ at $Q$. Prove that $\frac{PQ}{AQ} =\left( \frac{BC}{AB + AC}\right)^2$

Suppose that $X$ is a set with $n$ elements and $\mathcal F\subseteq X^{(k)}$ and $X_1,X_2,...,X_s$ is a partition of $X$. We know that for every $A,B\in \mathcal F$ and every $1\le j\le s$, $E=B\cap (\bigcup_{i=1}^{j}X_i)\neq A\cap (\bigcup_{i=1}^{j} X_i)=F$ shows that none of $E,F$ contains the other one. Prove that \[|\mathcal F|\le \max_{\sum\limits_{i=1}^{S}w_i=k}\prod_{j=1}^{s}\binom{|X_j|}{w_j}\] (15 points) Exam time was 5 hours and 20 minutes.

[b]p1.[/b] Zion, RJ, Cam, and Tre decide to start learning languages. The four most popular languages that Duke offers are Spanish, French, Latin, and Korean. If each friend wants to learn exactly three of these four languages, how many ways can they pick courses such that they all attend at least one course together? [b]p2. [/b] Suppose we wrote the integers between $0001$ and $2019$ on a blackboard as such: $$000100020003 · · · 20182019.$$ How many $0$’s did we write? [b]p3.[/b] Duke’s basketball team has made $x$ three-pointers, $y$ two-pointers, and $z$ one-point free throws, where $x, y, z$ are whole numbers. Given that $3|x$, $5|y$, and $7|z$, find the greatest number of points that Duke’s basketball team could not have scored. [b]p4.[/b] Find the minimum value of $x^2 + 2xy + 3y^2 + 4x + 8y + 12$, given that $x$ and $y$ are real numbers. Note: calculus is not required to solve this problem. [b]p5.[/b] Circles $C_1, C_2$ have radii $1, 2$ and are centered at $O_1, O_2$, respectively. They intersect at points $ A$ and $ B$, and convex quadrilateral $O_1AO_2B$ is cyclic. Find the length of $AB$. Express your answer as $x/\sqrt{y}$ , where $x, y$ are integers and $y$ is square-free. [b]p6.[/b] An infinite geometric sequence $\{a_n\}$ has sum $\sum_{n=0}^{\infty} a_n = 3$. Compute the maximum possible value of the sum $\sum_{n=0}^{\infty} a_{3n} $. [b]p7.[/b] Let there be a sequence of numbers $x_1, x_2, x_3,...$ such that for all $i$, $$x_i = \frac{49}{7^{\frac{i}{1010}} + 49}.$$ Find the largest value of $n$ such that $$\left\lfloor \sum_{i=1}{n} x_i \right\rfloor \le 2019.$$ [b]p8.[/b] Let $X$ be a $9$-digit integer that includes all the digits $1$ through $9$ exactly once, such that any $2$-digit number formed from adjacent digits of $X$ is divisible by $7$ or $13$. Find all possible values of $X$. [b]p9.[/b] Two $2025$-digit numbers, $428\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}571$ and $571\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}428$ , form the legs of a right triangle. Find the sum of the digits in the hypotenuse. [b]p10.[/b] Suppose that the side lengths of $\vartriangle ABC$ are positive integers and the perimeter of the triangle is $35$. Let $G$ the centroid and $I$ be the incenter of the triangle. Given that $\angle GIC = 90^o$ , what is the length of $AB$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Dave has a pile of fair standard six-sided dice. In round one, Dave selects eight of the dice and rolls them. He calculates the sum of the numbers face up on those dice to get $r_1$. In round two, Dave selects $r_1$ dice and rolls them. He calculates the sum of the numbers face up on those dice to get $r_2$. In round three, Dave selects $r_2$ dice and rolls them. He calculates the sum of the numbers face up on those dice to get $r_3$. Find the expected value of $r_3$.

Determine if for all natural $n$ there is a $n \times n$ matrix of real entries such that its determinant is 0 and that changing any entry produce another matrix with nonzero determinant.

Given $s,t$ are non-zero integers, $(x,y) $ is an integer pair , A transformation is to change pair $(x,y)$ into pair $(x+t,y-s)$ . If the two integers in a certain pair becoems relatively prime after several tranfomations , then we call the original integer pair "a good pair" . (1) Is $(s,t)$ a good pair ? (2) Prove :for any $s$ and $t$ , there exists pair $(x,y)$ which is " a good pair".

Let $f(x) = \frac{x^2}8$. Starting at the point $(7,3)$, what is the length of the shortest path that touches the graph of $f$, and then the $x$-axis? [i]Proposed by Sam Delatore[/i]

Determine all pairs $(p, q)$ of primes for which $p^{q+1}+q^{p+1}$ is a perfect square.

Suppose $a_3x^3 - x^2 + a_1x - 7 = 0$ is a cubic polynomial in x whose roots $\alpha,\beta, \gamma$ are positive real numbers satisfying $$\frac{225\alpha^2}{\alpha^2 +7}=\frac{144\beta^2}{\beta^2 +7}=\frac{100\gamma^2}{\gamma^2 +7}.$$ Find $a_1$.

There are $17$ people at a party, and each has a reputation that is either $1$, $2$, $3$, $4$, or $5$. Some of them split into pairs under the condition that within each pair, the two people’s reputations differ by at most $1$. Compute the largest value of $k$ such that no matter what the reputations of these people are, they are able to form $k$ pairs

Let \[ A = \lim_{n \rightarrow \infty} \sum_{i=0}^{2016} (-1)^i \cdot \frac{\binom{n}{i}\binom{n}{i+2}}{\binom{n}{i+1}^2} \] Find the largest integer less than or equal to $\frac{1}{A}$. The following decimal approximation might be useful: $ 0.6931 < \ln(2) < 0.6932$, where $\ln$ denotes the natural logarithm function.

Find all triplets of positive integers $(a,b,c)$ for which the number $3^a+3^b+3^c$ is a perfect square.

Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of 5? $ \textbf{(A) } \frac 12 \qquad \textbf{(B) } \frac 35 \qquad \textbf{(C) } \frac 23 \qquad \textbf{(D) } \frac 45 \qquad \textbf{(E) } 1$

Find all polynomials $P(x)$ with integer coefficients such that, for all integers $a$ and $b$, $P(a+b) - P(b)$ is a multiple of $P(a)$.