Given is a square $ABCD$ with $E \in BC$, arbitrarily. On $CD$ lies the point $F$ is such that $\angle EAF = 45^o$. Prove that $EF$ is tangent to the circle with center $A$ and radius $AB$.

Let $ABCD$ be a convex quadrilateral with $AB=5$, $BC=6$, $CD=7$, and $DA=8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Compute $MN^2-PQ^2$.

Tetrahedron $ABCD$ has sides of lengths, in increasing order, $7, 13, 18, 27, 36, 41$. If $AB=41$, then what is the length of $CD$?

How many four-digit multiples of $8$ are greater than $2008$?

We have \( n \) chips that are initially placed on the number line at position 0. On each move, we select a position \( x \in \mathbb{Z} \) where there are at least two chips; we take two of these chips, then place one at \( x-1 \) and the other at \( x+1 \). a) Prove that after a finite number of moves, regardless of how the moves are chosen, we will reach a final position where no two chips occupy the same number on the number line. b) For every possible final position, let \( \Delta \) represent the difference between the numbers where the rightmost and the leftmost chips are located. Find all possible values of \( \Delta \) in terms of \( n \).

In triangle $ABC$ the circumcircle has radius $R$ and center $O$ and the incircle has radius $r$ and center $I\ne O$ . Let $G$ denote the centroid of triangle $ABC$. Prove that $IG \perp BC$ if and only if $AB = AC$ or $AB + AC = 3BC$.

Let $f:\mathbb{C} \to \mathbb{C}$ be an entire function, and suppose that the sequence $f^{(n)}$ of derivatives converges pointwise. Prove that $f^{(n)}(z)\to Ce^z$ pointwise for a suitable complex number $C$.

How many sets of two positive prime numbers $\{p,q\}$ have the property that $p + q = 100$?

A quadrilateral $ABCD$ is circumscribed about a circle. Lines $AC$ and $DC$ meet at point $E$ and lines $DA$ and $BC$ meet at $F$, where $B$ is between $A$ and $E$ and between $C$ and $F$. Let $I_1, I_2$ and $I_3$ be the incenters of triangles $AFB, BEC$ and $ABC$, respectively. The line $I_1I_3$ intersects $EA$ at $K$ and $ED$ at $L$, whereas the line $I_2I_3$ intersects $FC$ at $M$ and $FD$ at $N$. Prove that $EK = EL$ if and only if $FM = FN$

Let $n\geq 3$ be a positive integer and a circle $\omega$ is given. A regular polygon(with $n$ sides) $P$ is drawn and your vertices are in the circle $\omega$ and these vertices are red. One operation is choose three red points $A,B,C$, such that $AB=BC$ and delete the point $B$. Prove that one can do some operations, such that only two red points remain in the circle.

Ten square cardboards of $3$ centimeters on a side are cut by a line, as indicated in the figure. After the cuts, there are $20$ pieces: $10$ triangles and $10$ trapezoids. Assemble a square that uses all $20$ pieces without overlaps or gaps. [img]https://cdn.artofproblemsolving.com/attachments/7/9/ec2242cca617305b02eef7a5409e6a6b482d66.gif[/img]

Let $E$ be a finite set, $P_E$ the family of its subsets, and $f$ a mapping from $P_E$ to the set of non-negative reals, such that for any two disjoint subsets $A,B$ of $E$, $f(A\cup B)=f(A)+f(B)$. Prove that there exists a subset $F$ of $E$ such that if with each $A \subset E$, we associate a subset $A'$ consisting of elements of $A$ that are not in $F$, then $f(A)=f(A')$ and $f(A)$ is zero if and only if $A$ is a subset of $F$.

Twelve friends met for dinner at Oscar's Overstuffed Oyster House, and each ordered one meal. The portions were so large, there was enough food for $18$ people. If they share, how many meals should they have ordered to have just enough food for the $12$ of them? $\textbf{(A)}\ 8\qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ 15\qquad \textbf{(E)}\ 18$

A set of $n$ lights, numbered $1$ to $n$, are initially off. At every moment, it is possible to perform one of the following operations: $\bullet$ change the state of lamp $1$, $\bullet$ change the state of lamp $2$, as long as lamp $1$ is on, $\bullet$ change the state of lamp $k > 2$, as long as lamp $k - 1$ is on and all lamps $1, . . . , k - 2$ are off. It shows that it is possible, after a certain number of operations, to have only the lamp left on.

The cost of $3$ hamburgers, $5$ milk shakes, and $1$ order of fries at a certain fast food restaurant is $\$23.50$. At the same restaurant, the cost of $5$ hamburgers, $9$ milk shakes, and $1$ order of fries is $\$39.50$. What is the cost of $2$ hamburgers, $2$ milk shakes and $2$ orders of fries at this restaurant?

Prove that among the elements of the sequence $\left( \left\lfloor n \sqrt 2 \right\rfloor + \left\lfloor n \sqrt 3 \right\rfloor \right)_{n \geq 0}$ are an infinity of even numbers and an infinity of odd numbers.

Let $ABC$ be a triangle with $AB < AC$, let $L$ be midpoint of arc $BC$(the point $A$ is not in this arc) of the circumcircle $w$($ABC$). Let $E$ be a point in $AC$ where $AE = \frac{AB + AC}{2}$, the line $EL$ intersects $w$ in $P$. If $M$ and $N$ are the midpoints of $AB$ and $BC$, respectively, prove that $AL, BP$ and $MN$ are concurrents

There is a parabola $y = x^2$ drawn on the coordinate plane. The axes are deleted. Can you restore them with the help of compass and ruler?

Sergei chooses two different natural numbers $a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$, $b$ and $b+2$. He then writes all six pairwise products of the numbers of notebook on the blackboard. Let $S$ be the number of perfect squares on the blackboard. Find the maximum value of $S$. [i]S. Berlov[/i]

Find, with proof, a polynomial $f(x,y,z)$ in three variables, with integer coefficients, such that for all $a,b,c$ the sign of $f(a,b,c)$ (that is, positive, negative, or zero) is the same as the sign of $a+b\sqrt[3]{2}+c\sqrt[3]{4}$.

How many perfect squares are greater than $0$ but less than or equal to $100$? $\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10$

Let $M$ be a subset of $\mathbb{R}$ such that the following conditions are satisfied: a) For any $x \in M, n \in \mathbb{Z}$, one has that $x+n \in \mathbb{M}$. b) For any $x \in M$, one has that $-x \in M$. c) Both $M$ and $\mathbb{R}$ \ $M$ contain an interval of length larger than $0$. For any real $x$, let $M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}$. Show that if $\alpha,\beta$ are reals such that $M(\alpha) = M(\beta)$, then we must have one of $\alpha + \beta$ and $\alpha - \beta$ to be rational.

Find all prime numbers $p$ such that $p^3$ divides the determinant \[\begin{vmatrix} 2^2 & 1 & 1 & \dots & 1\\1 & 3^2 & 1 & \dots & 1\\ 1 & 1 & 4^2 & & 1\\ \vdots & \vdots & & \ddots & \\1 & 1 & 1 & & (p+7)^2 \end{vmatrix}.\]

In the equation below, $x$ is a nonzero real number such that \[\frac1{729}\left(3^t\right)=3^x.\] Which of the following is equal to $t$? $\textbf{(A) } \dfrac16x\qquad\textbf{(B) } \dfrac13x\qquad\textbf{(C) } 6x\qquad\textbf{(D) } x-6\qquad\textbf{(E) } x+6$

[b]p1.[/b] If Suzie has $6$ coins worth $23$ cents, how many nickels does she have? [b]p2.[/b] Let $a * b = (a - b)/(a + b)$. If $8 * (2 * x) = 4/3$, what is $x$? [b]p3.[/b] How many digits does $x = 100000025^2 - 99999975^2$ have when written in decimal form? [b]p4.[/b] A paperboy’s route covers $8$ consecutive houses along a road. He does not necessarily deliver to all the houses every day, but he always traverses the road in the same direction, and he takes care never to skip over $2$ consecutive houses. How many possible routes can he take? [b]p5.[/b] A regular $12$-gon is inscribed in a circle of radius $5$. What is the sum of the squares of the distances from any one fixed vertex to all the others? [b]p6.[/b] In triangle $ABC$, let $D, E$ be points on $AB$, $AC$, respectively, and let $BE$ and $CD$ meet at point $P$. If the areas of triangles $ADE$, $BPD$, and $CEP$ are $5$, $8$, and $3$, respectively, find the area of triangle ABC. [b]p7.[/b] Bob has $11$ socks in his drawer: $3$ different matched pairs, and $5$ socks that don’t match with any others. Suppose he pulls socks from the drawer one at a time until he manages to get a matched pair. What is the probability he will need to draw exactly $9$ socks? [b]p8.[/b] Consider the unit cube $ABCDEFGH$. The triangle bound to $A$ is the triangle formed by the $3$ vertices of the cube adjacent to $A$ (and similarly for the other vertices of the cube). Suppose we slice a knife through each of the $8$ triangles bound to vertices of the cube. What is the volume of the remaining solid that contains the former center of the cube? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].