A square-shaped pizza with side length $30$ cm is cut into pieces (not necessarily rectangular). All cuts are parallel to the sides, and the total length of the cuts is $240$ cm. Show that there is a piece whose area is at least $36$ cm$^2$

Rectangle $ABCD$ has side lengths $AB = 3$ and $BC = 7$. Let $E$ be a point on $BC$, and let $F$ be the intersection of $DE$ and $AC$. Given that $[CDF] = 4$, find $\frac{DF}{FE}$ .

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[yf(x+1)=f(x+y-f(x))+f(x)f(f(y))\] for all $x,y \in \mathbb{R}$.

[b]p1.[/b] What is the units digit of the product of the first seven primes? [b]p2. [/b]In triangle $ABC$, $\angle BAC$ is a right angle and $\angle ACB$ measures $34$ degrees. Let $D$ be a point on segment $ BC$ for which $AC = CD$, and let the angle bisector of $\angle CBA$ intersect line $AD$ at $E$. What is the measure of $\angle BED$? [b]p3.[/b] Chad numbers five paper cards on one side with each of the numbers from $ 1$ through $5$. The cards are then turned over and placed in a box. Jordan takes the five cards out in random order and again numbers them from $ 1$ through $5$ on the other side. When Chad returns to look at the cards, he deduces with great difficulty that the probability that exactly two of the cards have the same number on both sides is $p$. What is $p$? [b]p4.[/b] Only one real value of $x$ satisfies the equation $kx^2 + (k + 5)x + 5 = 0$. What is the product of all possible values of $k$? [b]p5.[/b] On the Exeter Space Station, where there is no effective gravity, Chad has a geometric model consisting of $125$ wood cubes measuring $ 1$ centimeter on each edge arranged in a $5$ by $5$ by $5$ cube. An aspiring carpenter, he practices his trade by drawing the projection of the model from three views: front, top, and side. Then, he removes some of the original $125$ cubes and redraws the three projections of the model. He observes that his three drawings after removing some cubes are identical to the initial three. What is the maximum number of cubes that he could have removed? (Keep in mind that the cubes could be suspended without support.) [b]p6.[/b] Eric, Meena, and Cameron are studying the famous equation $E = mc^2$. To memorize this formula, they decide to play a game. Eric and Meena each randomly think of an integer between $1$ and $50$, inclusively, and substitute their numbers for $E$ and $m$ in the equation. Then, Cameron solves for the absolute value of $c$. What is the probability that Cameron’s result is a rational number? [b]p7.[/b] Let $CDE$ be a triangle with side lengths $EC = 3$, $CD = 4$, and $DE = 5$. Suppose that points $ A$ and $B$ are on the perimeter of the triangle such that line $AB$ divides the triangle into two polygons of equal area and perimeter. What are all the possible values of the length of segment $AB$? [b]p8.[/b] Chad and Jordan are raising bacteria as pets. They start out with one bacterium in a Petri dish. Every minute, each existing bacterium turns into $0, 1, 2$ or $3$ bacteria, with equal probability for each of the four outcomes. What is the probability that the colony of bacteria will eventually die out? [b]p9.[/b] Let $a = w + x$, $b = w + y$, $c = x + y$, $d = w + z$, $e = x + z$, and $f = y + z$. Given that $af = be = cd$ and $$(x - y)(x - z)(x - w) + (y - x)(y - z)(y - w) + (z - x)(z - y)(z - w) + (w - x)(w - y)(w - z) = 1,$$ what is $$2(a^2 + b^2 + c^2 + d^2 + e^2 + f^2) - ab - ac - ad - ae - bc - bd - bf - ce - cf - de - df - ef ?$$ [b]p10.[/b] If $a$ and $b$ are integers at least $2$ for which $a^b - 1$ strictly divides $b^a - 1$, what is the minimum possible value of $ab$? Note: If $x$ and $y$ are integers, we say that $x$ strictly divides $y$ if $x$ divides $y$ and $|x| \ne |y|$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Sunn and Tacks play a game alternately choosing a word among the following (German) words: ”bad”, ”binse”, ”kafig”, ”kosewort”, ”maitag”, ”name”, ”pol”, ”parade”, ”wolf”. Two words are said to compatible if they have exactly one consonant in common. In the first round, Sunn selects a word for herself and one for Tacks. In every consequent round, each player selects a word that is compatible with the one they chose in the previous round. Tacks wins the game if the two players successively select the same word. (a) Prove that Tacks can always win. How many rounds are necessary for that? (b) Upon Sunn’s desire, the word ”kafig” was replaced with the word ”feige”. Prove that Sunn can prevent Tacks from winning.

In each square of a $10\times 10$ board, there is an arrow pointing upwards, downwards, left, or right. Prove that it is possible to remove $50$ arrows from the board, such that no two remaining arrows point at each other.

How many three digit numbers are made up of three distinct digits?

Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$? [asy] unitsize(1cm); draw((0,1)--(2,2)--(4,1)--(2,0)--cycle); dot("$A$",(0,1),W); dot("$D$",(2,2),N); dot("$C$",(4,1),E); dot("$B$",(2,0),S); [/asy] $\textbf{(A) } 60 \qquad\textbf{(B) } 90 \qquad\textbf{(C) } 105 \qquad\textbf{(D) } 120 \qquad\textbf{(E) } 144$

Consider all arithmetical sequences of real numbers $(x_i)^{\infty}=1$ and $(y_i)^{\infty} =1$ with the common first term, such that for some $k > 1, x_{k-1}y_{k-1} = 42, x_ky_k = 30$, and $x_{k+1}y_{k+1} = 16$. Find all such pairs of sequences with the maximum possible $k$.

Let the maximum and minimum value of $f(x)=\cos \left(x^{2018}\right) \sin x$ are $M$ and $m$ respectively where $x \in[-2 \pi, 2 \pi] .$ Then $$ M+m= $$ [list=1] [*] $\frac{1}{2}$ [*] $-\frac{1}{\sqrt{2}}$ [*] $\frac{1}{2018}$ [*] Does not exists [/list]

If $p$ is an odd prime, prove that \[{k \choose p}\equiv \left\lfloor \frac{k}{p}\right\rfloor \pmod{p}.\]

Let ${a_n}$ and ${b_n}$ be positive real sequence that satisfy the following condition: (i) $a_1=b_1=1$ (ii) $b_n=a_n b_{n-1}-2$ (iii) $n$ is an integer larger than $1$. Let ${b_n}$ be a bounded sequence. Prove that $\sum_{n=1}^{\infty} \frac{1}{a_1a_2\cdots a_n}$ converges. Find the value of the sum.

Prove that there are no positive integers $n$ and $k\le n$ such that the numbers $$\binom nk,\binom n{k+1},\binom n{k+2},\binom n{k+3}$$in this order form an arithmetic progression.

Initially $A$ selects a graph with \( 2221 \) vertices such that each vertex is incident to at least one edge. Then $B$ deletes some of the edges (possibly none) from the chosen graph. Finally, $A$ pays $B$ one lev for each vertex that is incident to an odd number of edges. What is the maximum amount that $B$ can guarantee to earn?

The $8$ eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of $50$ mm and a length of $80$ mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectrangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least $200$ mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters. [asy] size(200); defaultpen(linewidth(0.7)); path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin; path laceR=reflect((75,0),(75,-240))*laceL; draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray); for(int i=0;i<=3;i=i+1) { path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5); unfill(circ1); draw(circ1); unfill(circ2); draw(circ2); } draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]

In three houses $A,B$ and $C$, forming a right triangle with the legs $AC=30$ and $CB=40$, live three beetles $a,b$ and $c$, capable of moving at speeds of $2, 3$ and $4$, respectively. Suppose that you simultaneously release these bugs from point $M$ and mark the time after which beetles reach their homes. Find on the plane such a point $M$, where is the last time to reach the house a bug would be minimal.

Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that: (i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and (ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.

A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$, $BC = 12$, $CA = 13$. The probability that a circle with radius $\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$.

Let $ n$ be a positive integer. Consider an $ n\times n$ matrix with entries $ 1,2,...,n^2$ written in order, starting at the top left and moving along each row in turn left-to-right. (e.g. for $ n \equal{} 3$ we get $ \left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$) We choose $ n$ entries of the matrix such that exactly one entry is chosen in each row and each column. What are the possible values of the sum of the selected entries?

Find the largest number formed by the digits 1 to 9, without repetition, that is divisible by 18.

Natural numbers from $1$ to $99$ (not necessarily distinct) are written on $99$ cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible by $100$. Show that all the cards contain the same number.

Determine all monic polynomial with integer coefficient $P$ such that for every integer $a,b$ there exists integer $c$ so that \[P(a)P(b)=P(c)\]

Solve the equation \[2^{x^{2}+x}+\log_{2}x = 2^{x+1}\]

Find the value of $a_2 + a_4 + a_6 + \dots + a_{98}$ if $a_1$, $a_2$, $a_3$, $\dots$ is an arithmetic progression with common difference 1, and $a_1 + a_2 + a_3 + \dots + a_{98} = 137$.

Find the self-intersection index of the surface $x^4+y^4=1$ in the projective plane $\text{CP}^2$.