A square with side length $6$ is rotated by $90^\circ$ about its center. What is the area of the region swept out by the perimeter of the square (that is, the four line segments forming the boundary of the square)?

The infinite checkered plane is divided into dominoes. If we move any horizontal domino of the partition by 49 cells to the right or left, we will also get a domino of the partition. If we move any vertical domino of the partition up or down by 49 cells, we will also get a domino of the partition. Can this happen?

$ABCD$ is a quadrilateral, $E,F,G,H$ are midpoints of $AB,BC,CD,DA$. Find the point P such that $area (PHAE) = area (PEBF) = area (PFCG) = area (PGDH)$.

The sum of the following seven numbers is exactly 19: \[a_1=2.56,\qquad a_2=2.61,\qquad a_3=2.65,\qquad a_4=2.71,\] \[a_5=2.79,\qquad a_6=2.82,\qquad a_7=2.86.\] It is desired to replace each $a_i$ by an integer approximation $A_i$, $1 \le i \le 7$, so that the sum of the $A_i$'s is also 19 and so that $M$, the maximum of the "errors" $|A_i - a_i|$, is as small as possible. For this minimum $M$, what is $100M$?

Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ meet at $P$. Let the area of triangle $APB$ be $24$ and let the area of triangle $CPD$ be $25$. What is the minimum possible area of quadrilateral $ABCD$?

Let the sum of the first $ n$ primes be denoted by $ S_n$. Prove that for any positive integer $ n$, there exists a perfect square between $ S_n$ and $ S_{n\plus{}1}$.

In $ \triangle ABC$, $ \angle ABC \equal{} 45^\circ$. Point $ D$ is on $ \overline{BC}$ so that $ 2 \cdot BD \equal{} CD$ and $ \angle DAB \equal{} 15^\circ$. Find $ \angle ACB$. [asy] pair A, B, C, D; A = origin; real Bcoord = 3*sqrt(2) + sqrt(6); B = Bcoord/2*dir(180); C = sqrt(6)*dir(120); draw(A--B--C--cycle); D = (C-B)/2.4 + B; draw(A--D); label("$A$", A, dir(0)); label("$B$", B, dir(180)); label("$C$", C, dir(110)); label("$D$", D, dir(130)); [/asy] $ \textbf{(A)} \ 54^\circ \qquad \textbf{(B)} \ 60^\circ \qquad \textbf{(C)} \ 72^\circ \qquad \textbf{(D)} \ 75^\circ \qquad \textbf{(E)} \ 90^\circ$

In the country of PUMACsboro, there are $n$ distinct cities labelled $1$ through $n$. There is a rail line going from city $i$ to city $j$ if and only if $i<j$; you can only take this rail line from city $i$ to city $j$. What is the smallest possible value of $n$ such that if each rail line's track is painted orange or black, you can always take the train between $2019$ cities on tracks that are all the same color? (This means there are some cities $c_1,c_2,\dots,c_{2019}$ such that there is a rail line going from city $c_i$ to $c_{i+1}$ for all $1\leq i\leq 2018$ and their rail lines' tracks are either all orange or all black.)

What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$ ${ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$

[b]p1.[/b] One day, Granny Smith bought a certain number of apples at Horock’s Farm Market. When she returned the next day she found that the price of the apples was reduced by $20\%$. She could therefore buy more apples while spending the same amount as the previous day. How many percent more? [b]p2.[/b] You are asked to move several boxes. You know nothing about the boxes except that each box weighs no more than $10$ tons and their total weight is $100$ tons. You can rent several trucks, each of which can carry no more than $30$ tons. What is the minimal number of trucks you can rent and be sure you will be able to carry all the boxes at once? [b]p3.[/b] The five numbers $1, 2, 3, 4, 5$ are written on a piece of paper. You can select two numbers and increase them by $1$. Then you can again select two numbers and increase those by $1$. You can repeat this operation as many times as you wish. Is it possible to make all numbers equal? [b]p4.[/b] There are $15$ people in the room. Some of them are friends with others. Prove that there is a person who has an even number of friends in the room. [u]Bonus Problem [/u] [b]p5.[/b] Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all polynomials $P(x)$ with real coefficients such that the polynomial $$Q(x) = (x + 1)P(x-1) -(x-1)P(x)$$ is constant.

Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\ldots $ by the following rules: [b](a)[/b] On every move of his $B$ passes $1$ coin from every box to an adjacent box. [b](b)[/b] On every move of hers $A$ chooses several coins that were [i]not[/i] involved in $B$'s previous move and are in different boxes. She passes every coin to an adjacent box. Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.

a) Show that there exist irrational numbers $a$, $b$, and $c$ such that the numbers $a+b\cdot c$, $b+a\cdot c$, and $c+a\cdot b$ are rational numbers. b) Show that if $a$, $b$, and $c$ are real numbers such that $a+b+c=1$, and the numbers $a+b\cdot c$, $b+a\cdot c$, and $c+a\cdot b$ are rational and non-zero, then $a$, $b$, and $c$ are rational numbers.

Find the number of functions $ f: \mathbb N\rightarrow\mathbb N$ which satisfying: (i) $ f(1) \equal{} 1$ (ii) $ f(n)f(n \plus{} 2) \equal{} f^2(n \plus{} 1) \plus{} 1997$ for every natural numbers n.

What is the value of \[\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16} + \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16}+\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16}+\tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?\] $\textbf{(A) } 28 \qquad \textbf{(B) } 68 \qquad \textbf{(C) } 70 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 84$

Two circles with radius $2$, $\omega_1$ and $\omega_2$, are centered at $O_1$ and $O_2$ respectively. The circles $\omega_1$ and $\omega_2$ are externally tangent to each other and internally tangent to a larger circle $\omega$ centered at $O$ at points $A$ and $B$, respectively. Let $M$ be the midpoint of minor arc $AB$. Let $P$ be the intersection of $\omega_1$ and $O_1M$, and let $Q$ be the intersection of $\omega_2$ and $O_2M$. Given that there is a point $R$ on $\omega$ such that $\triangle PQR$ is equilateral, the radius of $\omega$ can be written as $\frac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers and $a$ and $c$ are relatively prime. Find $a+b+c$.

There are three types of piece shown as below. Today Alice wants to cover a $100 \times 101$ board with these pieces without gaps and overlaps. Determine the minimum number of $1\times 1$ pieces should be used to cover the whole board and not exceed the board. (There are an infinite number of these three types of pieces.) [asy] size(9cm,0); defaultpen(fontsize(12pt)); draw((9,10) -- (59,10) -- (59,60) -- (9,60) -- cycle); draw((59,10) -- (109,10) -- (109,60) -- (59,60) -- cycle); draw((9,60) -- (59,60) -- (59,110) -- (9,110) -- cycle); draw((9,110) -- (59,110) -- (59,160) -- (9,160) -- cycle); draw((109,10) -- (159,10) -- (159,60) -- (109,60) -- cycle); draw((180,11) -- (230,11) -- (230,61) -- (180,61) -- cycle); draw((180,61) -- (230,61) -- (230,111) -- (180,111) -- cycle); draw((230,11) -- (280,11) -- (280,61) -- (230,61) -- cycle); draw((230,61) -- (280,61) -- (280,111) -- (230,111) -- cycle); draw((280,11) -- (330,11) -- (330,61) -- (280,61) -- cycle); draw((280,61) -- (330,61) -- (330,111) -- (280,111) -- cycle); draw((330,11) -- (380,11) -- (380,61) -- (330,61) -- cycle); draw((330,61) -- (380,61) -- (380,111) -- (330,111) -- cycle); draw((401,11) -- (451,11) -- (451,61) -- (401,61) -- cycle); [/asy] [i]Proposed by amano_hina[/i]

Let $ABC$ be a triangle with $AB = 9$, $BC = 10$, and $CA = 17$. Let $B'$ be the reflection of the point $B$ over the line $CA$. Let $G$ be the centroid of triangle $ABC$, and let $G'$ be the centroid of triangle $AB'C$. Determine the length of segment $GG'$.

Let $ABCD$ be a quadrilateral and let $O$ be the point of intersection of diagonals $AC$ and $BD$. Knowing that the area of triangle $AOB$ is equal to $ 1$, the area of triangle $BOC$ is equal to $2$, and the area of triangle $COD$ is equal to $4$, calculate the area of triangle $AOD$ and prove that $ABCD$ is a trapezoid.

Find all triples $(l, m, n)$ of distinct positive integers satisfying \[{\gcd(l, m)}^{2}= l+m, \;{\gcd(m, n)}^{2}= m+n, \; \text{and}\;\;{\gcd(n, l)}^{2}= n+l.\]

An acute triangle ABC is inscribed in a circle C. Tangents in A and C to circle C intersect at F. Segment bisector of AB intersects the line BC at E. Show that the lines FE and AB are parallel.

The image below consists of a large triangle divided into $13$ smaller triangles. Let $N$ be the number of ways to color each smaller triangle one of red, green, and blue such that if $T_1$ and $T_2$ are smaller triangles whose perimeters intersect at more than one point, $T_1$ and $T_2$ have two different colors. Compute the number of positive integer divisors of $N$. [asy] size(5 cm); draw((-4,0)--(4,0)--(0,6.928)--cycle); draw((0,0)--(2,3.464)--(-2,3.464)--cycle); draw((-2,0)--(-1,1.732)--(-3,1.732)--cycle); draw((2,0)--(1,1.732)--(3,1.732)--cycle); draw((0,3.464)--(1,5.196)--(-1,5.196)--cycle); [/asy] [i]2021 CCA Math Bonanza Individual Round #7[/i]

Let $ABCD$ be a trapezoid such that $AB \parallel CD$, $|BC|+|AD| = 7$, $|AB| = 9$ and $|BC| = 14$. What is the ratio of the area of the triangle formed by $CD$, angle bisector of $\widehat{BCD}$ and angle bisector of $\widehat{CDA}$ over the area of the trapezoid? $ \textbf{a)}\ \dfrac{9}{14} \qquad\textbf{b)}\ \dfrac{5}{7} \qquad\textbf{c)}\ \sqrt 2 \qquad\textbf{d)}\ \dfrac{49}{69} \qquad\textbf{e)}\ \dfrac 13 $

Outside the square $ABCD$, the rhombus $BCMN$ is constructed with angle $BCM$ obtuse . Let $P$ be the intersection point of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and the triangle $DPM$ is right isosceles .

For what values of the ratio $a/b$ is the limaçon $r = a - b \cos \theta$ a convex curve? $(a > b > 0)$