[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [u]Set 1[/u] [b]C.1 / G.1[/b] Daniel is exactly one year younger than his friend David. If David was born in the year $2008$, in what year was Daniel born? [b]C.2 / G.3[/b] Mr. Pham flips three coins. What is the probability that no two coins show the same side? [b]C.3 / G.2[/b] John has a sheet of white paper which is $3$ cm in height and $4$ cm in width. He wants to paint the sky blue and the ground green so the entire paper is painted. If the ground takes up a third of the page, how much space (in cm$^2$) does the sky take up? [b]C.4 / G.5[/b] Jihang and Eric are busy fidget spinning. While Jihang spins his fidget spinner at $15$ revolutions per second, Eric only manages $10$ revolutions per second. How many total revolutions will the two have made after $5$ continuous seconds of spinning? [b]C.5 / G.4[/b] Find the last digit of $1333337777 \cdot 209347802 \cdot 3940704 \cdot 2309476091$. [u]Set 2[/u] [b]C.6[/b] Evan, Chloe, Rachel, and Joe are splitting a cake. Evan takes $\frac13$ of the cake, Chloe takes $\frac14$, Rachel takes $\frac15$, and Joe takes $\frac16$. There is $\frac{1}{x}$ of the original cake left. What is $x$? [b]C.7[/b] Pacman is a $330^o$ sector of a circle of radius $4$. Pacman has an eye of radius $1$, located entirely inside Pacman. Find the area of Pacman, not including the eye. [b]C.8[/b] The sum of two prime numbers $a$ and $b$ is also a prime number. If $a < b$, find $a$. [b]C.9[/b] A bus has $54$ seats for passengers. On the first stop, $36$ people get onto an empty bus. Every subsequent stop, $1$ person gets off and $3$ people get on. After the last stop, the bus is full. How many stops are there? [b]C.10[/b] In a game, jumps are worth $1$ point, punches are worth $2$ points, and kicks are worth $3$ points. The player must perform a sequence of $1$ jump, $1$ punch, and $1$ kick. To compute the player’s score, we multiply the 1st action’s point value by $1$, the $2$nd action’s point value by $2$, the 3rd action’s point value by $3$, and then take the sum. For example, if we performed a punch, kick, jump, in that order, our score would be $1 \times 2 + 2 \times 3 + 3 \times 1 = 11$. What is the maximal score the player can get? [u]Set 3[/u] [b]C.11[/b] $6$ students are sitting around a circle, and each one randomly picks either the number $1$ or $2$. What is the probability that there will be two people sitting next to each other who pick the same number? [b]C.12 / G. 8[/b] You can buy a single piece of chocolate for $60$ cents. You can also buy a packet with two pieces of chocolate for $\$1.00$. Additionally, if you buy four single pieces of chocolate, the fifth one is free. What is the lowest amount of money you have to pay for $44$ pieces of chocolate? Express your answer in dollars and cents (ex. $\$3.70$). [b]C.13 / G.12[/b] For how many integers $k$ is there an integer solution $x$ to the linear equation $kx + 2 = 14$? [b]C.14 / G.9[/b] Ten teams face off in a swim meet. The boys teams and girls teams are ranked independently, each team receiving some number of positive integer points, and the final results are obtained by adding the points for the boys and the points for the girls. If Blair’s boys got $7$th place while the girls got $5$th place (no ties), what is the best possible total rank for Blair? [b]C.15 / G.11[/b] Arlene has a square of side length $1$, an equilateral triangle with side length $1$, and two circles with radius $1/6$. She wants to pack her four shapes in a rectangle without items piling on top of each other. What is the minimum possible area of the rectangle? PS. You should use hide for answers. C16-30/G10-15, G25-30 have been posted [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here[/url] and G16-25 [url=https://artofproblemsolving.com/community/c3h2790679p24540159]here [/url] . Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron?

Find all primes $p$, such that there exist positive integers $x$, $y$ which satisfy $$\begin{cases} p + 49 = 2x^2\\ p^2 + 49 = 2y^2\\ \end{cases}$$

A positive integer $n$ and a real number $a$ are given. Find all $n$-tuples $(x_1, ... ,x_n)$ of real numbers that satisfy the system of equations $$\sum_{i=1}^{n} x_i^k= a^k \,\,\,\, for \,\,\,\, k = 1,2, ... ,n$$

For all $k=1,2,\ldots,2008$,$a_k>0$.Prove that iff $\sum_{k=1}^{2008}a_k>1$,there exists a function $f:N\rightarrow R$ satisfying (1)$0=f(0)<f(1)<f(2)<\ldots$; (2)$f(n)$ has a finite limit when $n$ approaches infinity; (3)$f(n)-f({n-1})=\sum_{k=1}^{2008}a_kf({n+k})-\sum_{k=0}^{2007}a_{k+1}f({n+k})$,for all $n=1,2,3,\ldots$.

Starting with four given integers $a_1, b_1, c_1, d_1$ is defined recursively for all positive integers $n$: $$a_{n+1} := |a_n - b_n|, b_{n+1} := |b_n - c_n|, c_{n+1} := |c_n - d_n|, d_{n+1} := |d_n - a_n|.$$ Prove that there is a natural number $k$ such that all terms $a_k, b_k, c_k, d_k$ take the value zero.

For all positive numbers $a, b, c$ prove the inequality $2\sqrt{bc + ca + ab} \le \sqrt{3} \sqrt[3]{(b + c)(c + a)(a + b)}$.

Find all functions $f,g : N \to N$ such that for all $m ,n \in N$ the following relation holds: $$f(m ) - f(n) = (m - n)(g(m) + g(n))$$. Note: $N = \{0,1,2,...\}$

Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3$. The radii of $C_1$ and $C_2$ are $4$ and $10$, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2$. Given that the length of the chord is $\frac{m\sqrt{n}}{p}$ where $m,n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p$.

Find all functions $f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$, for which $f(k+1)>f(f(k)) \quad \forall k \geq 1$.

[b]a.)[/b] Let $a,b$ be real numbers. Define sequence $x_k$ and $y_k$ such that \[x_0 = 1, y_0 = 0, x_{k+1} = a \cdot x_k - b \cdot y_l, \quad y_{k+1} = x_k - a \cdot y_k \text{ for } k = 0,1,2, \ldots \] Prove that \[x_k = \sum^{[k/2]}_{l=0} (-1)^l \cdot a^{k - 2 \cdot l} \cdot \left(a^2 + b \right)^l \cdot \lambda_{k,l}\] where $\lambda_{k,l} = \sum^{[k/2]}_{m=l} \binom{k}{2 \cdot m} \cdot \binom{m}{l}$ [b]b.)[/b] Let $u_k = \sum^{[k/2]}_{l=0} \lambda_{k,l} $. For positive integer $m,$ denote the remainder of $u_k$ divided by $2^m$ as $z_{m,k}$. Prove that $z_{m,k},$ $k = 0,1,2, \ldots$ is a periodic function, and find the smallest period.

The smallest value of the function $f(x) =|x| +\left|\frac{1 - 2013x}{2013 - x}\right|$ where $x \in [-1, 1] $ is: (A): $\frac{1}{2012}$, (B): $\frac{1}{2013}$, (C): $\frac{1}{2014}$, (D): $\frac{1}{2015}$, (E): None of the above.

[b]p1.[/b] Compute $2020 \cdot \left( 2^{(0\cdot1)} + 9 - \frac{(20^1)}{8}\right)$. [b]p2.[/b] Nathan has five distinct shirts, three distinct pairs of pants, and four distinct pairs of shoes. If an “outfit” has a shirt, pair of pants, and a pair of shoes, how many distinct outfits can Nathan make? [b]p3.[/b] Let $ABCD$ be a rhombus such that $\vartriangle ABD$ and $\vartriangle BCD$ are equilateral triangles. Find the angle measure of $\angle ACD$ in degrees. [b]p4.[/b] Find the units digit of $2019^{2019}$. [b]p5.[/b] Determine the number of ways to color the four vertices of a square red, white, or blue if two colorings that can be turned into each other by rotations and reflections are considered the same. [b]p6.[/b] Kathy rolls two fair dice numbered from $1$ to $6$. At least one of them comes up as a $4$ or $5$. Compute the probability that the sumof the numbers of the two dice is at least $10$. [b]p7.[/b] Find the number of ordered pairs of positive integers $(x, y)$ such that $20x +19y = 2019$. [b]p8.[/b] Let $p$ be a prime number such that both $2p -1$ and $10p -1$ are prime numbers. Find the sum of all possible values of $p$. [b]p9.[/b] In a square $ABCD$ with side length $10$, let $E$ be the intersection of $AC$ and $BD$. There is a circle inscribed in triangle $ABE$ with radius $r$ and a circle circumscribed around triangle $ABE$ with radius $R$. Compute $R -r$ . [b]p10.[/b] The fraction $\frac{13}{37 \cdot 77}$ can be written as a repeating decimal $0.a_1a_2...a_{n-1}a_n$ with $n$ digits in its shortest repeating decimal representation. Find $a_1 +a_2 +...+a_{n-1}+a_n$. [b]p11.[/b] Let point $E$ be the midpoint of segment $AB$ of length $12$. Linda the ant is sitting at $A$. If there is a circle $O$ of radius $3$ centered at $E$, compute the length of the shortest path Linda can take from $A$ to $B$ if she can’t cross the circumference of $O$. [b]p12.[/b] Euhan and Minjune are playing tennis. The first one to reach $25$ points wins. Every point ends with Euhan calling the ball in or out. If the ball is called in, Minjune receives a point. If the ball is called out, Euhan receives a point. Euhan always makes the right call when the ball is out. However, he has a $\frac34$ chance of making the right call when the ball is in, meaning that he has a $\frac14$ chance of calling a ball out when it is in. The probability that the ball is in is equal to the probability that the ball is out. If Euhan won, determine the expected number of wrong callsmade by Euhan. [b]p13.[/b] Find the number of subsets of $\{1, 2, 3, 4, 5, 6,7\}$ which contain four consecutive numbers. [b]p14.[/b] Ezra and Richard are playing a game which consists of a series of rounds. In each round, one of either Ezra or Richard receives a point. When one of either Ezra or Richard has three more points than the other, he is declared the winner. Find the number of games which last eleven rounds. Two games are considered distinct if there exists a round in which the two games had different outcomes. [b]p15.[/b] There are $10$ distinct subway lines in Boston, each of which consists of a path of stations. Using any $9$ lines, any pair of stations are connected. However, among any $8$ lines there exists a pair of stations that cannot be reached from one another. It happens that the number of stations is minimized so this property is satisfied. What is the average number of stations that each line passes through? [b]p16.[/b] There exist positive integers $k$ and $3\nmid m$ for which $$1 -\frac12 + \frac13 - \frac14 +...+ \frac{1}{53}-\frac{1}{54}+\frac{1}{55}=\frac{3^k \times m}{28\times 29\times ... \times 54\times 55}.$$ Find the value $k$. [b]p17.[/b] Geronimo the giraffe is removing pellets from a box without replacement. There are $5$ red pellets, $10$ blue pellets, and $15$ white pellets. Determine the probability that all of the red pellets are removed before all the blue pellets and before all of the white pellets are removed. [b]p18.[/b] Find the remainder when $$70! \left( \frac{1}{4 \times 67}+ \frac{1}{5 \times 66}+...+ \frac{1}{66\times 5}+ \frac{1}{67\times 4} \right)$$ is divided by $71$. [b]p19.[/b] Let $A_1A_2...A_{12}$ be the regular dodecagon. Let $X$ be the intersection of $A_1A_2$ and $A_5A_{11}$. Given that $X A_2 \cdot A_1A_2 = 10$, find the area of dodecagon. [b]p20.[/b] Evaluate the following infinite series: $$\sum^{\infty}_{n=1}\sum^{\infty}_{m=1} \frac{n \sec^2m -m \tan^2 n}{3^{m+n}(m+n)}$$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Function $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(xf(y))=yf(x)$ for any $x,y$ are real numbers. Prove that $f(-x) = -f(x)$ for all real numbers $x$.

The headteacher wants to hire a certain number of new teachers in addition to existing teachers. If he hired an additional $10$ teachers, the number of school students would be reduced number per teacher by $5$. However, if the headmaster hired $20$ new teachers, the number of students per teacher would be reduced by $8$. How many students and how many there are teachers in this school? [img]https://cdn.artofproblemsolving.com/attachments/2/8/c0157ff43fd3d92138c87556a0fca2414e8a3f.png[/img]

For the real numbers $x$ and $y$, $[\sqrt{x}] = 10$ and $[\sqrt{y}] =14$. How large is $\left[\sqrt{[ \sqrt{x+y} ]}\right]$ ? (Note: the square roots are the positive values ​​and $[x]$ is the largest integer less than or equal to x.)

Find all sets $x,y,z$ of real numbers that satisfy $$\begin{cases} x^3 - y^2 = z^2 - x \\ y^3 -z^2 =x^2 -y \\z^3 -x^2 = y^2 -z \end{cases}$$

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Prove that if the polynomial $ f $ which is not identical to zero satisfies for every real $ x $ the equality $$ f(x)f(x + 3) = f(x^2 + x + 3), $$then it has no real roots .

In the cell $(i,j)$ of a table $n\times n$ is written the number $(i-1)n + j$. Determine all positive integers $n$ such that there are exactly $2025$ rows not containing a perfect square.

For any positive integer $n$, define $$c_n=\min_{(z_1,z_2,...,z_n)\in\{-1,1\}^n} |z_1\cdot 1^{2018} + z_2\cdot 2^{2018} + ... + z_n\cdot n^{2018}|.$$ Is the sequence $(c_n)_{n\in\mathbb{Z}^+}$ bounded?

Prove that : For each integer $n \ge 3$, there exists the positive integers $a_1<a_2< \cdots <a_n$ , such that for $ i=1,2,\cdots,n-2 $ , With $a_{i},a_{i+1},a_{i+2}$ may be formed as a triangle side length , and the area of the triangle is a positive integer.

$\boxed{A3}$The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$

In Kacs Aladár street, houses are only found on one side of the road, so that only odd house numbers are found along the street. There are an odd number of allotments, as well. The middle three allotments belong to Scrooge McDuck, so he only put up the smallest of the three house numbers. The numbering of the other houses is standard, and the numbering begins with $1$. What is the largest number in the street if the sum of house numbers put up is $3133$?

Let $\mathbb{N}$ be the set of positive integers. Determine all positive integers $k$ for which there exist functions $f:\mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N}\to \mathbb{N}$ such that $g$ assumes infinitely many values and such that $$ f^{g(n)}(n)=f(n)+k$$ holds for every positive integer $n$. ([i]Remark.[/i] Here, $f^{i}$ denotes the function $f$ applied $i$ times i.e $f^{i}(j)=f(f(\dots f(j)\dots ))$.)