Let $x,y,z$ be complex numbers satisfying \begin{align*} z^2 + 5x &= 10z \\ y^2 + 5z &= 10y \\ x^2 + 5y &= 10x \end{align*} Find the sum of all possible values of $z$. [i]Proposed by Aaron Lin[/i]

How many different solutions on the interval $[0, \pi]$ does the equation $$6\sqrt2 \sin x \cdot tgx - 2\sqrt2 tgx +3\sin x -1=0$$ have?

Determine all triples $(x,y,z)$ of real numbers such that the following system of equations holds true: \begin{align*}2^{\sqrt[3]{x^2}}\cdot 4^{\sqrt[3]{y^2}}\cdot 16^{\sqrt[3]{z^2}}&=128\\ \left(xy^2+z^4\right)^2&=4+\left(xy^2-z^4\right)^2\mbox{.}\end{align*}

The product of Albrecht’s three favorite numbers is $2022$, and if we add one to each number, their product will be $1514$. What is the sum of their squares, if we know their sum is $0$?

(a) Prove or find a counter-example: For every two complex numbers $z,w$ the following inequality holds: $$|z|+|w|\le|z+w|+|z-w|.$$(b) Prove that for all $z_1,z_2,z_3,z_4\in\mathbb C$: $$\sum_{k=1}^4|z_k|\le\sum_{1\le i<j\le4}|z_i+z_j|.$$

Let $x_1, x_2, ..., x_n$ be positive reals. Prove that $$\sum_{k=1}^n \frac{x_k(2x_k - x_{k+1} - x_{k+2})}{x_{k+1} + x_{k+2}} \ge 0$$ In the sum, cyclic indices have been taken, that is, $x_{n+1} = x_1$ and $x_{n+2} = x_2$.

[u]Round 1[/u] [b]1.1.[/b] A person asks for help every $3$ seconds. Over a time period of $5$ minutes, how many times will they ask for help? [b]1.2.[/b] In a big bag, there are $14$ red marbles, $15$ blue marbles, and$ 16$ white marbles. If Anuj takes a marble out of the bag each time without replacement, how many marbles does Anuj need to remove to be sure that he will have at least $3$ red marbles? [b]1.3.[/b] If Josh has $5$ distinct candies, how many ways can he pick $3$ of them to eat? [u]Round 2[/u] [b]2.1.[/b] Annie has a circular pizza. She makes $4$ straight cuts. What is the minimum number of slices of pizza that she can make? [b]2.2.[/b] What is the sum of the first $4$ prime numbers that can be written as the sum of two perfect squares? [b]2.3.[/b] Consider a regular octagon $ABCDEFGH$ inscribed in a circle of area $64\pi$. If the length of arc $ABC$ is $n\pi$, what is $n$? [u]Round 3[/u] [b]3.1.[/b] Let $ABCDEF$ be an equiangular hexagon with consecutive sides of length $6, 5, 3, 8$, and $3$. Find the length of the sixth side. [b]3.2.[/b] Jack writes all of the integers from $ 1$ to $ n$ on a blackboard except the even primes. He selects one of the numbers and erases all of its digits except the leftmost one. He adds up the new list of numbers and finds that the sum is $2020$. What was the number he chose? [b]3.3.[/b] Our original competition date was scheduled for April $11$, $2020$ which is a Saturday. The numbers $4116$ and $2020$ have the same remainder when divided by $x$. If $x$ is a prime number, find the sum of all possible $x$. [u]Round 4[/u] [b]4.1.[/b] The polynomials $5p^2 + 13pq + cq^2$ and $5p^2 + 13pq - cq^2$ where $c$ is a positive integer can both be factored into linear binomials with integer coefficients. Find $c$. [b]4.2.[/b] In a Cartesian coordinate plane, how many ways are there to get from $(0, 0)$ to $(2, 3)$ in $7$ moves, if each move consists of a moving one unit either up, down, left, or right? [b]4.3.[/b] Bob the Builder is building houses. On Monday he finds an empty field. Each day starting on Monday, he finishes building a house at noon. On the $n$th day, there is a $\frac{n}{8}$ chance that a storm will appear at $3:14$ PM and destroy all the houses on the field. At any given moment, Bob feels sad if and only if there is exactly $1$ house left on the field that is not destroyed. The probability that he will not be sad on Friday at $6$ PM can be expressed as $p/q$ in simplest form. Find $p + q$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784570p24468605]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all triples of real numbers $(x, y,z)$ such that $$\begin{cases} x^4 + y^2 + 4 = 5yz \\ y^4 + z^2 + 4 = 5zx \\ z^4 + x^2 + 4 = 5xy\end{cases}$$

Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ continuous functions such that $\lim_{x\rightarrow \infty} f(x) =\infty$ and $\forall x,y\in \mathbb{R}, |x-y|>\varphi, \exists n<\varphi^{2023}, n\in \mathbb{N}$ such that $$f^n(x)+f^n(y)=x+y$$

Suppose $p$ is a prime number and $a,b,c \in \mathbb Q^+$ are rational numbers; [b]a)[/b] Prove that $\mathbb Q(\sqrt[p]{a}+\sqrt[p]{b})=\mathbb Q(\sqrt[p]{a},\sqrt[p]{b})$. [b]b)[/b] If $\sqrt[p]{b} \in \mathbb Q(\sqrt[p]{a})$, prove that for a nonnegative integer $k$ we have $\sqrt[p]{\frac{b}{a^k}}\in \mathbb Q$. [b]c)[/b] If $\sqrt[p]{a}+\sqrt[p]{b}+\sqrt[p]{c} \in \mathbb Q$, then prove that numbers $\sqrt[p]{a},\sqrt[p]{b}$ and $\sqrt[p]{c}$ are rational.

Let $A =\left\{a = q + \frac{1}{q }/ q \in Q^*,q > 0 \right\}$, $A + A = \{a + b |a,b \in A\}$,$A \cdot A =\{a \cdot b | a, b \in A\}$. Prove that: i) $A + A \ne A \cdot A$ ii) $(A + A) \cap N = (A \cdot A) \cap N$. Vasile Pop

Let $n$ be a natural number and $P_1, P_2, ... , P_n$ are polynomials with integer coefficients, each of degree at least $2$. Let $S$ be the set of all natural numbers $N$ for which there exists a natural number $a$ and an index $1 \le i \le n$ such that $P_i(a) = N$. Prove, that there are infinitely many primes that do not belong to $S$.

Let $n$ be a natural number and $\alpha ,\beta ,\gamma$ be the angles of an acute triangle. Determine the least possible value of the sum: $T=tan^n \alpha+tan^n \beta+tan^n \gamma$.

Each of given $100$ numbers was increased by $1$. Then each number was increased by $1$ once more. Given that the first time the sum of the squares of the numbers was not changed find how this sum was changed the second time.

A sequence $x_0=A$ and $x_1=B$ and $x_{n+2}=x_{n+1} +x_n$ is called a Fibonacci type sequence. Call a number $C$ a repeated value if $x_t=x_s=c$ for $t$ different from $s$. Prove one can choose $A$ and $B$ to have as many repeated value as one likes but never infinite.

Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$, $$f(x+f(y))+xy=f(x)f(y)+f(x)+y.$$ [i]Andrew Carratu[/i]

Suppose the numbers $a_0, a_1, a_2, ... , a_n$ satisfy the following conditions: $a_0 =\frac12$, $a_{k+1} = a_k +\frac{1}{n}a_k^2$ for $k = 0, 1, ... , n - 1$. Prove that $1 - \frac{1}{n}< a_n < 1$

Let $r$, $s$ be real numbers, find maximum $t$ so that if $a_1, a_2, \ldots$ is a sequence of positive real numbers satisfying \[ a_1^r + a_2^r + \cdots + a_n^r \le 2023 \cdot n^t \] for all $n \ge 2023$ then the sum \[ b_n = \frac 1{a_1^s} + \cdots + \frac 1{a_n^s} \] is unbounded, i.e for all positive reals $M$ there is an $n$ such that $b_n > M$.

Find all positive integer $k$s for which such $f$ exists and unique: $f(mn)=f(n)f(m)$ for $n, m \in \mathbb{Z^+}$ $f^{n^k}(n)=n$ for all $n \in \mathbb{Z^+}$ for which $f^x (n)$ means the n times operation of function $f$(i.e. $f(f(...f(n))...)$)

$(SWE 4)$ Let $a_0, a_1, a_2, \cdots$ be determined with $a_0 = 0, a_{n+1} = 2a_n + 2^n$. Prove that if $n$ is power of $2$, then so is $a_n$

For each positive integer $n$, determine the smallest possible value of the polynomial $$ W_n(x)=x^{2n}+2x^{2n-1}+3x^{2n-2}+\ldots + (2n-1)x^2+2nx. $$

Let $x,y,z$ be a positive real numbers. Prove that $$\frac {x}{\sqrt {x + y}} + \frac {y}{\sqrt {y + z}} + \frac { z}{\sqrt {z + x}}\geq\sqrt [4]{\frac {27(yz + zx + xy)}{4}}$$ [i](dektep)[/i]

[b]p1.[/b] What is $(2 + 4 + ... + 20) - (1 + 3 + ...+ 19)$? [b]p2.[/b] Two ants start on opposite vertices of a dodecagon ($12$-gon). Each second, they randomly move to an adjacent vertex. What is the probability they meet after four moves? [b]p3.[/b] How many distinct $8$-letter strings can be made using $8$ of the $9$ letters from the words $FORK$ and $KNIFE$ (e.g., $FORKNIFE$)? [b]p4.[/b] Let $ABC$ be an equilateral triangle with side length $8$ and let $D$ be a point on segment $BC$ such that $BD = 2$. Given that $E$ is the midpoint of $AD$, what is the value of $CE^2 - BE^2$? [b][color=#f00](mistyped p4)[/color][/b] Let $ABC$ be an equilateral triangle with side length $8$ and let $D$ be a point on segment $BC$ such that $BD = 2$. Given that $E$ is the midpoint of $AD$, what is the value of $CE^2 + BE^2$? [b]p5.[/b] You have two fair six-sided dice, one labeled $1$ to $6$, and for the other one, each face is labeled $1$, $2$, $3$, or $4$ (not necessarily all numbers are used). Let $p$ be the probability that when the two dice are rolled, the number on the special die is smaller than the number on the normal die. Given that $p = 1/2$, how many distinct combinations of $1$, $2$, $3$, $4$ can appear on the special die? The arrangement of the numbers on the die does not matter. [b]p6.[/b] Let $\omega_1$ and $\omega_2$ be two circles with centers $A$ and $B$ and radii $3$ and $13$, respectively. Suppose $AB = 10$ and that $C$ is the midpoint of $AB$. Let $\ell$ be a line that passes through $C$ and is tangent to $\omega_1$ at $P$. Given that $\ell$ intersects $\omega_2$ at $X$ and $Y$ such that $XP < Y P$, what is $XP$? [b]p7.[/b] Let $f(x)$ be a cubic polynomial. Given that $f(1) = 13$, $f(4) = 19$, $f(7) = 7$, and $f(10) = 13$, find $f(13)$. [b]p8.[/b] For all integers $0 \le n \le 202$ not divisible by seven, define $f(n) = \{\sqrt{7n}\}$. For what value $n$ does $f(n)$ take its minimum value? (Note: $\{x\} = x - \lfloor x \rfloor$, where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) [b]p9.[/b] Let $ABC$ be a triangle with $AB = 14$ and $AC = 25$. Let the incenter of $ABC$ be $I$. Let line $AI$ intersect the circumcircle of $BIC$ at $D$ (different from $I$). Given that line $DC$ is tangent to the circumcircle of $ABC$, find the area of triangle $BCD$. [b]p10.[/b] Evaluate the infinite sum $$\frac{4^2 + 3}{1 \cdot 3 \cdot 5 \cdot 7} +\frac{6^2 + 3}{3 \cdot 5 \cdot 7 \cdot 9}+\frac{8^2 + 3}{5 \cdot 7 \cdot 9 \cdot 11}+ ...$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $r,s,t$ be the roots of the equation $x(x-2)(3x-7)=2$. Show that $r,s,t$ are real and positive and determine $\arctan r+\arctan s +\arctan t$.

A positive integer $n$ is such that $n(n+2013)$ is a perfect square. a) Show that $n$ cannot be prime. b) Find a value of $n$ such that $n(n+2013)$ is a perfect square.