Solve the system of equations $$\begin{cases} \sin^3 x+\sin^4 y=1 \\ \cos^4 x+\cos^5 y =1\end{cases}$$

Let $P(x)$ and $Q(x)$ be monic polynomials. Prove that the sum of the squares of the coeficients of the polynomial $P(x)Q(x)$ is not smaller than the sum of the squares of the free coefficients of $P(x)$ and $Q(x)$. [i]A. Galochkin, O. Ljashko[/i]

Let $a,b,c$ be positive reals. Find the minimum value of $$\dfrac{13a+13b+2c}{2a+2b}+\dfrac{24a-b+13c}{2b+2c}+\dfrac{(-a+24b+13c)}{2c+2a}$$. (1)What is the minimum value? (2)If the minimum value occurs when $(a,b,c)=(a_0,b_0,c_0)$,then find $\frac{b_0}{a_0}+\frac{c_0}{b_0}$.

Let $\displaystyle \mathcal K$ be a finite field. Prove that the following statements are equivalent: (a) $\displaystyle 1+1=0$; (b) for all $\displaystyle f \in \mathcal K \left[ X \right]$ with $\displaystyle \textrm{deg} \, f \geq 1$, $\displaystyle f \left( X^2 \right)$ is reducible.

Three bells begin to ring simultaneously. The intervals between strikes for these bells are, respectively, $\frac43$ seconds, $\frac53$ second and $2$ seconds. Impacts that coincide in time are perceived as one. How many beats will be heard in $1$ minute? (Include first and last.)

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]B16 / G11[/b] Let triangle $ABC$ be an equilateral triangle with side length $6$. If point $D$ is on $AB$ and point $E$ is on $BC$, find the minimum possible value of $AD + DE + CE$. [b]B17 / G12[/b] Find the smallest positive integer $n$ with at least seven divisors. [b]B18 / G13[/b] Square $A$ is inscribed in a circle. The circle is inscribed in Square $B$. If the circle has a radius of $10$, what is the ratio between a side length of Square $A$ and a side length of Square $B$? [b]B19 / G14[/b] Billy Bob has $5$ distinguishable books that he wants to place on a shelf. How many ways can he order them if he does not want his two math books to be next to each other? [b]B20 / G15[/b] Six people make statements as follows: Person $1$ says “At least one of us is lying.” Person $2$ says “At least two of us are lying.” Person $3$ says “At least three of us are lying.” Person $4$ says “At least four of us are lying.” Person $5$ says “At least five of us are lying.” Person $6$ says “At least six of us are lying.” How many are lying? [u]Set 5[/u] [b]B21 / G16[/b] If $x$ and $y$ are between $0$ and $1$, find the ordered pair $(x, y)$ which maximizes $(xy)^2(x^2 - y^2)$. [b]B22 / G17[/b] If I take all my money and divide it into $12$ piles, I have $10$ dollars left. If I take all my money and divide it into $13$ piles, I have $11$ dollars left. If I take all my money and divide it into $14$ piles, I have $12$ dollars left. What’s the least amount of money I could have? [b]B23 / G18[/b] A quadratic equation has two distinct prime number solutions and its coefficients are integers that sum to a prime number. Find the sum of the solutions to this equation. [b]B24 / G20[/b] A regular $12$-sided polygon is inscribed in a circle. Gaz then chooses $3$ vertices of the polygon at random and connects them to form a triangle. What is the probability that this triangle is right? [b]B25 / G22[/b] A book has at most $7$ chapters, and each chapter is either $3$ pages long or has a length that is a power of $2$ (including $1$). What is the least positive integer $n$ for which the book cannot have $n$ pages? [u]Set 6[/u] [b]B26 / G26[/b] What percent of the problems on the individual, team, and guts rounds for both divisions have integer answers? [b]B27 / G27[/b] Estimate $12345^{\frac{1}{123}}$. [b]B28 / G28[/b] Let $O$ be the center of a circle $\omega$ with radius $3$. Let $A, B, C$ be randomly selected on $\omega$. Let $M$, $N$ be the midpoints of sides $BC$, $CA$, and let $AM$, $BN$ intersect at $G$. What is the probability that $OG \le 1$? [b]B29 / G29[/b] Let $r(a, b)$ be the remainder when $a$ is divided by $b$. What is $\sum^{100}_{i=1} r(2^i , i)$? [b]B30 / G30[/b] Bongo flips $2023$ coins. Call a run of heads a sequence of consecutive heads. Say a run is maximal if it isn’t contained in another run of heads. For example, if he gets $HHHT T HT T HHHHT H$, he’d have maximal runs of length $3, 1, 4, 1$. Bongo squares the lengths of all his maximal runs and adds them to get a number $M$. What is the expected value of $M$? - - - - - - [b]G19[/b] Let $ABCD$ be a square of side length $2$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AD$. Let the intersection of $BN$ and $CM$ be $E$. Find the area of quadrilateral $NECD$. [b]G21[/b] Quadrilateral $ABCD$ has $\angle A = \angle D = 60^o$. If $AB = 8$, $CD = 10$, and $BC = 3$, what is length $AD$? [b]G23[/b] $\vartriangle ABC$ is an equilateral triangle of side length $x$. Three unit circles $\omega_A$, $\omega_B$, and $\omega_C$ lie in the plane such that $\omega_A$ passes through $A$ while $\omega_B$ and $\omega_C$ are centered at $B$ and $C$, respectively. Given that $\omega_A$ is externally tangent to both $\omega_B$ and $\omega_C$, and the center of $\omega_A$ is between point $A$ and line $\overline{BC}$, find $x$. [b]G24[/b] For some integers $n$, the quadratic function $f(x) = x^2 - (6n - 6)x - (n^2 - 12n + 12)$ has two distinct positive integer roots, exactly one out of which is a prime and at least one of which is in the form $2^k$ for some nonnegative integer $k$. What is the sum of all possible values of $n$? [b]G25[/b] In a triangle, let the altitudes concur at $H$. Given that $AH = 30$, $BH = 14$, and the circumradius is $25$, calculate $CH$ PS. You should use hide for answers. Rest problems have been posted [url=https://artofproblemsolving.com/community/c3h3132167p28376626]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] [b]B1[/b] What is the sum of the first $5$ positive integers? [b]B2[/b] Bread picks a number $n$. He finds out that if he multiplies $n$ by $23$ and then subtracts $20$, he gets $46279$. What is $n$? [b]B3[/b] A [i]Harshad [/i] Number is a number that is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$. Only one two-digit multiple of $9$ is not a [i]Harshad [/i] Number. What is this number? [b]B4 / G1[/b] There are $5$ red balls and 3 blue balls in a bag. Alice randomly picks a ball out of the bag and then puts it back in the bag. Bob then randomly picks a ball out of the bag. What is the probability that Alice gets a red ball and Bob gets a blue ball, assuming each ball is equally likely to be chosen? [b]B5[/b] Let $a$ be a $1$-digit positive integer and $b$ be a $3$-digit positive integer. If the product of $a$ and $b$ is a$ 4$-digit integer, what is the minimum possible value of the sum of $a$ and $b$? [b]B6 / G2[/b] A circle has radius $6$. A smaller circle with the same center has radius $5$. What is the probability that a dart randomly placed inside the outer circle is outside the inner circle? [b]B7[/b] Call a two-digit integer “sus” if its digits sum to $10$. How many two-digit primes are sus? [b]B8 / G3[/b] Alex and Jeff are playing against Max and Alan in a game of tractor with $2$ standard decks of $52$ cards. They take turns taking (and keeping) cards from the combined decks. At the end of the game, the $5$s are worth $5$ points, the $10$s are worth $10$ points, and the kings are worth 10 points. Given that a team needs $50$ percent more points than the other to win, what is the minimal score Alan and Max need to win? [b]B9 / G4[/b] Bob has a sandwich in the shape of a rectangular prism. It has side lengths $10$, $5$, and $5$. He cuts the sandwich along the two diagonals of a face, resulting in four pieces. What is the volume of the largest piece? [b]B10 / G5[/b] Aven makes a rectangular fence of area $96$ with side lengths $x$ and $y$. John makesva larger rectangular fence of area 186 with side lengths $x + 3$ and $y + 3$. What is the value of $x + y$? [b]B11 / G6[/b] A number is prime if it is only divisible by itself and $1$. What is the largest prime number $n$ smaller than $1000$ such that $n + 2$ and $n - 2$ are also prime? Note: $1$ is not prime. [b]B12 / G7[/b] Sally has $3$ red socks, $1$ green sock, $2$ blue socks, and $4$ purple socks. What is the probability she will choose a pair of matching socks when only choosing $2$ socks without replacement? [b]B13 / G8[/b] A triangle with vertices at $(0, 0)$,$ (3, 0)$, $(0, 6)$ is filled with as many $1 \times 1$ lattice squares as possible. How much of the triangle’s area is not filled in by the squares? [b]B14 / G10[/b] A series of concentric circles $w_1, w_2, w_3, ...$ satisfy that the radius of $w_1 = 1$ and the radius of $w_n =\frac34$ times the radius of $w_{n-1}$. The regions enclosed in $w_{2n-1}$ but not in $w_{2n}$ are shaded for all integers $n > 0$. What is the total area of the shaded regions? [b]B15 / G12[/b] $10$ cards labeled 1 through $10$ lie on a table. Kevin randomly takes $3$ cards and Patrick randomly takes 2 of the remaining $7$ cards. What is the probability that Kevin’s largest card is smaller than Patrick’s largest card, and that Kevin’s second-largest card is smaller than Patrick’s smallest card? [b]G9[/b] Let $A$ and $B$ be digits. If $125A^2 + B161^2 = 11566946$. What is $A + B$? [b]G11[/b] How many ordered pairs of integers $(x, y)$ satisfy $y^2 - xy + x = 0$? [b]G13[/b] $N$ consecutive integers add to $27$. How many possible values are there for $N$? [b]G14[/b] A circle with center O and radius $7$ is tangent to a pair of parallel lines $\ell_1$ and $\ell_2$. Let a third line tangent to circle $O$ intersect $\ell_1$ and $\ell_2$ at points $A$ and $B$. If $AB = 18$, find $OA + OB$. [b]G15[/b] Let $$ M =\prod ^{42}_{i=0}(i^2 - 5).$$ Given that $43$ doesn’t divide $M$, what is the remainder when M is divided by $43$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $(a_k)^{\infty}_{k=0}$ be a sequence of real numbers such that if $k$ is a non-negative integer, then $$a_{k+1} = 3a_k - \lfloor 2a_k \rfloor - \lfloor a_k \rfloor.$$ Definitely all positive integers $n$ such that if $a_0 = 1/n$, then this sequence is constant after a certain term.

$k$ is a fixed positive integer. Let $a_n$ be the number of maps $f$ from the subsets of $\{1, 2, ... , n\}$ to $\{1, 2, ... , k\}$ such that for all subsets $A, B$ of $\{1, 2, ... , n\}$ we have $f(A \cap B) = \min (f(A), f(B))$. Find $\lim_{n \to \infty} \sqrt[n]{a_n}$.

Three rational number $x,p,q$ satisfy $p^2-xq^2$=1. Prove that there are integers $a,b$ such that $p=\frac{a^2+xb^2}{a^2-xb^2}$ and $q=\frac{2ab}{a^2-xb^2}$.

Let $P$ be a polynom of degree $n \geq 5$ with integer coefficients given by $P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_0 \quad$ with $a_i \in \mathbb{Z}$, $a_n \neq 0$. Suppose that $P$ has $n$ different integer roots (elements of $\mathbb{Z}$) : $0,\alpha_2,\ldots,\alpha_n$. Find all integers $k \in \mathbb{Z}$ such that $P(P(k))=0$.

Given are two polynomial $f$ and $g$ of degree $100$ with real coefficients. For each positive integer $n$ there is an integer $k$ such that \[\frac{f(k)}{g(k)}=\frac{n+1}{n}.\] Prove that $f$ and $g$ have a common non-constant factor.

Find all $f: \mathbb{Q}_{+} \rightarrow \mathbb{R}$ such that \[ f(x)+f(y)+f(z)=1 \] holds for every positive rationals $x, y, z$ satisfying $x+y+z+1=4xyz$.

Let $A$ and $B$ be two sets of real numbers. Suppose that the elements of the set $AB = \{ab: a\in A, b\in B\}$ form a finite arithmetic progression. Prove that one of these sets contains no more than three elements

Let $a, b, c > 0$ be real numbers. Show the following inequality: $$a^2 \cdot \frac{a - b}{a + b}+ b^2\cdot \frac{b - c}{b + c}+ c^2\cdot \frac{c - a}{c + a} \ge 0 .$$ When does equality holds?

Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.

A real valued continuous function $f$ satisfies for all real $x$ and $y$ the functional equation $$ f(\sqrt{x^2 +y^2 })= f(x)f(y).$$ Prove that $$f(x) =f(1)^{x^{2}}.$$

Three positive reals $x , y , z $ satisfy \\ $x^2 + y^2 = 3^2 \\ y^2 + yz + z^2 = 4^2 \\ x^2 + \sqrt{3}xz + z^2 = 5^2 .$ \\ Find the value of $2xy + xz + \sqrt{3}yz$

The teacher wrote $5$ distinct real numbers on the board. After this, Petryk calculated the sums of each pair of these numbers and wrote them on the left part of the board, and Vasyl calculated the sums of each triple of these numbers and wrote them on the left part of the board (each of them wrote $10$ numbers). Could the multisets of numbers written by Petryk and Vasyl be identical?

Let $r$ be a given positive integer. Is is true that for every $r$-colouring of the natural numbers there exists a monochromatic solution of the equation $x+y=3z$? (proposed by B. Batbaysgalan, folklore)

Find all values ​​of the parameter $a$ for which the equation $$(a-1)^2x^4 + (a^2-a) x^3 + 3x - 1 = 0$$ has a unique solution and for these $a$ solve the equation.

Determine all real numbers $a, b$ and all integers $n\ge 1$ for which$ (a + b)^n = a^n + b^n$ holds.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

For every positive integer $k>1$ prove that there exist a real number $x$ so that for every positive integer $n<1398$: $$\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.$$ [i]Proposed by Mohammad Amin Sharifi[/i]

Factor $(b - c)^3 + (c - a)^3 + (a - b)^3$.