For a real number $a$, denote by $(a]$ the smallest integer that is not less than $a$. Find all real values of $x$ for which holds the equality $$(\sin x]^2 + (\cos x]^2 =|tg x| +|ctg x|.$$

Let $s(n)$ denote the sum of the digits of $n$. a) Do there exist distinct positive integers $a, b$, such that $2023a+s(a)=2023b+s(b)$? b) Do there exist distinct positive integers $a, b$, such that $a+2023s(a)=b+2023s(b)$?

Farmer Boso has a busy farm with lots of animals. He tends to $5b$ cows, $5a +7$ chickens, and $b^{a-5}$ insects. Note that each insect has $6$ legs. The number of cows is equal to the number of insects. The total number of legs present amongst his animals can be expressed as $\overline{LLL }+1$, where $L$ stands for a digit. Find $L$.

Find all integer solutions of the equation $a^3+3ab^2+7b^3=2011$.

For a positive integer $n$, let $\alpha(n)$ be the arithmetic mean of the divisors of $n$, and let $\beta(n)$ be the arithmetic mean of the numbers $k \le n$ with $\text{gcd}(k,n)=1$. Determine all positive integers $n$ with $\alpha(n)=\beta(n)$.

We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.

Consider the prime numbers $p_1,p_2,\dots ,p_{2021}$ such that the sum $$p_1^4+p_2^4+\dots +p_{2021}^4$$ is divisible by $6060$. Prove that at least $4$ of these prime numbers are less than $2021$. $\textit{Stefan Bălăucă}$

A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$. Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$. [i]Proposed by Carl Schildkraut and Holden Mui[/i]

Solve the equation $4xy-x-y=z^2$ in positive integers.

Compute the sum of all prime numbers $p$ with $p \ge 5$ such that $p$ divides $(p + 3)^{p-3} + (p + 5)^{p-5}$. .

Show that it is possible to color the points of $\mathbb Q\times\mathbb Q$ in two colors in such a way that any two points having distance $1$ have distinct colors.

Let $S$ be a nonempty set of primes satisfying the property that for each proper subset $P$ of $S$, all the prime factors of the number $\left(\prod_{p\in P}p\right)-1$ are also in $S$. Determine all possible such sets $S$.

A natural number $n$ is called breakable if there exist positive integers $a,b,x,y$ such that $a+b = n$ and $\frac{x}{a}+\frac{y}{b}= 1$. Find all breakable numbers.

For any given non-negative integer $m$, let $f(m)$ be the number of $1$'s in the base $2$ representation of $m$. Let $n$ be a positive integer. Prove that the integer $$\sum^{2^n - 1}_{m = 0} \Big( (-1)^{f(m)} \cdot 2^m \Big)$$ contains at least $n!$ positive divisors.

Let $\tau(n)$ be the number of positive divisors of $n$. Let $\tau_1(n)$ be the number of positive divisors of $n$ which have remainders $1$ when divided by $3$. Find all positive integral values of the fraction $\frac{\tau(10n)}{\tau_1(10n)}$.

[b]p1.[/b] Aidan walks into a skyscraper’s first floor lobby and takes the elevator up $50$ floors. After exiting the elevator, he takes the stairs up another $10$ floors, then takes the elevator down $30$ floors. Find the floor number Aidan is currently on. [b]p2.[/b] Jeff flips a fair coin twice and Kaylee rolls a standard $6$-sided die. The probability that Jeff flips $2$ heads and Kaylee rolls a $4$ is $P$. Find $\frac{1}{P}$ . [b]p3.[/b] Given that $a\odot b = a + \frac{a}{b}$ , find $(4\odot 2)\odot 3$. [b]p4.[/b] The following star is created by gluing together twelve equilateral triangles each of side length $3$. Find the outer perimeter of the star. [img]https://cdn.artofproblemsolving.com/attachments/e/6/ad63edbf93c5b7d4c7e5d68da2b4632099d180.png[/img] [b]p5.[/b] In Lexington High School’sMath Team, there are $40$ students: $20$ of whom do science bowl and $22$ of whom who do LexMACS. What is the least possible number of students who do both science bowl and LexMACS? [b]p6.[/b] What is the least positive integer multiple of $3$ whose digits consist of only $0$s and $1$s? The number does not need to have both digits. [b]p7.[/b] Two fair $6$-sided dice are rolled. The probability that the product of the numbers rolled is at least $30$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p8.[/b] At the LHSMath Team Store, $5$ hoodies and $1$ jacket cost $\$13$, and $5$ jackets and $1$ hoodie cost $\$17$. Find how much $15$ jackets and $15$ hoodies cost, in dollars. [b]p9.[/b] Eric wants to eat ice cream. He can choose between $3$ options of spherical ice cream scoops. The first option consists of $4$ scoops each with a radius of $3$ inches, which costs a total of $\$3$. The second option consists of a scoop with radius $4$ inches, which costs a total of $\$2$. The third option consists of $5$ scoops each with diameter $2$ inches, which costs a total of $\$1$. The greatest possible ratio of volume to cost of ice cream Eric can buy is nπ cubic inches per dollar. Find $3n$. [b]p10.[/b] Jen claims that she has lived during at least part of each of five decades. What is the least possible age that Jen could be? (Assume that age is always rounded down to the nearest integer.) [b]p11.[/b] A positive integer $n$ is called Maisylike if and only if $n$ has fewer factors than $n -1$. Find the sum of the values of $n$ that are Maisylike, between $2$ and $10$, inclusive. [b]p12.[/b] When Ginny goes to the nearby boba shop, there is a $30\%$ chance that the employee gets her drink order wrong. If the drink she receives is not the one she ordered, there is a $60\%$ chance that she will drink it anyways. Given that Ginny drank a milk tea today, the probability she ordered it can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find the value of $a +b$. [b]p13.[/b] Alex selects an integer $m$ between $1$ and $100$, inclusive. He notices there are the same number of multiples of $5$ as multiples of $7$ betweenm and $m+9$, inclusive. Find how many numbers Alex could have picked. [b]p14.[/b] In LMTown there are only rainy and sunny days. If it rains one day there’s a $30\%$ chance that it will rain the next day. If it’s sunny one day there’s a $90\%$ chance it will be sunny the next day. Over n days, as n approaches infinity, the percentage of rainy days approaches $k\%$. Find $10k$. [b]p15.[/b] A bag of letters contains $3$ L’s, $3$ M’s, and $3$ T’s. Aidan picks three letters at random from the bag with replacement, and Andrew picks three letters at random fromthe bag without replacement. Given that the probability that both Aidan and Andrew pick one each of L, M, and T can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. [b]p16.[/b] Circle $\omega$ is inscribed in a square with side length $2$. In each corner tangent to $2$ of the square’s sides and externally tangent to $\omega$ is another circle. The radius of each of the smaller $4$ circles can be written as $(a -\sqrt{b})$ where $a$ and $b$ are positive integers. Find $a +b$. [img]https://cdn.artofproblemsolving.com/attachments/d/a/c76a780ac857f745067a8d6c4433efdace2dbb.png[/img] [b]p17.[/b] In the nonexistent land of Lexingtopia, there are $10$ days in the year, and the Lexingtopian Math Society has $5$ members. The probability that no two of the LexingtopianMath Society’s members share the same birthday can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p18.[/b] Let $D(n)$ be the number of diagonals in a regular $n$-gon. Find $$\sum^{26}_{n=3} D(n).$$ [b]p19.[/b] Given a square $A_0B_0C_0D_0$ as shown below with side length $1$, for all nonnegative integers $n$, construct points $A_{n+1}$, $B_{n+1}$, $C_{n+1}$, and $D_{n+1}$ on $A_nB_n$, $B_nC_n$, $C_nD_n$, and $D_nA_n$, respectively, such that $$\frac{A_n A_{n+1}}{A_{n+1}B_n}=\frac{B_nB_{n+1}}{B_{n+1}C_n} =\frac{C_nC_{n+1}}{C_{n+1}D_n}=\frac{D_nD_{n+1}}{D_{n+1}A_n} =\frac34.$$ [img]https://cdn.artofproblemsolving.com/attachments/6/a/56a435787db0efba7ab38e8401cf7b06cd059a.png[/img] The sum of the series $$\sum^{\infty}_{i=0} [A_iB_iC_iD_i ] = [A_0B_0C_0D_0]+[A_1B_1C_1D_1]+[A_2B_2C_2D_2]...$$ where $[P]$ denotes the area of polygon $P$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p20.[/b] Let $m$ and $n$ be two real numbers such that $$\frac{2}{n}+m = 9$$ $$\frac{2}{m}+n = 1$$ Find the sum of all possible values of $m$ plus the sumof all possible values of $n$. [b]p21.[/b] Let $\sigma (x)$ denote the sum of the positive divisors of $x$. Find the smallest prime $p$ such that $$\sigma (p!) \ge 20 \cdot \sigma ([p -1]!).$$ [b]p22.[/b] Let $\vartriangle ABC$ be an isosceles triangle with $AB = AC$. Let $M$ be the midpoint of side $\overline{AB}$. Suppose there exists a point X on the circle passing through points $A$, $M$, and $C$ such that $BMCX$ is a parallelogram and $M$ and $X$ are on opposite sides of line $BC$. Let segments $\overline{AX}$ and $\overline{BC}$ intersect at a point $Y$ . Given that $BY = 8$, find $AY^2$. [b]p23.[/b] Kevin chooses $2$ integers between $1$ and $100$, inclusive. Every minute, Corey can choose a set of numbers and Kevin will tell him how many of the $2$ chosen integers are in the set. How many minutes does Corey need until he is certain of Kevin’s $2$ chosen numbers? [b]p24.[/b] Evaluate $$1^{-1} \cdot 2^{-1} +2^{-1} \cdot 3^{-1} +3^{-1} \cdot 4^{-1} +...+(2015)^{-1} \cdot (2016)^{-1} \,\,\, (mod \,\,\,2017).$$ [b]p25.[/b] In scalene $\vartriangle ABC$, construct point $D$ on the opposite side of $AC$ as $B$ such that $\angle ABD = \angle DBC = \angle BC A$ and $AD =DC$. Let $I$ be the incenter of $\vartriangle ABC$. Given that $BC = 64$ and $AD = 225$, find$ BI$ . [img]https://cdn.artofproblemsolving.com/attachments/b/1/5852dd3eaace79c9da0fd518cfdcd5dc13aecf.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many positive integers $n$ are there such that there are $20$ positive integers that are less than $n$ and relatively prime with $n$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \text{None}$

We have a positive integer $ n$ such that $ n \neq 3k$. Prove that there exists a positive integer $ m$ such that $ \forall_{k\in N \ k\geq m} \ k$ can be represented as a sum of digits of some multiplication of $ n$.

Let $n$'s positive divisors sum is $T(n)$. For all $n \geq 3$'s prove that, $$(T(n))^3<n^4$$

Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

Let $n$ be a given natural number. Determine the number of all orderer triples $(x, y, z)$ of non-negative integers $x, y, z$ that satisfy the equation $$x + 2y + 5z=10n.$$

Solve in integers the following equation: \[y=2x^2+5xy+3y^2\]

We call a natural number $n$ good if each of the numbers $n$, $ n+1$, $n+2$ and $n+3$ are divided by the sum of their digits. (For example, $n = 60398$ is good.) Does the penultimate digit of a good number ending in eight have to be nine?

Let $ a, b, c, m, n$ be positive integers. Consider the trinomial $ f (x) = ax^{2}+bx+c$. Show that there exist $ n$ consecutive natural numbers $ a_{1}, a_{2}, . . . , a_{n}$ such that each of the numbers $ f (a_{1}), f (a_{2}), . . . , f (a_{n})$ has at least $ m$ different prime factors.

Find all the integers $x,y$ satisfying equation $x^2+x=y^4+y^3+y^2+y$.