Let $m\ge 3$ and $n$ be positive integers such that $n>m(m-2)$. Find the largest positive integer $d$ such that $d\mid n!$ and $k\nmid d$ for all $k\in\{m,m+1,\ldots,n\}$.

[b]p1.[/b] The least prime factor of $a$ is $3$, the least prime factor of $b$ is $7$. Find the least prime factor of $a + b$. [b]p2.[/b] In a Cartesian coordinate system, the two tangent lines from $P = (39, 52)$ meet the circle defined by $x^2 + y^2 = 625$ at points $Q$ and $R$. Find the length $QR$. [b]p3.[/b] For a positive integer $n$, there is a sequence $(a_0, a_1, a_2,..., a_n)$ of real values such that $a_0 = 11$ and $(a_k + a_{k+1}) (a_k - a_{k+1}) = 5$ for every $k$ with $0 \le k \le n-1$. Find the maximum possible value of $n$. (Be careful that your answer isn’t off by one!) [b]p4.[/b] Persons $A$ and $B$ stand at point $P$ on line $\ell$. Point $Q$ lies at a distance of $10$ from point $P$ in the direction perpendicular to $\ell$. Both persons intially face towards $Q$. Person $A$ walks forward and to the left at an angle of $25^o$ with $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the right, and continues walking. Person $B$ walks forward and to the right at an angle of $55^o$ with line $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the left, and continues walking. Their paths cross at point $R$. Find the distance $PR$. [b]p5.[/b] Compute $$\frac{lcm (1,2, 3,..., 200)}{lcm (102, 103, 104, ..., 200)}.$$ [b]p6.[/b] There is a unique real value $A$ such that for all $x$ with $1 < x < 3$ and $x \ne 2$, $$\left| \frac{A}{x^2-x - 2} +\frac{1}{x^2 - 6x + 8} \right|< 1999.$$ Compute $A$. [b]p7.[/b] Nine poles of height $1, 2,..., 9$ are placed in a line in random order. A pole is called [i]dominant [/i] if it is taller than the pole immediately to the left of it, or if it is the pole farthest to the left. Count the number of possible orderings in which there are exactly $2$ dominant poles. [b]p8.[/b] $\tan (11x) = \tan (34^o)$ and $\tan (19x) = \tan (21^o)$. Compute $\tan (5x)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Positive integer $n$ is such that number $n^2-9$ has exactly $6$ positive divisors. Prove that GCD $(n-3, n+3)=1$

Calculate the value of $\sqrt{\dfrac{11^4+100^4+111^4}{2}}$ and answer in the form of an integer.

[b]p1.[/b] There are $2024$ apples in a very large basket. First, Julie takes away half of the apples in the basket; then, Diane takes away $202$ apples from the remaining bunch. How many apples remain in the basket? [b]p2.[/b] The set of all permutations (different arrangements) of the letters in ”ABMC” are listed in alphabetical order. The first item on the list is numbered $1$, the second item is numbered $2$, and in general, the kth item on the list is numbered $k$. What number is given to ”ABMC”? [b]p3.[/b] Daniel has a water bottle that is three-quarters full. After drinking $3$ ounces of water, the water bottle is three-fifths full. The density of water is $1$ gram per milliliter, and there are around $28$ grams per ounce. How many milliliters of water could the bottle fit at full capacity? [b]p4.[/b] How many ways can four distinct $2$-by-$1$ rectangles fit on a $2$-by-$4$ board such that each rectangle is fully on the board? [b]p5.[/b] Iris and Ivy start reading a $240$ page textbook with $120$ left-hand pages and $120$ right-hand pages. Iris takes $4$ minutes to read each page, while Ivy takes $5$ minutes to read a left-hand page and $3$ minutes to read a right-hand page. Iris and Ivy move onto the next page only when both sisters have completed reading. If a sister finishes reading a page first, the other sister will start reading three times as fast until she completes the page. How many minutes after they start reading will both sisters finish the textbook? [b]p6.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $24$. Then, let $M$ be the midpoint of $BC$. Define $P$ to be the set of all points $P$ such that $2PM = BC$. The minimum value of $AP$ can be expressed as $\sqrt{a}- b$, where $a$ and $b$ are positive integers. Find $a + b$. [b]p7.[/b] Jonathan has $10$ songs in his playlist: $4$ rap songs and $6$ pop songs. He will select three unique songs to listen to while he studies. Let $p$ be the probability that at least two songs are rap, and let $q$ be the probability that none of them are rap. Find $\frac{p}{q}$ . [b]p8.[/b] A number $K$ is called $6,8$-similar if $K$ written in base $6$ and $K$ written in base $8$ have the same number of digits. Find the number of $6,8$-similar values between $1$ and $1000$, inclusive. [b]p9.[/b] Quadrilateral $ABCD$ has $\angle ABC = 90^o$, $\angle ADC = 120^o$, $AB = 5$, $BC = 18$, and $CD = 3$. Find $AD^2$. [b]p10.[/b] Bob, Eric, and Raymond are playing a game. Each player rolls a fair $6$-sided die, and whoever has the highest roll wins. If players are tied for the highest roll, the ones that are tied reroll until one wins. At the start, Bob rolls a $4$. The probability that Eric wins the game can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [b]p11.[/b] Define the following infinite sequence $s$: $$s = \left\{\frac92,\frac{99}{2^2},\frac{999}{2^3} , ... , \overbrace{\frac{999...999}{2^k}}^{k\,\,nines}, ...\right\}$$ The sum of the first $2024$ terms in $s$, denoted $S$, can be expressed as $$S =\frac{5^a - b}{4}+\frac{1}{2^c},$$ where $a, b$, and $c$ are positive integers. Find $a + b + c$. [b]p12.[/b] Andy is adding numbers in base $5$. However, he accidentally forgets to write the units digit of each number. If he writes all the consecutive integers starting at $0$ and ending at $50$ (base $10$) and adds them together, what is the difference between Andy’s sum and the correct sum? (Express your answer in base-$10$.) [b]p13.[/b] Let $n$ be the positive real number such that the system of equations $$y =\frac{1}{\sqrt{2024 - x^2}}$$ $$y =\sqrt{x^2 - n}$$ has exactly two real solutions for $(x, y)$: $(a, b)$ and $(-a, b)$. Then, $|a|$ can be expressed as $j\sqrt{k}$, where $j$ and $k$ are integers such that $k$ is not divisible by any perfect square other than $1$. Find $j · k$. [b]p14.[/b] Nakio is playing a game with three fair $4$-sided dice. But being the cheater he is, he has secretly replaced one of the three die with his own $4$-sided die, such that there is a $1/2$ chance of rolling a $4$, and a $1/6$ chance to roll each number from $1$ to $3$. To play, a random die is chosen with equal probability and rolled. If Nakio guesses the number that is on the die, he wins. Unfortunately for him, Nakio’s friends have an anti-cheating mechanism in place: when the die is picked, they will roll it three times. If each roll lands on the same number, that die is thrown out and one of the two unused dice is chosen instead with equal probability. If Nakio always guesses $4$, the probability that he wins the game can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime. Find $m + n$. [b]p15.[/b] A particle starts in the center of a $2$m-by-$2$m square. It moves in a random direction such that the angle between its direction and a side of the square is a multiple of $30^o$. It travels in that direction at $1$ m/s, bouncing off of the walls of the square. After a minute, the position of the particle is recorded. The expected distance from this point to the start point can be written as $$\frac{1}{a}\left(b - c\sqrt{d}\right),$$ where $a$ and $b$ are relatively prime, and d is not divisible by any perfect square. Find $a + b + c + d$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f: \mathbb Z^+ \to \mathbb Z$ be a function such that [b]a)[/b] $f(p)=1$ for every prime $p$. [b]b)[/b] $f(xy)=xf(y)+yf(x)$ for every pair of positive integers $x,y$ Find the least number $n \ge 2021$ such that $f(n)=n$

Find all pairs $(x, y)$ of positive rational numbers such that $x^{y}=y^{x}$.

Prove that all positive rational numbers can be written as a fraction, which numerator and denominator are products of factorials of not necessarily different prime numbers. For example $\frac{10}{9}=\frac{2!5!}{3!3!3!}$.

Show that $n^n + (n + 1)^{n + 1}$ is composite for infinitely many positive integers $n$.

[b]p1.[/b] It takes $12$ months for Santa Claus to pack gifts. It would take $20$ months for his apprentice to do the job. If they work together, how long will it take for them to pack the gifts? [b]p2.[/b] All passengers on a bus sit in pairs. Exactly $2/5$ of all men sit with women, exactly $2/3$ of all women sit with men. What part of passengers are men? [b]p3.[/b] There are $100$ colored balls in a box. Every $10$-tuple of balls contains at least two balls of the same color. Show that there are at least $12$ balls of the same color in the box. [b]p4.[/b] There are $81$ wheels in storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that one can determine which wheel is incorrectly marked with four measurements. [b]p5.[/b] Remove from the figure below the specified number of matches so that there are exactly $5$ squares of equal size left: (a) $8$ matches (b) $4$ matches [img]https://cdn.artofproblemsolving.com/attachments/4/b/0c5a65f2d9b72fbea50df12e328c024a0c7884.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There exists a block of 1000 consecutive positive integers containing no prime numbers, namely, $1001!+2,1001!+3,...,1001!+1001$. Does there exist a block of 1000 consecutive positive intgers containing exactly five prime numbers?

Let $n$ be a positive integer greater than $1$, and let $p$ be a prime such that $n$ divides $p-1$ and $p$ divides $n^3-1$. Prove that $4p-3$ is a square.

Let be $n$ a positive integer. Denote all its (positive) divisors as $1=d_1<d_2<\cdots<d_{k-1}<d_k=n$. Find all values of $n$ satisfying $d_5-d_3=50$ and $11d_5+8d_7=3n$. (Day 1, 1st problem author: Matúš Harminc)

Suppose $a_1,a_2, \dots$ is an infinite strictly increasing sequence of positive integers and $p_1, p_2, \dots$ is a sequence of distinct primes such that $p_n \mid a_n$ for all $n \ge 1$. It turned out that $a_n-a_k=p_n-p_k$ for all $n,k \ge 1$. Prove that the sequence $(a_n)_n$ consists only of prime numbers.

A non-negative integer $n$ is said to be [i]squaredigital[/i] if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital.

For positive integers $a, b, c$ (not necessarily distinct), suppose that $a+bc, b+ac, c+ab$ are all perfect squares. Show that $$a^2(b+c)+b^2(a+c)+c^2(a+b)+2abc$$ can be written as sum of two squares.

Determine all sequences $ a_1,a_2,a_3,...$ of $ 1$ and $ \minus{}1$ such that $ a_{mn}\equal{}a_ma_n$ for all $ m,n$ and among any three successive terms $ a_n,a_{n\plus{}1},a_{n\plus{}2}$ both $ 1$ and $ \minus{}1$ occur.

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide] [b]R1.[/b] What is $11^2 - 9^2$? [b]R2.[/b] Write $\frac{9}{15}$ as a decimal. [b]R3.[/b] A $90^o$ sector of a circle is shaded, as shown below. What percent of the circle is shaded? [b]R4.[/b] A fair coin is flipped twice. What is the probability that the results of the two flips are different? [b]R5.[/b] Wayne Dodson has $55$ pounds of tungsten. If each ounce of tungsten is worth $75$ cents, and there are $16$ ounces in a pound, how much money, in dollars, is Wayne Dodson’s tungsten worth? [b]R6.[/b] Tenley Towne has a collection of $28$ sticks. With these $28$ sticks he can build a tower that has $1$ stick in the top row, $2$ in the next row, and so on. Let $n$ be the largest number of rows that Tenley Towne’s tower can have. What is n? [b]R7.[/b] What is the sum of the four smallest primes? [b]R8 / P1.[/b] Let $ABC$ be an isosceles triangle such that $\angle B = 42^o$. What is the sum of all possible degree measures of angle $A$? [b]R9.[/b] Consider a line passing through $(0, 0)$ and $(4, 8)$. This line passes through the point $(2, a)$. What is the value of $a$? [b]R10 / P2.[/b] Brian and Stan are playing a game. In this game, Brian rolls a fair six-sided die, while Stan rolls a fair four-sided die. Neither person shows the other what number they rolled. Brian tells Stan, “The number I rolled is guaranteed to be higher than the number you rolled.” Stan now has to guess Brian’s number. If Stan plays optimally, what is the probability that Stan correctly guesses the number that Brian rolled? [b]R11.[/b] Guang chooses $4$ distinct integers between $0$ and $9$, inclusive. How many ways can he choose the integers such that every pair of chosen integers sums up to an even number? [b]R12 / P4.[/b] David is trying to write a problem for MBMT. He assigns degree measures to every interior angle in a convex $n$-gon, and it so happens that every angle he assigned is less than $144$ degrees. He tells Pratik the value of $n$ and the degree measures in the $n$-gon, and to David’s dismay, Pratik claims that such an $n$-gon does not exist. What is the smallest value of $n \ge 3$ such that Pratik’s claim is necessarily true? [b]R13 / P3.[/b] Consider a triangle $ABC$ with side lengths of $5$, $5$, and $2\sqrt5$. There exists a triangle with side lengths of $5, 5$, and $x$ ($x \ne 2\sqrt5$) which has the same area as $ABC$. What is the value of $x$? [b]R14 / P5.[/b] A mother has $11$ identical apples and $9$ identical bananas to distribute among her $3$ kids. In how many ways can the fruits be allocated so that each child gets at least one apple and one banana? [b]R15 / P7.[/b] Find the sum of the five smallest positive integers that cannot be represented as the sum of two not necessarily distinct primes. [b]P6.[/b] Srinivasa Ramanujan has the polynomial $P(x) = x^5 - 3x^4 - 5x^3 + 15x^2 + 4x - 12$. His friend Hardy tells him that $3$ is one of the roots of $P(x)$. What is the sum of the other roots of $P(x)$? [b]P8.[/b] $ABC$ is an equilateral triangle with side length $10$. Let $P$ be a point which lies on ray $\overrightarrow{BC}$ such that $PB = 20$. Compute the ratio $\frac{PA}{PC}$. [b]P9.[/b] Let $ABC$ be a triangle such that $AB = 10$, $BC = 14$, and $AC = 6$. The median $CD$ and angle bisector $CE$ are both drawn to side $AB$. What is the ratio of the area of triangle $CDE$ to the area of triangle $ABC$? [b]P10.[/b] Find all integer values of $x$ between $0$ and $2017$ inclusive, which satisfy $$2016x^{2017} + 990x^{2016} + 2x + 17 \equiv 0 \,\,\, (mod \,\,\, 2017).$$ [b]P11.[/b] Let $x^2 + ax + b$ be a quadratic polynomial with positive integer roots such that $a^2 - 2b = 97$. Compute $a + b$. [b]P12.[/b] Let $S$ be the set $\{2, 3, ... , 14\}$. We assign a distinct number from $S$ to each side of a six-sided die. We say a numbering is predictable if prime numbers are always opposite prime numbers and composite numbers are always opposite composite numbers. How many predictable numberings are there? (Rotations of a die are not distinct) [b]P13.[/b] In triangle $ABC$, $AB = 10$, $BC = 21$, and $AC = 17$. $D$ is the foot of the altitude from $A$ to $BC$, $E$ is the foot of the altitude from $D$ to $AB$, and $F$ is the foot of the altitude from $D$ to $AC$. Find the area of the smallest circle that contains the quadrilateral $AEDF$. [b]P14.[/b] What is the greatest distance between any two points on the graph of $3x^2 + 4y^2 + z^2 - 12x + 8y + 6z = -11$? [b]P15.[/b] For a positive integer $n$, $\tau (n)$ is defined to be the number of positive divisors of $n$. Given this information, find the largest positive integer $n$ less than $1000$ such that $$\sum_{d|n} \tau (d) = 108.$$ In other words, we take the sum of $\tau (d)$ for every positive divisor $d$ of $n$, which has to be $108$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $v_0 = 0, v_1 = 1$ and $v_{n+1} = 8 \cdot v_n - v_{n-1},$ $n = 1,2, ...$. Prove that in the sequence $\{v_n\}$ there aren't terms of the form $3^{\alpha} \cdot 5^{\beta}$ with $\alpha, \beta \in \mathbb{N}.$

We have an a sequence such that $a_n = 2 \cdot 10^{n + 1} + 19$. Determine all the primes $p$, with $p \le 19$, for which there exists some $n \ge 1$ such that $p$ divides $a_n$.

The prime numbers $p$ and $q$ and the integer $a$ are chosen such that $p> 2$ and $a \not\equiv 1$ (mod $q$), but $a^p \equiv 1$ (mod $q$). Prove that $(1 + a^1)(1 + a^2)...(1 + a^{p - 1})\equiv 1$ (mod $q$) .

Let $ x,y,z$ be positive integers. If $ 7$ divides $ (x\plus{}6y)(2x\plus{}5y)(3x\plus{}4y)$ than prove that $ 343$ also divides it.

Determine all pairs $(a,b)$ of positive integers with the following property: all of the terms of the sequence $(a^n+b^n+1)_{n\geqslant 1}$ have a greatest common divisor $d>1.$

A natural number $n$ is said to be [i]abundant [/i] if the sum of its divisors is greater than $2n$. For example, $18$ is abundant because the sum of its divisors, $1 + 2 + 3 + 6 + 9 + 18$, is greater than $36$. Write every even number greater than $46$ as a sum of two numbers abundant.

Define the sequence of positive prime numbers. $p_1,p_2,p_3,...$. Let set $A$ be the infinite set of positive integers whose prime divisor does not exceed $p_n$. How many at least members must be selected from the set $A$ , such that we ensures that there are $2$ numbers whose products are perfect squares? [i](PP-nine)[/i]