[b]p25.[/b] Compute: $$\frac{ \sum^{\infty}_{i=0}\frac{(2\pi)^{4i+1}}{(4i+1)!}}{\sum^{\infty}_{i=0}\frac{(2\pi)^{4i+1}}{(4i+3)!}}$$ [b]p26.[/b] Suppose points $A, B, C, D$ lie on a circle $\omega$ with radius $4$ such that $ABCD$ is a quadrilateral with $AB = 6$, $AC = 8$, $AD = 7$. Let $E$ and $F$ be points on $\omega$ such that $AE$ and $AF$ are respectively the angle bisectors of $\angle BAC$ and $\angle DAC$. Compute the area of quadrilateral $AECF$. [b]p27.[/b] Let $P(x) = x^2 - ax + 8$ with a a positive integer, and suppose that $P$ has two distinct real roots $r$ and $s$. Points $(r, 0)$, $(0, s)$, and $(t, t)$ for some positive integer t are selected on the coordinate plane to form a triangle with an area of $2021$. Determine the minimum possible value of $a + t$. [b]p28.[/b] A quartic $p(x)$ has a double root at $x = -\frac{21}{4}$ , and $p(x) - 1344x$ has two double roots each $\frac14$ less than an integer. What are these two double roots? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $m \in {\mathbb R}$ and $$x^2+(m-4)x+(m^2-3m+3)=0$$ equations roots are $x_1$ and $x_2$ and $x_1^2+x_2^2=6$. Find all $m$ values.

Find the least natural number $n$, which has at least 6 different divisors $1=d_1<d_2<d_3<d_4<d_5<d_6<...$, for which $d_3+d_4=d_5+6$ and $d_4+d_5=d_6+7$.

Find all triples $ \left(p,x,y\right)$ such that $ p^x\equal{}y^4\plus{}4$, where $ p$ is a prime and $ x$ and $ y$ are natural numbers.

Determine all positive integers $a,b,c$ such that $ab + ac + bc$ is a prime number and $$\frac{a+b}{a+c}=\frac{b+c}{b+a}.$$

Let $L$ be the number formed by $2022$ digits equal to $1$, that is, $L=1111\dots 111$. Compute the sum of the digits of the number $9L^2+2L$.

The positive integers $a,b,c$ are less than $99$ and satisfy $a^2+b^2=c^2+99^2$. . Find the minimum and maximum value of $a+b+c$.

[u]Set 1 [/u] [b]1.1[/b] Compute the number of real numbers x such that the sequence $x$, $x^2$, $x^3$,$ x^4$, $x^5$, $...$ eventually repeats. (To be clear, we say a sequence “eventually repeats” if there is some block of consecutive digits that repeats past some point—for instance, the sequence $1$, $2$, $3$, $4$, $5$, $6$, $5$, $6$, $5$, $6$, $...$ is eventually repeating with repeating block $5$, $6$.) [b]1.2[/b] Let $T$ be the answer to the previous problem. Nicole has a broken calculator which, when told to multiply $a$ by $b$, starts by multiplying $a$ by $b$, but then multiplies that product by b again, and then adds $b$ to the result. Nicole inputs the computation “$k \times k$” into the calculator for some real number $k$ and gets an answer of $10T$. If she instead used a working calculator, what answer should she have gotten? [b]1.3[/b] Let $T$ be the answer to the previous problem. Find the positive difference between the largest and smallest perfect squares that can be written as $x^2 + y^2$ for integers $x, y$ satisfying $\sqrt{T} \le x \le T$ and $\sqrt{T} \le y \le T$. PS. You should use hide for answers.

Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$

[b]p1.[/b] How many two-digit primes have a units digit of $3$? [b]p2.[/b] How many ways can you arrange the letters $A$, $R$, and $T$ such that it makes a three letter combination? Each letter is used once. [b]p3.[/b] Hanna and Kevin are running a $100$ meter race. If Hanna takes $20$ seconds to finish the race and Kevin runs $15$ meters per second faster than Hanna, by how many seconds does Kevin finish before Hanna? [b]p4.[/b] It takes an ant $3$ minutes to travel a $120^o$ arc of a circle with radius $2$. How long (in minutes) would it take the ant to travel the entirety of a circle with radius $2022$? [b]p5.[/b] Let $\vartriangle ABC$ be a triangle with angle bisector $AD$. Given $AB = 4$, $AD = 2\sqrt2$, $AC = 4$, find the area of $\vartriangle ABC$. [b]p6.[/b] What is the coefficient of $x^5y^2$ in the expansion of $(x + 2y + 4)^8$? [b]p7.[/b] Find the least positive integer $x$ such that $\sqrt{20475x}$ is an integer. [b]p8.[/b] What is the value of $k^2$ if $\frac{x^5 + 3x^4 + 10x^2 + 8x + k}{x^3 + 2x + 4}$ has a remainder of $2$? [b]p9.[/b] Let $ABCD$ be a square with side length $4$. Let $M$, $N$, and $P$ be the midpoints of $\overline{AB}$, $\overline{BC}$ and $\overline{CD}$, respectively. The area of the intersection between $\vartriangle DMN$ and $\vartriangle ANP$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p10.[/b] Let $x$ be all the powers of two from $2^1$ to $2^{2023}$ concatenated, or attached, end to end ($x = 2481632...$). Let y be the product of all the powers of two from $2^1$ to $2^{2023}$ ($y = 2 \cdot 4 \cdot 8 \cdot 16 \cdot 32... $ ). Let 2a be the largest power of two that divides $x$ and $2^b$ be the largest power of two that divides $y$. Compute $\frac{b}{a}$ . [b]p11.[/b] Larry is making a s’more. He has to have one graham cracker on the top and one on the bottom, with eight layers in between. Each layer can made out of chocolate, more graham crackers, or marshmallows. If graham crackers cannot be placed next to each other, how many ways can he make this s’more? [b]p12.[/b] Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, $AC = 5$. Circle $O$ is centered at $B$ and has radius $\frac{8\sqrt{3}}{5}$ . The area inside the triangle but not inside the circle can be written as $\frac{a-b\sqrt{c}-d\pi}{e}$ , where $gcd(a, b, d, e) =1$ and $c$ is squarefree. Find $a + b + c + d + e$. [b]p13.[/b] Let $F(x)$ be a quadratic polynomial. Given that $F(x^2 - x) = F (2F(x) - 1)$ for all $x$, the sum of all possible values of $F(2022)$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p14.[/b] Find the sum of all positive integers $n$ such that $6\phi (n) = \phi (5n)+8$, where $\phi$ is Euler’s totient function. Note: Euler’s totient $(\phi)$ is a function where $\phi (n)$ is the number of positive integers less than and relatively prime to $n$. For example, $\phi (4) = 2$ since only $1$, $3$ are the numbers less than and relatively prime to $4$. [b]p15.[/b] Three numbers $x$, $y$, and $z$ are chosen at random from the interval $[0, 1]$. The probability that there exists an obtuse triangle with side lengths $x$, $y$, and $z$ can be written in the form $\frac{a\pi-b}{c}$ , where $a$, $b$, $c$ are positive integers with $gcd(a, b, c) = 1$. Find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

a) Let $a,n,m$ be natural numbers, $a > 1$. Prove that if $(a^m + 1)$ is divisible by $(a^n + 1)$ than $m$ is divisible by $n$. b) Let $a,b,n,m$ be natural numbers, $a>1, a$ and $b$ are relatively prime. Prove that if $(a^m+b^m)$ is divisible by $(a^n+b^n)$ than $m$ is divisible by $n$.

An infinite sequence of integers, $a_0,a_1,a_2,\dots,$ with $a_0>0$, has the property that for $n\ge 0$, $a_{n+1}=a_n-b_n$, where $b_n$ is the number having the same sign as $a_n$, but having the digits written in the reverse order. For example if $a_0=1210,a_1=1089$ and $a_2=-8712$, etc. Find the smallest value of $a_0$ so that $a_n\neq 0$ for all $n\ge 1$.

a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers. b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?

Find the smallest natural $ k $ such that among any $ k $ distinct and pairwise coprime naturals smaller than $ 2018, $ a prime can be found. [i]Vlad Robu[/i]

For an integer $n \geq 3$ we define the sequence $\alpha_1, \alpha_2, \ldots, \alpha_k$ as the sequence of exponents in the prime factorization of $n! = p_1^{\alpha_1}p_2^{\alpha_2} \ldots p_k^{\alpha_k}$, where $p_1 < p_2 < \ldots < p_k$ are primes. Determine all integers $n \geq 3$ for which $\alpha_1, \alpha_2, \ldots, \alpha_k$ is a geometric progression.

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

Let $p>5$ be a prime number. Prove that there exist $m,n\in \mathbb{N}$ for which $m+n<p$ and $2^m 3^n-1$ is a multiple of $p$.

How many numbers are there that appear both in the arithmetic sequence $10, 16, 22, 28, ... 1000$ and the arithmetic sequence $10, 21, 32, 43, ..., 1000?$

The set $A{}$ of positive integers satisfies the following conditions: [list=1] [*]If a positive integer $n{}$ belongs to $A{}$, then $2n$ also belongs to $A{}$; [*]For any positive integer $n{}$ there exists an element of $A{}$ divisible by $n{}$; [*]There exist finite subsets of $A{}$ with arbitrarily large sums of reciprocals of elements. [/list]Prove that for any positive rational number $r{}$ there exists a finite subset $B\subset A$ such that \[\sum_{x\in B}\frac{1}{x}=r.\]

Determine all quadruplets ($x, y, z, t$) of positive integers, such that $12^x + 13^y - 14^z = 2013^t$.

Find the smallest three-digit number such that the following holds: If the order of digits of this number is reversed and the number obtained by this is added to the original number, the resulting number consists of only odd digits.

Compute $$ 2022^{\left(2022^{\cdot ^ {\cdot ^{\cdot ^{\left(2022^{2022}\right)}}}}\right)} \pmod{111} $$ where there are $2022$ $2022$s. (Give the answer as an integer from $0$ to $110$). [i]Proposed by David Tang[/i]

[b](1)[/b] Find the number of positive integer solutions $(m,n,r)$ of the indeterminate equation $mn+nr+mr=2(m+n+r)$. [b](2)[/b] Given an integer $k (k>1)$, prove that indeterminate equation $mn+nr+mr=k(m+n+r)$ has at least $3k+1$ positive integer solutions $(m,n,r)$.

Determine the sixth number after the decimal point in the number $(\sqrt{1978} +\lfloor\sqrt{1978}\rfloor)^{20}$

In the beginning there are $100$ integers in a row on the blackboard. Kain and Abel then play the following game: A [i]move[/i] consists in Kain choosing a chain of consecutive numbers; the length of the chain can be any of the numbers $1,2,\dots,100$ and in particular it is allowed that Kain only chooses a single number. After Kain has chosen his chain of numbers, Abel has to decide whether he wants to add $1$ to each of the chosen numbers or instead subtract $1$ from of the numbers. After that the next move begins, and so on. If there are at least $98$ numbers on the blackboard that are divisible by $4$ after a move, then Kain has won. Prove that Kain can force a win in a finite number of moves.