Let $m$ be a positive integer. Prove that there are integers $a, b, k$, such that both $a$ and $b$ are odd, $k\geq0$ and \[2m=a^{19}+b^{99}+k\cdot2^{1999}\]

Find the highest power of 2 that is a factor of the number $$ L_n = (n+1)(n+2)... 2n,$$ where $n$is a natural number.

Given a natural number $a_0$, we construct the sequence $\{a_n\}$ as follows $a_{n+1} = a^2_n-5$ if $a_n$ is odd, and $\frac{a_n}{2}$ if $a_n$ is even. Prove that for any odd $a_0 > 5$ in the sequence $\{a_n\}$ arbitrarily large numbers will occur.

A palindrome is a number that remains the same when its digits are reversed. Let $n$ be a product of distinct primes not divisible by $10$. Prove that infinitely many multiples of $n$ are palindromes.

Determine all pairs $(a, b)$ of integers having the following property: there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.

Prove that if the odd prime $p$ divides $a^{b}-1$, where $a$ and $b$ are positive integers, then $p$ appears to the same power in the prime factorization of $b(a^{d}-1)$, where $d=\gcd(b,p-1)$.

The natural numbers $a$ and $b$ are such that $ \frac{a+1}{b}+ \frac{b+1}{a}$ is an integer. Show that the greatest common divisor of a and b is not greater than $\sqrt{a+b}$.

Find all pairs of non-negative integers $(x,y)$ such that \[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\]

Let $v(n)$ be the order of $2$ in $n!$. Prove that for any positive integers $a$ and $m$ there exists $n$ ($n>1$) such that $v(n) \equiv a (\mod m)$. I have a book with Mongolian problems from this year, and this problem appeared in it. Perhaps I am terribly misinterpreting this problem, but it seems like it is wrong. Any ideas?

Find remainder when dividing upon $2012$ number $$A=1\cdot2+2\cdot3+3\cdot4+...+2009\cdot2010+2010\cdot2011$$

Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$

Define a $\textit{tasty residue}$ of $n$ to be an integer $1<a<n$ such that there exists an integer $m>1$ satisfying \[a^m\equiv a\pmod n.\] Find the number of tasty residues of $2016$.

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.

[b]p1.[/b] Susan plays a game in which she rolls two fair standard six-sided dice with sides labeled one through six. She wins if the number on one of the dice is three times the number on the other die. If Susan plays this game three times, compute the probability that she wins at least once. [b]p2.[/b] In triangles $\vartriangle ABC$ and $\vartriangle DEF$, $DE = 4AB$, $EF = 4BC$, and $FD = 4CA$. The area of $\vartriangle DEF$ is $360$ units more than the area of $\vartriangle ABC$. Compute the area of $\vartriangle ABC$. [b]p3.[/b] Andy has $2010$ square tiles, each of which has a side length of one unit. He plans to arrange the tiles in an $m\times n$ rectangle, where $mn = 2010$. Compute the sum of the perimeters of all of the different possible rectangles he can make. Two rectangles are considered to be the same if one can be rotated to become the other, so, for instance, a $1\times 2010$ rectangle is considered to be the same as a $2010\times 1$ rectangle. [b]p4.[/b] Let $$S = \log_2 9 \log_3 16 \log_4 25 ... \log_{999} 1000000.$$ Compute the greatest integer less than or equal to $\log_2 S$. [b]p5.[/b] Let $A$ and $B$ be fixed points in the plane with distance $AB = 1$. An ant walks on a straight line from point $A$ to some point $C$ in the plane and notices that the distance from itself to B always decreases at any time during this walk. Compute the area of the region in the plane containing all points where point $C$ could possibly be located. [b]p6.[/b] Lisette notices that $2^{10} = 1024$ and $2^{20} = 1 048 576$. Based on these facts, she claims that every number of the form $2^{10k}$ begins with the digit $1$, where k is a positive integer. Compute the smallest $k$ such that Lisette's claim is false. You may or may not find it helpful to know that $ln 2 \approx 0.69314718$, $ln 5 \approx 1.60943791$, and $log_{10} 2 \approx 0:30103000$. [b]p7.[/b] Let $S$ be the set of all positive integers relatively prime to $6$. Find the value of $\sum_{k\in S}\frac{1}{2^k}$ . [b]p8.[/b] Euclid's algorithm is a way of computing the greatest common divisor of two positive integers $a$ and $b$ with $a > b$. The algorithm works by writing a sequence of pairs of integers as follows. 1. Write down $(a, b)$. 2. Look at the last pair of integers you wrote down, and call it $(c, d)$. $\bullet$ If $d \ne 0$, let r be the remainder when c is divided by d. Write down $(d, r)$. $\bullet$ If $d = 0$, then write down c. Once this happens, you're done, and the number you just wrote down is the greatest common divisor of a and b. 3. Repeat step 2 until you're done. For example, with $a = 7$ and $b = 4$, Euclid's algorithm computes the greatest common divisor in $4$ steps: $$(7, 4) \to (4, 3) \to (3, 1) \to (1, 0) \to 1$$ For $a > b > 0,$ compute the least value of a such that Euclid's algorithm takes $10$ steps to compute the greatest common divisor of $a$ and $b$. [b]p9.[/b] Let $ABCD$ be a square of unit side length. Inscribe a circle $C_0$ tangent to all of the sides of the square. For each positive integer $n$, draw a circle Cn that is externally tangent to $C_{n-1}$ and also tangent to sides $AB$ and $AD$. Suppose $r_i$ is the radius of circle $C_i$ for every nonnegative integer $i$. Compute $\sqrt[200]{r_0/r_{100}}$. [b]p10.[/b] Rachel and Mike are playing a game. They start at $0$ on the number line. At each positive integer on the number line, there is a carrot. At the beginning of the game, Mike picks a positive integer $n$ other than $30$. Every minute, Rachel moves to the next multiple of $30$ on the number line that has a carrot on it and eats that carrot. At the same time, every minute, Mike moves to the next multiple of $n$ on the number line that has a carrot on it and eats that carrot. Mike wants to pick an $n$ such that, as the game goes on, he is always within $1000$ units of Rachel. Compute the average (arithmetic mean) of all such $n$. [b]p11.[/b] Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$. He knows the die is weighted so that one face comes up with probability $1/2$ and the other five faces have equal probability of coming up. He unfortunately does not know which side is weighted, but he knows each face is equally likely to be the weighted one. He rolls the die 5 times and gets a $1$, $2$, $3$, $4$ and $5$ in some unspecified order. Compute the probability that his next roll is a $6$. [b]p12.[/b] Let $F_0 = 1$, $F_1 = 1$ and $F_k = F_{k-1} + F_{k-2}$. Let $P(x) =\sum^{99}_{k=0} x^{F_k}$ . The remainder when $P(x)$ is divided by $x^3 - 1$ can be expressed as $ax^2 + bx + c$. Find $2a + b$. [b]p13.[/b] Let $\theta \ne 0$ be the smallest acute angle for which $\sin \theta$, $\sin (2\theta)$, $\sin (3\theta)$, when sorted in increasing order, form an arithmetic progression. Compute $\cos (\theta/2)$. [b]p14.[/b] A $4$-dimensional hypercube of edge length 1 is constructed in $4$-space with its edges parallel to the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given by $(a, b, c, d)$, where each of $a$, $b$, $c$, and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane given by $x + y + z + w = 2$ intersects the hypercube at $6$ of its vertices. Compute the $3$-dimensional volume of the solid formed by the intersection. [b]p15.[/b] A student puts $2010$ red balls and $1957$ blue balls into a box. Weiqing draws randomly from the box one ball at a time without replacement. She wins if, at anytime, the total number of blue balls drawn is more than the total number of red balls drawn. Assuming Weiqing keeps drawing balls until she either wins or runs out, ompute the probability that she eventually wins. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve in N $a^2 = 2^b3^c + 1$.

Ankit wants to create a pseudo-random number generator using modular arithmetic. To do so he starts with a seed $x_0$ and a function $f(x) = 2x + 25$ (mod $31$). To compute the $k$-th pseudo random number, he calls $g(k)$ dened as follows: $$g(k) = \begin{cases} x_0 \,\,\, \text{if} \,\,\, k = 0 \\ f(g(k- 1)) \,\,\, \text{if} \,\,\, k > 0 \end{cases}$$ If $x_0$ is $2017$, compute $\sum^{2017}_{j=0} g(j)$ (mod $31$).

$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$

When a number is divided by $2$ it has quotient $x$ and remainder $1$. Whereas, when the same number is divided by $3$ it has quotient $y$ and remainder $2$. What is the remainder when $x+y$ is divided by $5$?

Let $n$ be an integer greater than 1. If all digits of $97n$ are odd, find the smallest possible value of $n$.

Consider real numbers $A$, $B$, \dots, $Z$ such that \[ EVIL = \frac{5}{31}, \; LOVE = \frac{6}{29}, \text{ and } IMO = \frac{7}{3}. \] If $OMO = \tfrac mn$ for relatively prime positive integers $m$ and $n$, find the value of $m+n$. [i]Proposed by Evan Chen[/i]

Let $p$ and $q$ be different prime numbers. Solve the following system in integers: \[\frac{z+ p}x+\frac{z-p}y= q,\\ \frac{z+ p}y -\frac{z-p}x= q.\]

Find all positive integer solutions $(a, b, c)$ to the function $a^{2} + b^{2} + c^{2} = 2005$, where $a \leq b \leq c$.

Let $N$ be the number of ordered pairs $(x,y)$ of integers such that $x^2+xy+y^2 \le 2007$. Remember, integers may be positive, negative, or zero! (a) Prove that $N$ is odd. (b) Prove that $N$ is not divisible by $3$.

$(MON 4)$ Let $p$ and $q$ be two prime numbers greater than $3.$ Prove that if their difference is $2^n$, then for any two integers $m$ and $n,$ the number $S = p^{2m+1} + q^{2m+1}$ is divisible by $3.$