Given positive integers $k$ and $m$, show that $m$ and $\binom{n}{k}$ are coprime for infinitely many integers $n\geq k$.

Let $F_k(a,b)=(a+b)^k-a^k-b^k$ and let $S={1,2,3,4,5,6,7,8,9,10}$. For how many ordered pairs $(a,b)$ with $a,b\in S$ and $a\leq b$ is $\frac{F_5(a,b)}{F_3(a,b)}$ an integer?

Find all finite sets of positive integers with at least two elements such that for any two numbers $ a$, $ b$ ($ a > b$) belonging to the set, the number $ \frac {b^2}{a \minus{} b}$ belongs to the set, too.

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

The sequences of natural numbers $p_n$ and $q_n$ are given such that $$p_1 = 1,\ q_1 = 1,\ p_{n + 1} = 2q_n^2-p_n^2,\ q_{n + 1} = 2q_n^2+p_n^2 $$ Prove that $p_n$ and $q_m$ are coprime for any m and n.

Find the number of ordered triples of positive integers $(a, b, c)$ such that \[6a + 10b + 15c = 3000.\]

Pedro and Cecilia play the following game: Pedro chooses a positive integer number $a$ and Cecilia wins if she finds a positive integrer number $b$, prime with $a$, such that, in the factorization of $a^3+b^3$ will appear three different prime numbers. Prove that Cecilia can always win.

Let $n$ be a positive integer. Determine integers, $n+1 \leq r \leq 3n+2$ such that for all integers $a_1,a_2,\dots,a_m,b_1,b_2,\dots,b_m$ satisfying the equations \[ a_1b_1^k+a_2b_2^k+\dots + a_mb_m^k=0 \] for every $1 \leq k \leq n$, the condition \[ r \mid a_1b_1^r+a_2b_2^r+\dots + a_mb_m^r=0 \] also holds.

Two distinct positive integers $a,b$ are written on the board. The smaller of them is erased and replaced with the number $\frac{ab}{|a-b|}$. This process is repeated as long as the two numbers are not equal. Prove that eventually the two numbers on the board will be equal.

A nonnegative integer $n$ is said to be $\textit{squarish}$ is it satisfies the following conditions: $\textbf{(i)}$ it contains no digits of $9$; $\textbf{(ii)}$ no matter how we increase exactly one of its digits with $1$, the outcome is a square. Find all squarish nonnegative integers. $\textit{(Vlad Robu)}$

Find all prime $p$, for each of which there are such natural $ x$ and $y$ such that $p^x = y^3 + 1$.

Find all pairs of positive integers $(a,b)$ such that $a-b$ is a prime number and $ab$ is a perfect square.

Let $n\geq 3$ be an odd integer. On a line, $n$ points are marked in such a way that the distance between any two of them is an integer. It turns out that each marked point has an even sum of distances to the remaining $n-1$ marked points. Prove that the distance between any two marked points is even.

The sequence $(a_k)$, $k> 0$ is Fibonacci, with $a_0 = a_1 = 1$. Calculate the value of $$\sum_{j = 0}^{\infty} \frac{a_j}{2^j}$$

Let $(G,\cdot)$ be a finite group with the identity element, $e$. The smallest positive integer $n$ with the property that $x^{n}= e$, for all $x \in G$, is called the [i]exponent[/i] of $G$. (a) For all primes $p \geq 3$, prove that the multiplicative group $\mathcal G_{p}$ of the matrices of the form $\begin{pmatrix}\hat 1 & \hat a & \hat b \\ \hat 0 & \hat 1 & \hat c \\ \hat 0 & \hat 0 & \hat 1 \end{pmatrix}$, with $\hat a, \hat b, \hat c \in \mathbb Z \slash p \mathbb Z$, is not commutative and has [i]exponent[/i] $p$. (b) Prove that if $\left( G, \circ \right)$ and $\left( H, \bullet \right)$ are finite groups of [i]exponents[/i] $m$ and $n$, respectively, then the group $\left( G \times H, \odot \right)$ with the operation given by $(g,h) \odot \left( g^\prime, h^\prime \right) = \left( g \circ g^\prime, h \bullet h^\prime \right)$, for all $\left( g,h \right), \, \left( g^\prime, h^\prime \right) \in G \times H$, has the [i]exponent[/i] equal to $\textrm{lcm}(m,n)$. (c) Prove that any $n \geq 3$ is the [i]exponent[/i] of a finite, non-commutative group. [i]Ion Savu[/i]

Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S$, the probability that it is divisible by $9$ is $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

[u]Round 1[/u] [b]p1.[/b] How many powers of $2$ are greater than $3$ but less than $2013$? [b]p2.[/b] What number is equal to six greater than three times the answer to this question? [b]p3.[/b] Surya Cup-a-tea-raju goes to Starbucks Coffee to sip coffee out of a styrofoam cup. The cup is a cylinder, open on one end, with base radius $3$ centimeters and height $10$ centimeters. What is the exterior surface area of the styrofoam cup? [u]Round 2[/u] [b]p4.[/b] Andrew has two $6$-foot-length sticks that he wishes to make into two of the sides of the entrance to his fort, with the ground being the third side. If he wants to make his entrance in the shape of a triangle, what is the largest area that he can make the entrance? [b]p5.[/b] Ethan and Devin met a fairy who told them “if you have less than $15$ dollars, I will give you cake”. If both had integral amounts of dollars, and Devin had 5 more dollars than Ethan, but only Ethan got cake, how many different amounts of money could Ethan have had? [b]p6.[/b] If $2012^x = 2013$, for what value of $a$, in terms of $x$, is it true that $2012^a = 2013^2$? [u]Round 3[/u] [b]p7.[/b] Find the ordered triple $(L, M, T)$ of positive integers that makes the following equation true: $$1 + \dfrac{1}{L + \dfrac{1}{M+\dfrac{1}{T}}}=\frac{79}{43}.$$ [b]p8.[/b] Jonathan would like to start a banana plantation so he is saving up to buy an acre of land, which costs $\$600,000$. He deposits $\$300,000$ in the bank, which gives $20\%$ interest compounded at the end of each year. At this rate, how many years will Jonathan have to wait until he can buy the acre of land? [b]p9.[/b] Arul and Ethan went swimming at their town pool and started to swim laps to see who was in better shape. After one hour of swimming at their own paces, Ethan completed $32$ more laps than Arul. However, after that, Ethan got tired and swam at half his original speed while Arul’s speed didn’t change. After one more hour, Arul swam a total of $320$ laps. How many laps did Ethan swim after two hours? [u]Round 4[/u] [b]p10.[/b] A right triangle with a side length of $6$ and a hypotenuse of 10 has circles of radius $1$ centered at each vertex. What is the area of the space inside the triangle but outside all three circles? [b]p11.[/b] In isosceles trapezoid $ABCD$, $\overline{AB} \parallel\overline{CD}$ and the lengths of $\overline{AB}$ and $\overline{CD}$ are $2$ and $6$, respectively. Let the diagonals of the trapezoid intersect at point $E$. If the distance from $E$ to $\overline{CD}$ is $9$, what is the area of triangle $ABE$? [b]p12.[/b] If $144$ unit cubes are glued together to form a rectangular prism and the perimeter of the base is $54$ units, what is the height? PS. You should use hide for answers. Rounds 6-8 are [url=https://artofproblemsolving.com/community/c3h3136014p28427163]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3137069p28442224]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Matt has a twenty dollar bill and buys two items worth $\$7:99$ each. How much change does he receive, in dollars? [b]p2.[/b] The sum of two distinct numbers is equal to the positive difference of the two numbers. What is the product of the two numbers? [b]p3.[/b] Evaluate $$\frac{1 + 2 + 3 + 4 + 5 + 6 + 7}{8 + 9 + 10 + 11 + 12 + 13 + 14}.$$ [b]p4.[/b] A sphere with radius $r$ has volume $2\pi$. Find the volume of a sphere with diameter $r$. [b]p5.[/b] Yannick ran $100$ meters in $14.22$ seconds. Compute his average speed in meters per second, rounded to the nearest integer. [b]p6.[/b] The mean of the numbers $2, 0, 1, 5,$ and $x$ is an integer. Find the smallest possible positive integer value for $x$. [b]p7.[/b] Let $f(x) =\sqrt{2^2 - x^2}$. Find the value of $f(f(f(f(f(-1)))))$. [b]p8.[/b] Find the smallest positive integer $n$ such that $20$ divides $15n$ and $15$ divides $20n$. [b]p9.[/b] A circle is inscribed in equilateral triangle $ABC$. Let $M$ be the point where the circle touches side $AB$ and let $N$ be the second intersection of segment $CM$ and the circle. Compute the ratio $\frac{MN}{CN}$ . [b]p10.[/b] Four boys and four girls line up in a random order. What is the probability that both the first and last person in line is a girl? [b]p11.[/b] Let $k$ be a positive integer. After making $k$ consecutive shots successfully, Andy's overall shooting accuracy increased from $65\%$ to $70\%$. Determine the minimum possible value of $k$. [b]p12.[/b] In square $ABCD$, $M$ is the midpoint of side $CD$. Points $N$ and $P$ are on segments $BC$ and $AB$ respectively such that $ \angle AMN = \angle MNP = 90^o$. Compute the ratio $\frac{AP}{PB}$ . [b]p13.[/b] Meena writes the numbers $1, 2, 3$, and $4$ in some order on a blackboard, such that she cannot swap two numbers and obtain the sequence $1$, $2$, $3$, $4$. How many sequences could she have written? [b]p14.[/b] Find the smallest positive integer $N$ such that $2N$ is a perfect square and $3N$ is a perfect cube. [b]p15.[/b] A polyhedron has $60$ vertices, $150$ edges, and $92$ faces. If all of the faces are either regular pentagons or equilateral triangles, how many of the $92$ faces are pentagons? [b]p16.[/b] All positive integers relatively prime to $2015$ are written in increasing order. Let the twentieth number be $p$. The value of $\frac{2015}{p}-1$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Compute $a + b$. [b]p17.[/b] Five red lines and three blue lines are drawn on a plane. Given that $x$ pairs of lines of the same color intersect and $y$ pairs of lines of different colors intersect, find the maximum possible value of $y - x$. [b]p18.[/b] In triangle $ABC$, where $AC > AB$, $M$ is the midpoint of $BC$ and $D$ is on segment $AC$ such that $DM$ is perpendicular to $BC$. Given that the areas of $MAD$ and $MBD$ are $5$ and $6$, respectively, compute the area of triangle $ABC$. [b]p19.[/b] For how many ordered pairs $(x, y)$ of integers satisfying $0 \le x, y \le 10$ is $(x + y)^2 + (xy - 1)^2$ a prime number? [b]p20.[/b] A solitaire game is played with $8$ red, $9$ green, and $10$ blue cards. Totoro plays each of the cards exactly once in some order, one at a time. When he plays a card of color $c$, he gains a number of points equal to the number of cards that are not of color $c$ in his hand. Find the maximum number of points that he can obtain by the end of the game. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that $m^7- m$ is divisible by $42$ for any positive integer $m$.

Find all the values of $n \in N$ such that $n^2 = 2^n$.

[b]p1.[/b] Find the largest possible number of consecutive $9$’s in which an integer between $10,000,000$ and $13,371,337$ can end. For example, $199$ ends in two $9$’s, while $92,999$ ends in three $9$’s. [b]p2.[/b] Let $ABCD$ be a square of side length $2$. Equilateral triangles $ABP$, $BCQ$, $CDR$, and $DAS$ are constructed inside the square. Compute the area of quadrilateral $PQRS$. [b]p3.[/b] Evaluate the expression $7 \cdot 11 \cdot 13 \cdot 1003 - 3 \cdot 17 \cdot 59 \cdot 331$. [b]p4.[/b] Compute the number of positive integers $c$ such that there is a non-degenerate obtuse triangle with side lengths $21$, $29$, and $c$. [b]p5.[/b] Consider a $5\times 5$ board, colored like a chessboard, such that the four corners are black. Determine the number of ways to place $5$ rooks on black squares such that no two of the rooks attack one another, given that the rooks are indistinguishable and the board cannot be rotated. (Two rooks attack each other if they are in the same row or column.) [b]p6.[/b] Let $ABCD$ be a trapezoid of height $6$ with bases $AB$ and $CD$. Suppose that $AB = 2$ and $CD = 3$, and let $F$ and $G$ be the midpoints of segments $AD$ and $BC$, respectively. If diagonals $AC$ and $BD$ intersect at point $E$, compute the area of triangle $FGE$. [b]p7.[/b] A regular octahedron is a solid with eight faces that are congruent equilateral triangles. Suppose that an ant is at the center of one face of a regular octahedron of edge length $10$. The ant wants to walk along the surface of the octahedron to reach the center of the opposite face. (Two faces of an octahedron are said to be opposite if they do not share a vertex.) Determine the minimum possible distance that the ant must walk. [b]p8.[/b] Let $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$, and $D_1D_2D_3$ be triangles in the plane. All the sides of the four triangles are extended into lines. Determine the maximum number of pairs of these lines that can meet at $60^o$ angles. [b]p9.[/b] For an integer $n$, let $f_n(x)$ denote the function $f_n(x) =\sqrt{x^2 - 2012x + n}+1006$. Determine all positive integers $a$ such that $f_a(f_{2012}(x)) = x$ for all $x \ge 2012$. [b]p10.[/b] Determine the number of ordered triples of integers $(a, b, c)$ such that $(a + b)(b + c)(c + a) = 1800$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Replace $*$’s by an arithmetic operations (addition, subtraction, multiplication or division) to obtain true equality $$2*0*1*6*7=1.$$ [b]p2.[/b] The interval of length $88$ cm is divided into three unequal parts. The distance between middle points of the left and right parts is $46$ cm. Find the length of the middle part. [b]p3.[/b] A $5\times 6$ rectangle is drawn on a square grid. Paint some cells of the rectangle in such a way that every $3\times 2$ sub‐rectangle has exactly two cells painted. [b]p4.[/b] There are $8$ similar coins. $5$ of them are counterfeit. A detector can analyze any set of coins and show if there are counterfeit coins in this set. The detector neither determines which coins nare counterfeit nor how many counterfeit coins are there. How to run the detector twice to find for sure at least one counterfeit coin? [b]p5.[/b] There is a set of $20$ weights of masses $1, 2, 3,...$ and $20$ grams. Can one divide this set into three groups of equal total masses? [b]p6.[/b] Replace letters $A,B,C,D,E,F,G$ by the digits $0,1,...,9$ to get true equality $AB+CD=EF * EG$ (different letters correspond to different digits, same letter means the same digit, $AB$, $CD$, $EF$, and $EG$ are two‐digit numbers). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For every positive integer $n{}$, consider the numbers $a_1=n^2-10n+23, a_2=n^2-9n+31, a_3=n^2-12n+46.$ a) Prove that $a_1+a_2+a_3$ is even. b) Find all positive integers $n$ for which $a_1, a_2$ and $a_3$ are primes.

Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that $$k\le2\lfloor\sqrt n\rfloor.$$

Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.