Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.

We distribute $n\ge1$ labelled balls among nine persons $A,B,C, \dots , I$. How many ways are there to do this so that $A$ gets the same number of balls as $B,C,D$ and $E$ together?

Let $ABC$ and $A'B'C'$ be any two coplanar triangles. Let $L$ be a point such that $AL || BC, A'L || B'C'$ , and $M,N$ similarly defined. The line $BC$ meets $B'C'$ at $P$, and similarly defined are $Q$ and $R$. Prove that $PL, QM, RN$ are concurrent.

Find all positive integers $n, k$ such that $(n-1)!=n^{k}-1$.

Find all pairs of cubic equations $x^3 +ax^2 +bx +c =0$ and $x^3 +bx^2 + ax +c = 0$ where $a,b,c$ are integers, such that each equation has three integer roots and both the equations have exactly one common root.

Find the number of positive integers $n < 2018$ such that $25^n + 9^n$ is divisible by $13$.

$AD$ and $BE$ are the altitudes of the triangle $ABC$. It turned out that the point $C'$, symmetric to the vertex $C$ wrt to the midpoint of the segment $DE$, lies on the side $AB$. Prove that $AB$ is tangent to the circle circumscribed around the triangle $DEC'$.

Let $ a, b, c>0 $ such that $ ab+bc+ca=3 $. Prove that \[ \sum_{cyc} { \frac{ (a+b)^{3} }{ {(2(a+b)(a^2 + b^2))}^{\frac{1}{3}}} \ge 12 }\]

Let $a, b$ and $c$ be the sides of a triangle, that is: $a + b > c$, $b + c > a$ and $c + a > b$. Show that: $$\frac{ab+ 1}{a^2 + ca + 1} +\frac{bc + 1}{b^2 + ab + 1} +\frac{ca + 1}{c^2 + bc + 1} > \frac32$$

A cuboid has an integer volume. Three of the faces have different areas, namely $7, 27$, and $L$. What is the smallest possible integer value for $L$?

The unit of a screw is listed as $0.2$ cents. When a group of screws is sold to a customer, the total cost of the screws is computed with the listed price and then rounded to the nearest cent. If Al has $50$ cents and wishes to only make one purchase, what is the maximum possible number of screws he can buy?

An odd positive integer $n$ can be expressed as the sum of two or more consecutive integers in exactly $2023$ ways. Find the greatest possible nonnegative integer $k$ such that $3^k$ is a factor of the least possible value of $n$.

Is it true that when all the faces of a tetrahedron have the same area, they are congruent triangles?

Let $ABCD$ be a quadrilateral, with an inscribed circle with center $I$. Through $A$ are constructed perpendiculars to $AB$ and $AD$, which intersect $BI$ and $DI$ in points $M$ and $N$ respectively. Prove that $MN\perp AC$.

Given positive integer $ m \geq 17$, $ 2m$ contestants participate in a circular competition. In each round, we devide the $ 2m$ contestants into $ m$ groups, and the two contestants in one group play against each other. The groups are re-divided in the next round. The contestants compete for $ 2m\minus{}1$ rounds so that each contestant has played a game with all the $ 2m\minus{}1$ players. Find the least possible positive integer $ n$, so that there exists a valid competition and after $ n$ rounds, for any $ 4$ contestants, non of them has played with the others or there have been at least $ 2$ games played within those $ 4$.

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $m,n$ we have : $$f(n)+1400m^2|n^2+f(f(m))$$

Alice and Bob play a game on a strip of $n \ge 3$ squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. The figure shows the starting position for a strip of $ n = 7$ squares. [img]https://cdn.artofproblemsolving.com/attachments/1/7/c636115180fd624cbeec0c6adda31b4b89ed60.png[/img] The players alternate. In each move, they advance their own game piece by one or two squares in the direction of the opponent’s piece. The piece has to land on an empty square without jumping over the opponent’s piece. Alice makes the first move with her own piece. If a player cannot move, they lose. For which $n$ can Bob ensure a win no matter how Alice plays? For which $n$ can Alice ensure a win no matter how Bob plays? [i](Karl Czakler)[/i]

Bowling Pins is a game played between two players in the following way: We start with $ 14 $ bowling pins in a line: $ X\quad X \quad X \quad X\quad X\quad X\quad X\quad X\quad X\quad X\quad X\quad X\quad X\quad X $ Players alternate turns. On each turn, the player can knock down one, two or three consecutive pins at a time. For example: Jing Jing bowls: $ X \quad X \quad\quad \quad\quad \:\:\:\: X \quad X \quad X \quad X \quad X \quad X \quad X \quad X \quad X \quad X $ Soumya bowls: $ X \quad X \quad\quad \quad\quad \:\:\:\: X \quad X \quad X \quad \quad\quad X \quad X \quad X \quad X \quad X \quad X $ Jing Jing bowls again: $ X \quad X \quad\quad \quad\quad \:\:\:\: X \quad X \quad X \quad \quad\quad X \quad X \quad \quad \quad\quad\quad \quad\:\: X $ The player who knocks down the last pin wins. In the above game, it is Soumya’s turn. If he plays perfectly from here, he has a winning strategy (In fact, he has four different winning moves.) Imagine it’s Jing Jing’s turn to play and the game looks as follows $ X \quad \quad \quad\quad\quad \quad\:\: X\dots $ with 1 X on the left and a string of $ k $ consecutive X’s on the right. For what values of $ k $ from $ 1 $ to $ 10 $ does she have a winning strategy?

Let $n \geqslant 3$ be integer. Given convex $n-$polygon $\mathcal{P}$. A $3-$coloring of the vertices of $\mathcal{P}$ is called [i]nice[/i] such that every interior point of $\mathcal{P}$ is inside or on the bound of a triangle formed by polygon vertices with pairwise distinct colors. Determine the number of different nice colorings. ([I]Two colorings are different as long as they differ at some vertices. [/i])

Suppose $x,y,z$ are positive numbers such that $x+y+z=1$. Prove that (a) $xy+yz+xz\ge 9xyz$; (b) $xy+yz+xz<\frac{1}{4}+3xyz$;

Suppose $P(n) $ is a nonconstant polynomial where all of its coefficients are nonnegative integers such that \[ \sum_{i=1}^n P(i) | nP(n+1) \] for every $n \in \mathbb{N}$. Prove that there exists an integer $k \ge 0$ such that \[ P(n) = \binom{n+k}{n-1} P(1) \] for every $n \in \mathbb{N}$.

Is it possible to divide a circle by three chords, different from diameters, into several equal parts?

Let $a$ and $b$ be positive integers such that $\frac{5a^4 + a^2}{b^4 + 3b^2 + 4}$ is an integer. Show that $a$ is not prime.

[b]p1.[/b] Calculate $\left( \frac12 + \frac13 + \frac14 \right)^2$. [b]p2.[/b] Find the $2010^{th}$ digit after the decimal point in the expansion of $\frac17$. [b]p3.[/b] If you add $1$ liter of water to a solution consisting of acid and water, the new solutions will contain of $30\%$ water. If you add another $5$ liters of water to the new solution, it will contain $36\frac{4}{11}\%$ water. Find the number of liters of acid in the original solution. [b]p4.[/b] John places $5$ indistinguishable blue marbles and $5$ indistinguishable red marbles into two distinguishable buckets such that each bucket has at least one blue marble and one red marble. How many distinguishable marble distributions are possible after the process is completed? [b]p5.[/b] In quadrilateral $PEAR$, $PE = 21$, $EA = 20$, $AR = 15$, $RE = 25$, and $AP = 29$. Find the area of the quadrilateral. [b]p6.[/b] Four congruent semicircles are drawn within the boundary of a square with side length $1$. The center of each semicircle is the midpoint of a side of the square. Each semicircle is tangent to two other semicircles. Region $R$ consists of points lying inside the square but outside of the semicircles. The area of $R$ can be written in the form $a - b\pi$, where $a$ and $b$ are positive rational numbers. Compute $a + b$. [b]p7.[/b] Let $x$ and $y$ be two numbers satisfying the relations $x\ge 0$, $y\ge 0$, and $3x + 5y = 7$. What is the maximum possible value of $9x^2 + 25y^2$? [b]p8.[/b] In the Senate office in Exie-land, there are $6$ distinguishable senators and $6$ distinguishable interns. Some senators and an equal number of interns will attend a convention. If at least one senator must attend, how many combinations of senators and interns can attend the convention? [b]p9.[/b] Evaluate $(1^2 - 3^2 + 5^2 - 7^2 + 9^2 - ... + 2009^2) -(2^2 - 4^2 + 6^2 - 8^2 + 10^2- ... + 2010^2)$. [b]p10.[/b] Segment $EA$ has length $1$. Region $R$ consists of points $P$ in the plane such that $\angle PEA \ge 120^o$ and $PE <\sqrt3$. If point $X$ is picked randomly from the region$ R$, the probability that $AX <\sqrt3$ can be written in the form $a - \frac{\sqrt{b}}{c\pi}$ , where $a$ is a rational number, $b$ and $c$ are positive integers, and $b$ is not divisible by the square of a prime. Find the ordered triple $(a, b, c)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $A_{1}, ..., A_{n}$ be different subsets of an $n$-element set $X$. Show that there exists $x\in X$ such that the sets $A_{1}-\{x\}, A_{2}-\{x\}, ..., A_{n}-\{x\}$ are all different.