Oleksiy writes all the digits from $0$ to $9$ on the board, after which Vlada erases one of them. Then he writes $10$ nine-digit numbers on the board, each consisting of all the nine digits written on the board (they don't have to be distinct). It turned out that the sum of these $10$ numbers is a ten-digit number, all of whose digits are distinct. Which digit could have been erased by Vlada? [i]Proposed by Oleksii Masalitin[/i]

Handy Aaron helped a neighbor $1\frac{1}{4}$ hours on Monday, $50$ minutes on Tuesday, from $8:20$ to $10:45$ on Wednesday morning, and a half-hour on Friday. He is paid $\$3$ per hour. How much did he earn for the week? $\textbf{(A)}\ 8\qquad \textbf{(B)}\ 9\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 15$

Let $x_1<x_2< \ldots <x_{2024}$ be positive integers and let $p_i=\prod_{k=1}^{i}(x_k-\frac{1}{x_k})$ for $i=1,2, \ldots, 2024$. What is the maximal number of positive integers among the $p_i$?

Prove that any convex pentagon whose vertices (no three of which are collinear) have integer coordinates must have area greater than or equal to $ \dfrac {5}{2} $.

There were three times as many red candies as blue candies on a table. After Darrel took the same number of red candies and blue candies, there were four times as many red candies as blue candies left on the table. Then after Cloe took $12$ red candies and $12$ blue candies, there were five times as many red candies as blue candies left on the table. Find the total number of candies that Darrel took.

Let $P$ and $P '$ be two unequal regular $n-$gons and $A$ and $A'$two points inside $P$ and$ P '$, respectively.Suppose $\{ d_1 , d_2 , \cdots d_n \}$ are the distances from $A $ to the vertices of $P$ and $\{ d'_1 , d'_2 , \cdots d'_n \}$ are defines similarly for $P',A'$. Is it possible for $\{ d'_1 , d'_2 , \cdots d'_n \}$ to be a permutation of $\{ d_1 , d_2 , \cdots d_n \}$ ?

Prove that for every positive integer $n \geq 2$ the following inequality holds: $e^{n-1}n!<n^{n+\frac{1}{2}}$

A right triangle $ABC$ with $\angle C=90^o$ is inscribed in a circle. Suppose that $K$ is the midpoint of the arc $BC$ that does not contain $A$. Let $N$ be the midpoint of the segment $AC$, and $M$ be the intersection point of the ray $KN$ and the circle.The tangents to the circle drawn at $A$ and $C$ meet at $E$. prove that $\angle EMK = 90^o$

Given is a positive integer $n$. There are $2n$ mutually non-attacking rooks placed on a grid $2n \times 2n$. The grid is splitted into two connected parts, symmetric with respect to the center of the grid. What is the largest number of rooks that could lie in the same part?

With about six hours left on the van ride home from vacation, Wendy looks for something to do. She starts working on a project for the math team. There are sixteen students, including Wendy, who are about to be sophomores on the math team. Elected as a math team officer, one of Wendy's jobs is to schedule groups of the sophomores to tutor geometry students after school on Tuesdays. The way things have been done in the past, the same number of sophomores tutor every week, but the same group of students never works together. Wendy notices that there are even numbers of groups she could select whether she chooses $4$ or $5$ students at a time to tutor geometry each week: \begin{align*}\dbinom{16}4&=1820,\\\dbinom{16}5&=4368.\end{align*} Playing around a bit more, Wendy realizes that unless she chooses all or none of the students on the math team to tutor each week that the number of possible combinations of the sophomore math teamers is always even. This gives her an idea for a problem for the $2008$ Jupiter Falls High School Math Meet team test: \[\text{How many of the 2009 numbers on Row 2008 of Pascal's Triangle are even?}\] Wendy works the solution out correctly. What is her answer?

The number of real solutions of the equation $\left(\frac{9}{10}\right)^x=-3+x-x^2$ is [list=1] [*] 2 [*] 0 [*] 1 [*] None of these [/list]

Find all quadruples of positive integers $(k,a,b,c)$ such that $2^k=a!+b!+c!$ and $a\geq b\geq c$.

Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee? $ \textbf{(A) } \frac 67 \qquad \textbf{(B) } \frac {13}{14} \qquad \textbf{(C) } 1 \qquad \textbf{(D) } \frac {14}{13} \qquad \textbf{(E) } \frac 76$

Let $x,y,z\in \mathbb{R}^+_0$ such that $xy+yz+zx=1$. Prove that $$\frac{1}{\sqrt{x+y}}+\frac{1}{\sqrt{y+z}}+\frac{1}{\sqrt{z+x}}\ge 2+\frac{1}{\sqrt{2}}.$$ [i](Anonymous314)[/i]

Compute the minimum value of $$\frac{x^4+2x^3+3x^2+2x+10}{x^2+x+1}$$ where $x$ can be any real number.

Let $a_1<a_2<...<a_t$ be $t$ given positive integers where no three form an arithmetic progression. For $k=t,t+1,...$ define $a_{k+1}$ to be the smallest positive integer larger than $a_k$ satisfying the condition that no three of $a_1,a_2,...,a_{k+1}$ form an arithmetic progression. For any $x\in\mathbb{R}^+$ define $A(x)$ to be the number of terms in $\{a_i\}_{i\ge 1}$ that are at most $x$. Show that there exist $c>1$ and $K>0$ such that $A(x)\ge c\sqrt{x}$ for any $x>K$.

Find the max. value of $ M$,such that for all $ a,b,c>0$: $ a^{3}+b^{3}+c^{3}-3abc\geq M(|a-b|^{3}+|a-c|^{3}+|c-b|^{3})$

[b]p1.[/b] Find all real solutions of the system $$x^5 + y^5 = 1$$ $$x^6 + y^6 = 1$$ [b]p2.[/b] The centers of three circles of the radius $R$ are located in the vertexes of equilateral triangle. The length of the sides of the triangle is $a$ and $\frac{a}{2}< R < a$. Find the distances between the intersection points of the circles, which are outside of the triangle. [b]p3.[/b] Prove that no positive integer power of $2$ ends with four equal digits. [b]p4.[/b] A circle is divided in $10$ sectors. $90$ coins are located in these sectors, $9$ coins in each sector. At every move you can move a coin from a sector to one of two neighbor sectors. (Two sectors are called neighbor if they are adjoined along a segment.) Is it possible to move all coins into one sector in exactly$ 2004$ moves? [b]p5.[/b] Inside a convex polygon several points are arbitrary chosen. Is it possible to divide the polygon into smaller convex polygons such that every one contains exactly one given point? Justify your answer. [b]p6.[/b] A troll tried to spoil a white and red $8\times 8$ chessboard. The area of every square of the chessboard is one square foot. He randomly painted $1.5\%$ of the area of every square with black ink. A grasshopper jumped on the spoiled chessboard. The length of the jump of the grasshopper is exactly one foot and at every jump only one point of the chessboard is touched. Is it possible for the grasshopper to visit every square of the chessboard without touching any black point? Justify your answer. PS. You should use hide for answers.

On the sides $AD , BC$ of the square $ABCD$ the points $M, N$ are selected $N$, respectively, such that $AM = BN$. Point $X$ is the foot of the perpendicular from point $D$ on the line $AN$. Prove that the angle $MXC$ is right. (Mirchev Borislav)

Three main diagonals of a hegular hexagon divide the hegular hexagon into six regular triangles. Note them $A,B,C,D,E,F$. In one part, grow one kind of plant. Also, adjacent parts must be grown with different plants. If we are given four kinds of plants, then the number of wanys to grow plants is________.

On a isosceles triangle $\triangle ABC$ with $AB=BC$ let $K,M$ be the midpoints of $AB,AC$ respectivily. Let $(CKB)$ intersect $BM$ at $N \ne M$, the line through $N$ parallel to $AC$ intersects $(ABC)$ at $A_1,C_1$. Show that $\triangle A_1BC_1$ is equilateral.

We call a function $g$ [i]special [/i] if $g(x)=a^{f(x)}$ (for all $x$) where $a$ is a positive integer and $f$ is polynomial with integer coefficients such that $f(n)>0$ for all positive integers $n$. A function is called an [i]exponential polynomial[/i] if it is obtained from the product or sum of special functions. For instance, $2^{x}3^{x^{2}+x-1}+5^{2x}$ is an exponential polynomial. Prove that there does not exist a non-zero exponential polynomial $f(x)$ and a non-constant polynomial $P(x)$ with integer coefficients such that $$P(n)|f(n)$$ for all positive integers $n$.

$f$ is a function defined on integers and satisfies $f(x)+f(x+3)=x^2$ for every integer $x$. If $f(19)=94$, then calculate $f(94)$.

Find all integers $m$ and $n$ satisfying $m^3 -n^3 - 9mn = 27$.

With $28$ points, a “triangular grid” of equal sides is formed, as shown in the figure. One operation consists of choosing three points that are the vertices of an equilateral triangle and removing these three points from the grid. If after performing several of these operations there is only one point left, in what positions can that point remain? Give all the possibilities and indicate in each case the operations carried out. Justify why the remaining point cannot be in another position. [img]https://cdn.artofproblemsolving.com/attachments/f/c/1cedfe0e1c5086b77151538265f8e253e93d2e.gif[/img]