A natural number $x$ is written on the board. In one move, we can take the number on the board and between any two of its digits in its decimal notation we can we put a sign $+$, or we may not put it, then we calculate the obtained result and we write it on the board in place of $x$. For example, from the number $819$. we can get $18$ by $8 + 1 + 9$, $90$ by $81 + 9$, and $27$ by $8 + 19$. Prove that no matter what $x$ is, we can reach a single digit number with at most $4$ moves.

Let for all $k \in {\mathbb N}$ $k$'s sum of the digits is $T(k)$. If a natural number $n$ such that $T(n)=T(1997n)$, prove that $$9\mid n$$

Let $a$ and $b$ be two numbers in the interval $(0,1)$ such that $a$ is rational and [center]$\{na\} \ge \{nb\},$ for every nonnegative integer $n.$[/center] Prove that $a=b.$ (Note: $\{x\}$ is the fractional part of $x.$)

[b]p1.[/b] At $8$ AM Black Widow and Hawkeye began to move towards each other from two cities. They were planning to meet at the midpoint between two cities, but because Black Widow was driving $100$ mi/h faster than Hawkeye, they met at the point that is located $120$ miles from the midpoint. When they met Black Widow said ”If I knew that you drive so slow I would have started one hour later, and then we would have met exactly at the midpoint”. Find the distance between cities. [b]p2.[/b] Solve the inequality: $\frac{x-1}{x-2} \le \frac{x-2}{x-1}$. [b]p3.[/b] Solve the equation: $(x - y - z)^2 + (2x - 3y + 2z + 4)^2 + (x + y + z - 8)^2 = 0$. [b]p4.[/b] Three camps are located in the vertices of an equilateral triangle. The roads connecting camps are along the sides of the triangle. Captain America is inside the triangle and he needs to know the distances between camps. Being able to see the roads he has found that the sum of the shortest distances from his location to the roads is $50$ miles. Can you help Captain America to evaluate the distances between the camps. [b]p5.[/b] $N$ regions are located in the plane, every pair of them have a nonempty overlap. Each region is a connected set, that means every two points inside the region can be connected by a curve all points of which belong to the region. Iron Man has one charge remaining to make a laser shot. Is it possible for him to make the shot that goes through all $N$ regions? [b]p6.[/b] Numbers $1, 2, . . . , 100$ are randomly divided in two groups $50$ numbers in each. In the first group the numbers are written in increasing order and denoted $a_1$, $a_2$, $...$ , $a_{50}$. In the second group the numbers are written in decreasing order and denoted $b_1$, $b_2$, $...$, $b_{50}$. Thus, $a_1 < a_2 < ... < a_{50}$ and $b_1 > b2_ > ... > b_{50}$. Evaluate $|a_1 - b_1| + |a_2 - b_2| + ... + |a_{50} - b_{50}|$. PS. You should use hide for answers.

a) Prove that, whatever the real number x would be, the following inequality takes place ${{x}^{4}}-{{x}^{3}}-x+1\ge 0.$ b) Solve the following system in the set of real numbers: ${{x}_{1}}+{{x}_{2}}+{{x}_{3}}=3,x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=x_{1}^{4}+x_{2}^{4}+x_{3}^{4}$. The Mathematical Gazette

A boy buys oranges at $ 3$ for $ 10$ cents. He will sell them at $ 5$ for $ 20$ cents. In order to make a profit of $ \$ 1.00$, he must sell: $ \textbf{(A)}\ 67 \text{ oranges} \qquad\textbf{(B)}\ 150 \text{ oranges} \qquad\textbf{(C)}\ 200 \text{ oranges} \\ \textbf{(D)}\ \text{an infinite number of oranges} \qquad\textbf{(E)}\ \text{none of these}$

For which value of $m$, there is no integer pair $(x,y)$ satisfying the equation $3x^2-10xy-8y^2=m^{19}$? $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 3$

Let $A$ be a finite set of positive real numbers satisfying the property: [i]For any real numbers a > 0, the sets $\{x \in A | x > a\}$ and $\{x \in A | x < \frac{1}{a}\}$ have the cardinals of the same parity.[/i] Show that the product of all elements in $A$ is equal to $1$.

Investigate the convergence of the integral \[\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{dxdy}{1+x^4y^4}\]

What is the largest prime $p$ that makes $\sqrt{17p+625}$ an integer? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 67 \qquad\textbf{(C)}\ 101 \qquad\textbf{(D)}\ 151 \qquad\textbf{(E)}\ 211 $

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [b]D1.[/b] Triangle $ABC$ has $AB = 3$, $BC = 4$, and $\angle B = 90^o$. Find the area of triangle $ABC$. [b]D2 / L1.[/b] Let $ABCDEF$ be a regular hexagon. Given that $AD = 5$, find $AB$. [b]D3.[/b] Caroline glues two pentagonal pyramids to the top and bottom of a pentagonal prism so that the pentagonal faces coincide. How many edges does Caroline’s figure have? [b]D4 / L3.[/b] The hour hand of a clock is $6$ inches long, and the minute hand is $10$ inches long. Find the area of the region swept out by the hands from $8:45$ AM to $9:15$ AM of a single day, in square inches. [b]D5 / L2.[/b] Circles $A$, $B$, and $C$ are all externally tangent, with radii $1$, $10$, and $100$, respectively. What is the radius of the smallest circle entirely containing all three circles? [b]D6.[/b] Four parallel lines are drawn such that they are equally spaced and pass through the four vertices of a unit square. Find the distance between any two consecutive lines. [b]D7 / L4.[/b] In rectangle $ABCD$, $AB = 2$ and $AD > AB$. Two quarter circles are drawn inside of $ABCD$ with centers at $A$ and $C$ that pass through $B$ and $D$, respectively. If these two quarter circles are tangent, find the area inside of $ABCD$ that is outside both of the quarter circles. [b]D8 / L6.[/b] Triangle $ABC$ is equilateral. A circle passes through $A$ and is tangent to side $BC$. It intersects sides AB and $AC$ again at $E$ and $F$, respectively. If $AE = 10$ and $AF = 11$, find $AB$. [b]L5.[/b] Find the area of a triangle with side lengths $\sqrt{2}$, $\sqrt{58}$, and $2\sqrt{17}$. [b]L7.[/b] Triangle $ABC$ has area $80$. Point $D$ is in the interior of $\vartriangle ABC$ such that $AD =6$, $BD = 4$, $CD = 16$, and the area of $\vartriangle ADC = 48$. Determine the area of $\vartriangle ADB$. [b]L8. [/b]Given two points $A$ and $B$ in the plane with $AB = 1$, define $f(C)$ to be the circumcenter of triangle $ABC$, if it exists. Find the number of points $X$ so that $f^{2019}(X) = X$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the circle is 5, that $BC = 6$, and that $AD$ is bisected by $BC$. Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$. It follows that the sine of the minor arc $AB$ is a rational number. If this fraction is expressed as a fraction $m/n$ in lowest terms, what is the product $mn$? [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=dir(200), D=dir(95), M=midpoint(A--D), C=dir(30), BB=C+2*dir(C--M), B=intersectionpoint(M--BB, Circle(origin, 1)); draw(Circle(origin, 1)^^A--D^^B--C); real r=0.05; pair M1=midpoint(M--D), M2=midpoint(M--A); draw((M1+0.1*dir(90)*dir(A--D))--(M1+0.1*dir(-90)*dir(A--D))); draw((M2+0.1*dir(90)*dir(A--D))--(M2+0.1*dir(-90)*dir(A--D))); pair point=origin; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D));[/asy]

Consider $\vartriangle ABC$ where $\angle ABC= 60 ^o$. Points $M$ and $D$ are on the sides $(AC)$, respectively $(AB)$, such that $\angle BCA = 2 \angle MBC$, and $BD = MC$. Determine $\angle DMB$.

Let $n$ and $\ell$ be integers such that $n \ge 3$ and $1 < \ell < n$. A country has $n$ cities. Between any two cities $A$ and $B$, either there is no flight from $A$ to $B$ and also none from $B$ to $A$, or there is a unique [i]two-way trip[/i] between them. A [i]two-way trip[/i] is a flight from $A$ to $B$ and a flight from $B$ to $A$. There exist two cities such that the least possible number of flights required to travel from one of them to the other is $\ell$. Find the maximum number of two-way trips among the $n$ cities.

On the table there are $2016$ coins. Two players play the following game making alternating moves. In one move it is allowed to take $1, 2$ or $3$ coins. The player who takes the last coin wins. Which player has a winning strategy?

Find all triples of positive integers $(x, y, z)$ that satisfy the equation $$2020^x + 2^y = 2024^z.$$ [i]Proposed by Ognjen Tešić, Serbia[/i]

Let $n$ to be a positive integer. Given a set $\{ a_1, a_2, \ldots, a_n \} $ of integers, where $a_i \in \{ 0, 1, 2, 3, \ldots, 2^n -1 \},$ $\forall i$, we associate to each of its subsets the sum of its elements; particularly, the empty subset has sum of its elements equal to $0$. If all of these sums have different remainders when divided by $2^n$, we say that $\{ a_1, a_2, \ldots, a_n \} $ is [i]$n$-complete[/i]. For each $n$, find the number of [i]$n$-complete[/i] sets.

Determine all real numbers $x$ which satisfy \[ x = \sqrt{a - \sqrt{a+x}} \] where $a > 0$ is a parameter.

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

Distinct points $A$ and $B$ are given on the plane. Consider all triangles $ABC$ in this plane on whose sides $BC,CA$ points $D,E$ respectively can be taken so that (i) $\frac{BD}{BC}=\frac{CE}{CA}=\frac{1}{3}$; (ii) points $A,B,D,E$ lie on a circle in this order. Find the locus of the intersection points of lines $AD$ and $BE$.

a. Prove that there are infinitely many positive integers $t$ such that both $2012t+1$ and $2013t+1$ are perfect squares. b. Suppose that $m,n$ are positive integers such that both $mn+1$ and $mn+n+1$ are perfect squares. Prove that $8(2m+1)$ divides $n$.

If $a, b, c$ represent the lengths of the sides of a triangle, prove that: $$\frac{a}{b-a+c}+ \frac{b}{b-a+c}+ \frac{c}{b-a+c} \ge 3$$

Regular hexagon $ ABCDEF$ has vertices $ A$ and $ C$ at $ (0,0)$ and $ (7,1)$, respectively. What is its area? $ \textbf{(A) } 20\sqrt {3} \qquad \textbf{(B) } 22\sqrt {3} \qquad \textbf{(C) } 25\sqrt {3} \qquad \textbf{(D) } 27\sqrt {3} \qquad \textbf{(E) } 50$

In a $\triangle {ABC}$, consider a point $E$ in $BC$ such that $AE \perp BC$. Prove that $AE=\frac{bc}{2r}$, where $r$ is the radio of the circle circumscripte, $b=AC$ and $c=AB$.

A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)