[b]p1.[/b] $ x,y,z$ are required to be non-negative whole numbers, find all solutions to the pair of equations $$x+y+z=40$$ $$2x + 4y + 17z = 301.$$ [b]p2.[/b] Let $P$ be a point lying between the sides of an acute angle whose vertex is $O$. Let $A,B$ be the intersections of a line passing through $P$ with the sides of the angle. Prove that the triangle $AOB$ has minimum area when $P$ bisects the line segment $AB$. [b]p3.[/b] Find all values of $x$ for which $|3x-2|+|3x+1|=3$. [b]p4.[/b] Prove that $x^2+y^2+z^2$ cannot be factored in the form $$(ax + by + cz) (dx + ey + fz),$$ $a, b, c, d, e, f$ real. [b]p5.[/b] Let $f(x)$ be a continuous function for all real values of $x$ such that $f(a)\le f(b)$ whenever $a\le b$. Prove that, for every real number $r$, the equation $$x + f(x) = r$$ has exactly one solution. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the tribe of Zimmer, being able to hike long distances and knowing the roads through the forest are both extremely important, so a boy who reaches the age of manhood is not designated as a man by the tribe until he completes an interesting rite of passage. The man must go on a sequence of hikes. The first hike is a $5$ kilometer hike down the main road. The second hike is a $5\frac{1}{4}$ kilometer hike down a secondary road. Each hike goes down a different road and is a quarter kilometer longer than the previous hike. The rite of passage is completed at the end of the hike where the cumulative distance walked by the man on all his hikes exceeds $1000$ kilometers. So in the tribe of Zimmer, how many roads must a man walk down, before you call him a man?

Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to factorials of some positive integers. Proposed by [i]Nikola Velov, Macedonia[/i]

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

Is it possible to put together $27$ equal cubes, $9$ red, $9$ blue and $9$ white, so as to obtain a big cube in which each row (parallel to an arbitrary edge of the cube) contains three cubes with exactly two different colours? (S. Fomin, Leningrad)

In a certain magical country, there are banknotes in denominations of $2^0, 2^1, 2^2, \ldots$ UAH. Businessman Victor has to make cash payments to $44$ different companies totaling $44000$ UAH, but he does not remember how much he has to pay to each company. What is the smallest number of banknotes Victor should withdraw from an ATM (totaling exactly $44000$ UAH) to guarantee that he would be able to pay all the companies without leaving any change? [i]Proposed by Oleksii Masalitin[/i]

On a given $ 2008 \times 2008$ chessboard, each unit square is colored in a different color. Every unit square is filled with one of the letters C, G, M, O. The resulting board is called [i]harmonic[/i] if every $ 2 \times 2$ subsquare contains all four different letters. How many harmonic boards are there?

Find the least natural number $n\geq 5$, for which $x^n\equiv 16\, (mod\, p)$ has a solution for any prime number $p$.

Is it possible to place within a square an equilateral triangle whose area is larger than $9/ 20$ of the area of the square?

In an acute triangle $ABC$, point $D$ is on the segment $AC$ such that $\overline{AD}=\overline{BC}$ and $\overline{AC}^2-\overline{AD}^2=\overline{AC}\cdot\overline{AD}$. The line that is parallel to the bisector of $\angle{ACB}$ and passes the point $D$ meets the segment $AB$ at point $E$. Prove, if $\overline{AE}=\overline{CD}$, $\angle{ADB}=3\angle{BAC}$.

A function \[\text{f:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] is called contract if, for every numbers $x,y\in \text{(0,}\infty \text{)}$ we have, $\underset{n\to \infty }{\mathop{\lim }}\,\left( {{f}^{n}}\left( x \right)-{{f}^{n}}\left( y \right) \right)=0$ where ${{f}^{n}}=\underbrace{f\circ f\circ ...\circ f}_{n\ f\text{'s}}$ a) Consider \[f:\text{(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] a function contract, continue with the property that has a fixed point, that existing ${{x}_{0}}\in \text{(0,}\infty \text{) }$ there so that $f\left( {{x}_{0}} \right)={{x}_{0}}.$ Show that $f\left( x \right)>x,$ for every $x\in \text{(0,}{{x}_{0}}\text{)}\,$ and $f\left( x \right)<x$, for every $x\in \text{(}{{x}_{0}}\text{,}\infty \text{)}\,$. b) Show that the given function \[f\text{:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] given by $f\left( x \right)=x+\frac{1}{x}$ is contracted but has no fix number.

There are non-constant polynom $P(x)$ with integral coefficients and natural number $n$. Suppose that $a_0=n$, $a_k=P(a_{k-1})$ for any natural $k$. Finally, for every natural $b$ there is number in sequence $a_0, a_1, a_2, \ldots$ that is $b$-th power of some natural number that is more than 1. Prove that $P(x)$ is linear polynom.

The symmetric difference of two homothetic triangles $T_1$ and $T_2$ consists of six triangles $t_1, \ldots, t_6$ with circumcircles $\omega_1, \omega_2, \ldots, \omega_6$ (counterclockwise, no two intersect). Circle $\Omega_1$ with center $O_1$ is externally tangent to $\omega_1, \omega_3,$ and $\omega_5$; circle $\Omega_2$ with center $O_2$ is externally tangent to $\omega_2, \omega_4,$ and $\omega_6$; circle $\Omega_3$ with center $O_3$ is internally tangent to $\omega_1, \omega_3,$ and $\omega_5$; circle $\Omega_4$ with center $O_4$ is internally tangent to $\omega_2, \omega_4,$ and $\omega_6$. Prove that $O_1O_3 = O_2O_4$. [i]Proposed by Ilya Zamotorin[/i]

A number $m$ is randomly selected from the set $\{11,13,15,17,19\}$, and a number $n$ is randomly selected from $\{1999,2000,2001,\ldots,2018\}$. What is the probability that $m^n$ has a units digit of $1$? $\textbf{(A) } \frac{1}{5} \qquad \textbf{(B) } \frac{1}{4} \qquad \textbf{(C) } \frac{3}{10} \qquad \textbf{(D) } \frac{7}{20} \qquad \textbf{(E) } \frac{2}{5} $

Positive integers are written on all the faces of a cube, one on each. At each corner of the cube, the product of the numbers on the faces that meet at the vertex is written. The sum of the numbers written on the corners is 2004. If T denotes the sum of the numbers on all the faces, find the possible values of T.

Suppose the real line is decomposed into two uncountable Borel sets. Prove that a suitable translated copy of the first set intersects the second in an uncountable set.

[b]p1.[/b] Evaluate $6^4 +5^4 +3^4 +2^4$. [b]p2.[/b] What digit is most frequent between $1$ and $1000$ inclusive? [b]p3.[/b] Let $n = gcd \, (2^2 \cdot 3^3 \cdot 4^4,2^4 \cdot 3^3 \cdot 4^2)$. Find the number of positive integer factors of $n$. [b]p4.[/b] Suppose $p$ and $q$ are prime numbers such that $13p +5q = 91$. Find $p +q$. [b]p5.[/b] Let $x = (5^3 -5)(4^3 -4)(3^3 -3)(2^3 -2)(1^3 -1)$. Evaluate $2018^x$ . [b]p6.[/b] Liszt the lister lists all $24$ four-digit integers that contain each of the digits $1,2,3,4$ exactly once in increasing order. What is the sum of the $20$th and $18$th numbers on Liszt’s list? [b]p7.[/b] Square $ABCD$ has center $O$. Suppose $M$ is the midpoint of $AB$ and $OM +1 =OA$. Find the area of square $ABCD$. [b]p8.[/b] How many positive $4$-digit integers have at most $3$ distinct digits? [b]p9.[/b] Find the sumof all distinct integers obtained by placing $+$ and $-$ signs in the following spaces $$2\_3\_4\_5$$ [b]p10.[/b] In triangle $ABC$, $\angle A = 2\angle B$. Let $I$ be the intersection of the angle bisectors of $B$ and $C$. Given that $AB = 12$, $BC = 14$,and $C A = 9$, find $AI$ . [b]p11.[/b] You have a $3\times 3\times 3$ cube in front of you. You are given a knife to cut the cube and you are allowed to move the pieces after each cut before cutting it again. What is the minimumnumber of cuts you need tomake in order to cut the cube into $27$ $1\times 1\times 1$ cubes? p12. How many ways can you choose $3$ distinct numbers fromthe set $\{1,2,3,...,20\}$ to create a geometric sequence? [b]p13.[/b] Find the sum of all multiples of $12$ that are less than $10^4$ and contain only $0$ and $4$ as digits. [b]p14.[/b] What is the smallest positive integer that has a different number of digits in each base from $2$ to $5$? [b]p15.[/b] Given $3$ real numbers $(a,b,c)$ such that $$\frac{a}{b +c}=\frac{b}{3a+3c}=\frac{c}{a+3b},$$ find all possible values of $\frac{a +b}{c}$. [b]p16.[/b] Let S be the set of lattice points $(x, y, z)$ in $R^3$ satisfying $0 \le x, y, z \le 2$. How many distinct triangles exist with all three vertices in $S$? [b]p17.[/b] Let $\oplus$ be an operator such that for any $2$ real numbers $a$ and $b$, $a \oplus b = 20ab -4a -4b +1$. Evaluate $$\frac{1}{10} \oplus \frac19 \oplus \frac18 \oplus \frac17 \oplus \frac16 \oplus \frac15 \oplus \frac14 \oplus \frac13 \oplus \frac12 \oplus 1.$$ [b]p18.[/b] A function $f :N \to N$ satisfies $f ( f (x)) = x$ and $f (2f (2x +16)) = f \left(\frac{1}{x+8} \right)$ for all positive integers $x$. Find $f (2018)$. [b]p19.[/b] There exists an integer divisor $d$ of $240100490001$ such that $490000 < d < 491000$. Find $d$. [b]p20.[/b] Let $a$ and $b$ be not necessarily distinct positive integers chosen independently and uniformly at random from the set $\{1,2, 3, ... ,511,512\}$. Let $x = \frac{a}{b}$ . Find the probability that $(-1)^x$ is a real number. [b]p21[/b]. In $\vartriangle ABC$ we have $AB = 4$, $BC = 6$, and $\angle ABC = 135^o$. $\angle ABC$ is trisected by rays $B_1$ and $B_2$. Ray $B_1$ intersects side $C A$ at point $F$, and ray $B_2$ intersects side $C A$ at point $G$. What is the area of $\vartriangle BFG$? [b]p22.[/b] A level number is a number which can be expressed as $x \cdot \lfloor x \rfloor \cdot \lceil x \rceil$ where $x$ is a real number. Find the number of positive integers less than or equal to $1000$ which are also level numbers. [b]p23.[/b] Triangle $\vartriangle ABC$ has sidelengths $AB = 13$, $BC = 14$, $C A = 15$ and circumcenter $O$. Let $D$ be the intersection of $AO$ and $BC$. Compute $BD/DC$. [b]p24.[/b] Let $f (x) = x^4 -3x^3 +2x^2 +5x -4$ be a quartic polynomial with roots $a,b,c,d$. Compute $$\left(a+1 +\frac{1}{a} \right)\left(b+1 +\frac{1}{b} \right)\left(c+1 +\frac{1}{c} \right)\left(d+1 +\frac{1}{d} \right).$$ [b]p25.[/b] Triangle $\vartriangle ABC$ has centroid $G$ and circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $BC$. If $AD = 2018$, $BD =20$, and $CD = 18$, find the area of triangle $\vartriangle DOG$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$, $k$ be positive integers such that $n$ is not divisible by $3$ and $k\ge n$. Prove that there exists a positive integer m which is divisible by $n$ and the sum of its digits in the decimal representation is $k$.

The positive integers from 1 to \(n\) are arranged in a sequence, initially in ascending order. In one move, we can swap the positions of two of the numbers, provided they share a common divisor greater than 1. Let \(s_n\) be the number of sequences that can be obtained with a finite number of moves. Prove that \(s_n = a_n!\), where the sequence of positive integers \((a_n)_{n\geq 1}\) is such that for any \(\delta > 0\), there exists an integer \(N\), for which for all \(n\geq N\), the following is true: \[ n - \left(\frac{1}{2}+\delta\right)\frac{n}{\log n} < a_n < n - \left(\frac{1}{2}-\delta\right)\frac{n}{\log n}. \]

Assume that $a,b$ are integers and $n$ is a natural number. $2^na+b$ is a perfect square for every $n$.Prove that $a=0$.

The points of space are painted in two colors. Prove that there is a tetrahedron such that all its vertices and its centroid are of the same color.

Define $f(x) = |x-1|$. Determine the number of real numbers $x$ such that $f(f(\cdots f(f(x))\cdots )) = 0$, where there are $2018$ $f$'s in the equation. [i]Proposed by Yannick Yao

Functions $f$ and $g$ are defined so that $f(1) = 4$, $g(1) = 9$, and for each integer $n \ge 1$, $f(n+1) = 2f(n) + 3g(n) + 2n $ and $g(n+1) = 2g(n) + 3 f(n) + 5$. Find $f(2005) - g(2005)$.

Show that for a triangle we have \[ \max \{am_a,bm_b,cm_c\} \leq sR \] where $m_a$ denotes the length of median of side $BC$ and $s$ is half of the perimeter of the triangle.

A ship sails $ 10$ miles in a straight line from $ A$ to $ B$, turns through an angle between $ 45^{\circ}$ and $ 60^{\circ}$, and then sails another $ 20$ miles to $ C$. Let $ AC$ be measured in miles. Which of the following intervals contains $ AC^2$? [asy]unitsize(2mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair B=(0,0), A=(-10,0), C=20*dir(50); draw(A--B--C); draw(A--C,linetype("4 4")); dot(A); dot(B); dot(C); label("$10$",midpoint(A--B),S); label("$20$",midpoint(B--C),SE); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE);[/asy]$ \textbf{(A)}\ [400,500] \qquad \textbf{(B)}\ [500,600] \qquad \textbf{(C)}\ [600,700] \qquad \textbf{(D)}\ [700,800]$ $ \textbf{(E)}\ [800,900]$