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]$

Given integer $n\geq 2$ and positive real number $a$, find the smallest real number $M=M(n,a)$, such that for any positive real numbers $x_1,x_2,\cdots,x_n$ with $x_1 x_2\cdots x_n=1$, the following inequality holds: \[\sum_{i=1}^n \frac {1}{a+S-x_i}\leq M\] where $S=\sum_{i=1}^n x_i$.

Let $ABCD$ be a convex quadrilateral inscribed in a circle with center $O$ which does not lie on either diagonal. If the circumcentre of triangle $AOC$ lies on the line $BD$, prove that the circumcentre of triangle $BOD$ lies on the line $AC$.

Inside the triangle $ ABC $ a point $ M $ is given. The line $ BM $ meets the side $ AC $ at $ N $. The point $ K $ is symmetrical to $ M $ with respect to $ AC $. The line $ BK $ meets $ AC $ at $ P $. If $ \angle AMP = \angle CMN $, prove that $ \angle ABP=\angle CBN $.

Let $\displaystyle{n \geqslant 3}$ and let $\displaystyle{{x_1},{x_2},...,{x_n}}$ be nonnegative real numbers. Define $\displaystyle{A = \sum\limits_{i = 1}^n {{x_i}} ,B = \sum\limits_{i = 1}^n {x_i^2} ,C = \sum\limits_{i = 1}^n {x_i^3} }$. Prove that: \[\displaystyle{\left( {n + 1} \right){A^2}B + \left( {n - 2} \right){B^2} \geqslant {A^4} + \left( {2n - 2} \right)AC}.\] [i]Proposed by Géza Kós, Eötvös University, Budapest.[/i]

Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.

In the tetrahedron $OABC$ we denote by $\alpha,\beta,\gamma$ the measures of the angles $\angle BOC,\angle COA,$ and $\angle AOB$, respectively. Prove the inequality \[\cos^2\alpha+\cos^2\beta+\cos^2\gamma<1+2\cos\alpha\cos\beta\cos\gamma \]

Let $\triangle ABC$ be an acute-angled triangle with $AB<AC$ and let $(c)$ be its circumcircle with center $O$. Let $M$ be the midpoint of $BC$. The line $AM$ meets the circle $(c)$ again at the point $D$. The circumcircle $(c_1)$ of triangle $\triangle MDC$ intersects the line $AC$ at the points $C$ and $I$, and the circumcircle $(c_2)$ of $\triangle AMI$ intersects the line $AB$ at the points $A$ and $Z$. If $N$ is the foot of the perpendicular from $B$ on $AC$, and $P$ is the second point of intersection of $ZN$ with $(c_2)$, prove that the quadrilateral with vertices the points $N, P, I$ and $M$ is a parallelogram.

Let $ABC$ be an isosceles triangle with $AB=AC$. Circle $\Gamma$ lies outside $ABC$ and touches line $AC$ at point $C$. The point $D$ is chosen on circle $\Gamma$ such that the circumscribed circle of the triangle $ABD$ touches externally circle $\Gamma$. The segment $AD$ intersects circle $\Gamma$ at a point $E$ other than $D$. Prove that $BE$ is tangent to circle $\Gamma$ .

If $(2^x - 4^x) + (2^{-x} - 4^{-x}) = 3$, find the numerical value of the expression $$(8^x + 3\cdot 2^x) + (8^{-x} + 3\cdot 2^{-x}).$$

Find all relatively distinct integers $m, n, p\in \mathbb{Z}_{\ne 0}$ such that the polynomial $$F(x) = x(x - m)(x - n)(x - p) + 1$$is reducible in $\mathbb{Z}[x].$

Determine all real numbers $x>1$ such that \[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \] for any positive integer $n$.

A chessboard is covered with $32$ dominos. Each domino covers two adjacent squares. Show that the number of horizontal dominos with a white square on the left equals the number with a white square on the right.

A lamp is situated at point $A$ and shines inside the cube. A (massive) square is hung on the midpoints of the 4 vertical faces. What's the area of its shadow? [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=285[/img]