Let $f: N \to N$ be a function satisfying (a) $1\le f(x)-x \le 2019$ $\forall x \in N$ (b) $f(f(x))\equiv x$ (mod $2019$) $\forall x \in N$ Show that $\exists x \in N$ such that $f^k(x)=x+2019 k, \forall k \in N$

Let $ X_1,X_2,\ldots, X_n$ be independent, identically distributed, nonnegative random variables with a common continuous distribution function $ F$. Suppose in addition that the inverse of $ F$, the quantile function $ Q$, is also continuous and $ Q(0)=0$. Let $ 0=X_{0: n} \leq X_{1: n} \leq \ldots \leq X_{n: n}$ be the ordered sample from the above random variables. Prove that if $ EX_1$ is finite, then the random variable \[ \Delta = \sup_{0\leq y \leq 1} \left| \frac 1n \sum_{i=1}^{\lfloor ny \rfloor +1} (n+1-i)(X_{i: n}-X_{i-1: n})- \int_0^y (1-u)dQ(u) \right|\] tends to zero with probability one as $ n \rightarrow \infty$. [i]S. Csorgp, L. Horvath[/i]

[i]20 problems for 20 minutes.[/i] [b]p1.[/b] Determine how many digits the number $10^{10}$ has. [b]p2.[/b] Let $ABC$ be a triangle with $\angle ABC = 60^o$ and $\angle BCA = 70^o$. Compute $\angle CAB$ in degrees. [b]p3.[/b] Given that $x : y = 2012 : 2$ and $y : z = 1 : 2013$, compute $x : z$. Express your answer as a common fraction. [b]p4.[/b] Determine the smallest perfect square greater than $2400$. [b]p5.[/b] At $12:34$ and $12:43$, the time contains four consecutive digits. Find the next time after 12:43 that the time contains four consecutive digits on a 24-hour digital clock. [b]p6.[/b] Given that $ \sqrt{3^a \cdot 9^a \cdot 3^a} = 81^2$, compute $a$. [b]p7.[/b] Find the number of positive integers less than $8888$ that have a tens digit of $4$ and a units digit of $2$. [b]p8.[/b] Find the sum of the distinct prime divisors of $1 + 2012 + 2013 + 2011 \cdot 2013$. [b]p9.[/b] Albert wants to make $2\times 3$ wallet sized prints for his grandmother. Find the maximum possible number of prints Albert can make using one $4 \times 7$ sheet of paper. [b]p10.[/b] Let $ABC$ be an equilateral triangle, and let $D$ be a point inside $ABC$. Let $E$ be a point such that $ADE$ is an equilateral triangle and suppose that segments $DE$ and $AB$ intersect at point $F$. Given that $\angle CAD = 15^o$, compute $\angle DFB$ in degrees. [b]p11.[/b] A palindrome is a number that reads the same forwards and backwards; for example, $1221$ is a palindrome. An almost-palindrome is a number that is not a palindrome but whose first and last digits are equal; for example, $1231$ and $1311$ are an almost-palindromes, but $1221$ is not. Compute the number of $4$-digit almost-palindromes. [b]p12.[/b] Determine the smallest positive integer $n$ such that the sum of the digits of $11^n$ is not $2^n$. [b]p13.[/b] Determine the minimum number of breaks needed to divide an $8\times 4$ bar of chocolate into $1\times 1 $pieces. (When a bar is broken into pieces, it is permitted to rotate some of the pieces, stack some of the pieces, and break any set of pieces along a vertical plane simultaneously.) [b]p14.[/b] A particle starts moving on the number line at a time $t = 0$. Its position on the number line, as a function of time, is $$x = (t-2012)^2 -2012(t-2012)-2013.$$ Find the number of positive integer values of $t$ at which time the particle lies in the negative half of the number line (strictly to the left of $0$). [b]p15.[/b] Let $A$ be a vertex of a unit cube and let $B$,$C$, and $D$ be the vertices adjacent to A. The tetrahedron $ABCD$ is cut off the cube. Determine the surface area of the remaining solid. [b]p16.[/b] In equilateral triangle $ABC$, points $P$ and $R$ lie on segment $AB$, points $I$ and $M$ lie on segment $BC$, and points $E$ and $S$ lie on segment $CA$ such that $PRIMES$ is a equiangular hexagon. Given that $AB = 11$, $PS = 2$, $RI = 3$, and $ME = 5$, compute the area of hexagon $PRIMES$. [b]p17.[/b] Find the smallest odd positive integer with an odd number of positive integer factors, an odd number of distinct prime factors, and an odd number of perfect square factors. [b]p18.[/b] Fresh Mann thinks that the expressions $2\sqrt{x^2 -4} $and $2(\sqrt{x^2} -\sqrt4)$ are equivalent to each other, but the two expressions are not equal to each other for most real numbers $x$. Find all real numbers $x$ such that $2\sqrt{x^2 -4} = 2(\sqrt{x^2} -\sqrt4)$. [b]p19.[/b] Let $m$ be the positive integer such that a $3 \times 3$ chessboard can be tiled by at most $m$ pairwise incongruent rectangles with integer side lengths. If rotations and reflections of tilings are considered distinct, suppose that there are $n$ ways to tile the chessboard with $m$ pairwise incongruent rectangles with integer side lengths. Find the product $mn$. [b]p20.[/b] Let $ABC$ be a triangle with $AB = 4$, $BC = 5$, and $CA = 6$. A triangle $XY Z$ is said to be friendly if it intersects triangle $ABC$ and it is a translation of triangle $ABC$. Let $S$ be the set of points in the plane that are inside some friendly triangle. Compute the ratio of the area of $S$ to the area of triangle $ABC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Chords $AB$ and $CD$ of a circle intersect inside the circle at $P$ such that $AP = 8, P B = 6, P D = 3$ and $\angle AP C = 60^{\circ}$. Find the area of $\bigtriangleup AP C$.

Javiera and Claudio play on a board consisting of a row with $2019$ cells. Claudio starts by placing a token anywhere on the board. Next Javiera says a natural number $k$, $1 \le k \le n$ and Claudio must move the token to the right or to the left at your choice $k$ squares and so on. Javiera wins if she manages to remove the piece that Claudio moves from the board. Determine the smallest $n$ such that Javiera always wins after a finite number of moves.

Let the $f:\mathbb{N}\rightarrow\mathbb{N}$ be the function such that (i) For all positive integers $n,$ $f(n+f(n))=f(n)$ (ii) $f(n_o)=1$ for some $n_0$ Prove that $f(n)\equiv 1.$

Compute the number of positive integers $n\le1000$ such that $\text{lcm}(n,9)$ is a perfect square. (Recall that $\text{lcm}$ denotes the least common multiple.)

Let $f(x)$ and $g(x)$ be degree $n$ polynomials, and $x_0,x_1,\ldots,x_n$ be real numbers such that $$f(x_0)=g(x_0),f'(x_1)=g'(x_1),f''(x_2)=g''(x_2),\ldots,f^{(n)}(x_n)=g^{(n)}(x_n).$$Prove that $f(x)=g(x)$ for all $x$.

Let $n$ be a positive integer greater than $1$. Evaluate the following summation: \[ \sum_{k=0}^{n-1} \frac{1}{1 + 8 \sin^2 \left( \frac{k \pi}{n} \right)}. \]

A $3$ by $3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant.

Let $m, n$ be positive integers. Let $S(n,m)$ be the number of sequences of length $n$ and consisting of $0$ and $1$ in which there exists a $0$ in any consecutive $m$ digits. Prove that \[S(2015n,n).S(2015m,m)\ge S(2015n,m).S(2015m,n)\]

In a sports tournament involving $N$ teams, each team plays every other team exactly one. At the end of every match, the winning team gets $1$ point and losing team gets $0$ points. At the end of the tournament, the total points received by the individual teams are arranged in decreasing order as follows: \[x_1 \ge x_2 \ge \cdots \ge x_N . \] Prove that for any $1\le k \le N$, \[\frac{N - k}{2} \le x_k \le N - \frac{k+1}{2}\]

The cricles $\Gamma_1$ and $\Gamma_2$ intersect in $ M $ and $N.$ A line that goes through $ M $ intersects the cricles $\Gamma_1$ and $\Gamma_2$ in $ A$ and $B$, such that $M\in(AB)$. The bisector of angle $ AMN $ intersects the circle $\Gamma_1$ in $D,$ and the bisector of angle $BMN$ intersects the circle $\Gamma_2$ in $C.$ Prove that the circle with diameter $CD$ splits the segment $AB$ in half.

Let \(m\) be a positive integer. Find, in terms of \(m\), all polynomials \(P(x)\) with integer coefficients such that for every integer \(n\), there exists an integer \(k\) such that \(P(k)=n^m\). [i]Proposed by Raymond Feng[/i]

[b]p1.[/b] Trung took five tests this semester. For his first three tests, his average was $60$, and for the fourth test he earned a $50$. What must he have earned on his fifth test if his final average for all five tests was exactly $60$? [b]p2.[/b] Find the number of pairs of integers $(a, b)$ such that $20a + 16b = 2016 - ab$. [b]p3.[/b] Let $f : N \to N$ be a strictly increasing function with $f(1) = 2016$ and $f(2t) = f(t) + t$ for all $t \in N$. Find $f(2016)$. [b]p4.[/b] Circles of radius $7$, $7$, $18$, and $r$ are mutually externally tangent, where $r = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p5.[/b] A point is chosen at random from within the circumcircle of a triangle with angles $45^o$, $75^o$, $60^o$. What is the probability that the point is closer to the vertex with an angle of $45^o$ than either of the two other vertices? [b]p6.[/b] Find the largest positive integer $a$ less than $100$ such that for some positive integer $b$, $a - b$ is a prime number and $ab$ is a perfect square. [b]p7.[/b] There is a set of $6$ parallel lines and another set of six parallel lines, where these two sets of lines are not parallel with each other. If Blythe adds $6$ more lines, not necessarily parallel with each other, find the maximum number of triangles that could be made. [b]p8.[/b] Triangle $ABC$ has sides $AB = 5$, $AC = 4$, and $BC = 3$. Let $O$ be any arbitrary point inside $ABC$, and $D \in BC$, $E \in AC$, $F \in AB$, such that $OD \perp BC$, $OE \perp AC$, $OF \perp AB$. Find the minimum value of $OD^2 + OE^2 + OF^2$. [b]p9.[/b] Find the root with the largest real part to $x^4-3x^3+3x+1 = 0$ over the complex numbers. [b]p10.[/b] Tony has a board with $2$ rows and $4$ columns. Tony will use $8$ numbers from $1$ to $8$ to fill in this board, each number in exactly one entry. Let array $(a_1,..., a_4)$ be the first row of the board and array $(b_1,..., b_4)$ be the second row of the board. Let $F =\sum^{4}_{i=1}|a_i - b_i|$, calculate the average value of $F$ across all possible ways to fill in. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle, and $M$ an interior point such that $\angle MAB=10^\circ$, $\angle MBA=20^\circ$, $\angle MAC=40^\circ$ and $\angle MCA=30^\circ$. Prove that the triangle is isosceles.

A container is filled with a total of $51$ red and white balls and has at least $1$ red ball and $1$ white ball. The probability of picking up $3$ red balls and $1$ white ball, without replacement, is equivalent to the probability of picking up $1$ red ball and $2$ white balls, without replacement. Compute the original number of red balls in the container.

Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.

A positive natural number $n$ is said to be [i]comico[/i] if its prime factorization is $n = p_1p_2...p_k$, with $k\ge 3$, and also the primes $p_1,..., p_k$ they fulfill that $p_1 + p_2 = c^2_1$ $p_1 + p_2 + p_3 = c^2_2$ $...$ $p_1 + p_2 + ...+ p_n = c^2_{n-1}$ where $c_1, c_2, ..., c_{n-1}$ are positive integers where $c_1$ is not divisible by $7$. Find all comico numbers less than $10,000$.

Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedron is exactly edge $AB$.

Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$, \[a^n + a^{n-1} +
\cdots
+ a^1 + 1 \mid a^{n!} + a^{(n-1)!} +
\cdots

+ a^{1!} + 1.\] [i]Proposed by Milos Milosavljevic[/i]

Given that $x$ and $y$ are positive real numbers such that $\frac{5}{x}=\frac{y}{13}=\frac{x}{y}$, find the value of $x^3 + y^3$. Proposed by Ephram Chun

Let $ABC$ be a triangle, and let $D$ and $D_1$ be points on segment $BC$ such that $BD = CD_1$. Construct point $E$ such that $EC\perp BC$ and $ED\perp AC$. Similarly, construct point $F$ such that $FB\perp BC$ and $FD\perp AB$. Prove that $EF\perp AD_1$.

$n$ numbers are written down along the circumference. Their sum equals to zero, and one of them equals $1$. a) Prove that there are two neighbours with their difference not less than $n/4$. b) Prove that there is a number that differs from the arithmetic mean of its two neighbours not less than on $8/(n^2)$. c) Try to improve the previous estimation, i.e what number can be used instead of $8$? d) Prove that for $n=30$ there is a number that differs from the arithmetic mean of its two neighbours not less than on $2/113$, give an example of such $30$ numbers along the circumference, that not a single number differs from the arithmetic mean of its two neighbours more than on $2/113$.

Find the number of all real functions $f$ which map the sum of $n$ elements into the sum of their images, such that $f^{n-1}$ is a constant function and $f^{n-2}$ is not. Here $f^0(x) = x$ and $f^k = f \circ f^{k-1}$ for $k \ge 1$.