Suppose that $p, q \in \mathbb{N}$ satisfy the inequality \[\exp(1)\cdot( \sqrt{p+q}-\sqrt{q})^{2}<1.\] Show that $\ln \left(1+\frac{p}{q}\right)$ is irrational.

Construct a right triangle with given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.

Let $T = \{1,2,...,10\}$. Find the number of bijective functions $f : T\to T$ that satises the following for all $x \in T$: $f(f(x)) = x$ $|f(x) - x| \ge 2$

If $\log_MN=\log_NM$, $M\ne N$, $MN>0$, $M\ne 1$, $N\ne 1$, then $MN$ equals: $\text{(A)} \ \frac12 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 10 \qquad \text{(E)} \ \text{a number greater than 2 and less than 10}$

Find all pairs of positive integers $m, n \ge 3$ for which there exist infinitely many positive integers $a$ such that \[\frac{a^{m}+a-1}{a^{n}+a^{2}-1}\] is itself an integer.

The following isosceles trapezoid consists of equal equilateral triangles with side length $1$. The side $A_1E$ has length $3$ while the larger base $A_1A_n$ has length $n-1$. Starting from the point $A_1$ we move along the segments which are oriented to the right and up(obliquely right or left). Calculate (in terms of $n$ or not) the number of all possible paths we can follow, in order to arrive at points $B,\Gamma,\Delta, E$, if $n$ is an integer greater than $3$. [color=#00CCA7][Need image][/color]

Find all integers $a, m, n, k,$ such that $(a^m+1)(a^n-1)=15^k.$

[u]Round 5[/u] [b]p13.[/b] Two closed disks of radius $\sqrt2$ are drawn centered at the points $(1,0)$ and $(-1, 0)$. Let P be the region belonging to both disks. Two congruent non-intersecting open disks of radius $r$ have all of their points in $P$ . Find the maximum possible value of $r$ . [b]p14.[/b] A rectangle has positive integer side lengths. The sum of the numerical values of its perimeter and area is $2017$. Find the perimeter of the rectangle. [b]p15.[/b] Find all ordered triples of real numbers $(a,b,c)$ which satisfy $$a +b +c = 6$$ $$a \cdot (b +c) = 6$$ $$(a +b) \cdot c = 6$$ [u]Round 6[/u] [b]p16.[/b] A four digit positive integer is called confused if it is written using the digits $2$, $0$, $1$, and $7$ in some order, each exactly one. For example, the numbers $7210$ and $2017$ are confused. Find the sum of all confused numbers. [b]p17.[/b] Suppose $\vartriangle ABC$ is a right triangle with a right angle at $A$. Let $D$ be a point on segment $BC$ such that $\angle BAD = \angle CAD$. Suppose that $AB = 20$ and $AC = 17$. Compute $AD$. [b]p18.[/b] Let $x$ be a real number. Find the minimum possible positive value of $\frac{|x -20|+|x -17|}{x}$. [u]Round 7[/u] [b]p19.[/b] Find the sum of all real numbers $0 < x < 1$ that satisfy $\{2017x\} = \{x\}$. [b]p20.[/b] Let $a_1,a_2, ,,, ,a_{10}$ be real numbers which sum to $20$ and satisfy $\{a_i\} <0.5$ for $1 \le i\le 10$. Find the sum of all possible values of $\sum_{ 1 \le i <j\le 10} \lfloor a_i +a_j \rfloor .$ Here, $\lfloor x \rfloor$ denotes the greatest integer $x_0$ such that $x_0 \le x$ and $\{x\} =x -\lfloor x \rfloor$. [b]p21.[/b] Compute the remainder when $20^{2017}$ is divided by $17$. [u]Round 8[/u] [b]p22.[/b] Let $\vartriangle ABC$ be a triangle with a right angle at $B$. Additionally, letM be the midpoint of $AC$. Suppose the circumcircle of $\vartriangle BCM$ intersects segment $AB$ at a point $P \ne B$. If $CP = 20$ and $BP = 17$, compute $AC$. [b]p23.[/b] Two vertices on a cube are called neighbors if they are distinct endpoints of the same edge. On a cube, how many ways can a nonempty subset $S$ of the vertices be chosen such that for any vertex $v \in S$, at least two of the three neighbors of $v$ are also in $S$? Reflections and rotations are considered distinct. [b]p24.[/b] Let $x$ be a real number such that $x +\sqrt[4]{5-x^4}=2$. Find all possible values of $x\sqrt[4]{5-x^4}$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158491p28715220]here[/url].and 9-12 [url=https://artofproblemsolving.com/community/c3h3162362p28764144]here[/url] Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What is the ones digit of \[222{,}222-22{,}222-2{,}222-222-22-2?\] $\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad\textbf{(E) }8$

How many unordered pairs of edges of a given cube determine a plane? $\textbf{(A) } 21 \qquad\textbf{(B) } 28 \qquad\textbf{(C) } 36 \qquad\textbf{(D) } 42 \qquad\textbf{(E) } 66$

Several distinct real numbers are written on a blackboard. Peter wants to make an expression such that its values are exactly these numbers. To make such an expression, he may use any real numbers, brackets, and usual signs $+$ , $-$ and $\times$. He may also use a special sign $\pm$: computing the values of the resulting expression, he chooses values $+$ or $-$ for every $\pm$ in all possible combinations. For instance, the expression $5 \pm 1$ results in $\{4, 6 \}$, and $(2 \pm 0.5) \pm 0.5$ results in $\{1, 2, 3 \}$. Can Pete construct such an expression: $a)$ If the numbers on the blackboard are $1, 2, 4$; $b)$ For any collection of $100$ distinct real numbers on a blackboard?

Let $s(n)$ be the number of 1's in the binary representation of $n$. Find the number of ordered pairs of integers $(a,b)$ with $0 \leq a < 64, 0 \leq b < 64$ and $s(a+b) = s(a) + s(b) - 1$. [i]Author:Anderson Wang[/i]

For $100$ straight lines on a plane, let $T$ be the set of all right-angled triangles bounded by some $3$ lines. Determine, with proof, the maximum value of $|T|$.

What is the sum of the distinct prime integer divisors of $2016$? $\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }16\qquad\textbf{(D) }49\qquad \textbf{(E) }63$

Find $p+r$ if $p$ and $q$ are primes and $r$ is an integer such that \[ \left( r^2 + pr + 1 \right) \cdot \left( r^2 + \left( p^2 - q \right) r - p \right) = pq. \]

Does there exist a piecewise linear, continuous, surjective mapping $f: [0,1]\to [0,1]$ such that $f(0)=f(1)=0$, and for all positive integer $n$, $$2.0001^{(n-10)} <P_n(f)<2.9999^{(n+10)}$$holds, where $P_n(f)$ is the number of points $x$ such that $\underbrace{f(\dotsc f}_n(x)\dotsc )=x$?

Let $P(x)$ be a non-constant polynomial with integer coefficients and let $n$ be a positive integer. The sequence $a_0,a_1,\ldots$ is defined as follows: $a_0=n$ and $a_k=P(a_{k-1})$ for all positive integers $k.$ Assume that for every positive integer $b$ the sequence contains a $b$th power of an integer greater than $1.$ Show that $P(x)$ is linear.

Find the minimum possible sum of lengths of edges of a prism all of whose edges are tangent of a unit sphere. [Muller-Pfeiffer].

Let $ y_1(x)$ be an arbitrary, continuous, positive function on $ [0,A]$, where $ A$ is an arbitrary positive number. Let \[ y_{n+1}=2 \int_0^x \sqrt{y_n(t)}dt \;(n=1,2,...)\ .\] Prove that the functions $ y_n(x)$ converge to the function $ y=x^2$ uniformly on $ [0,A]$.

Consider the sequence of rational numbers defined by $x_1=\frac{4}{3}$, and $x_{n+1}=\frac{x_n^2}{x_n^2-x_n+1}$. Show that the nu,erator of the lowest term expression of each sum $x_1+x_2+...+x_k$ is a perfect square.

If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible? $\textbf{(A) }n$ and $m$ are even $\qquad\textbf{(B) }n$ and $m$ are odd $\qquad\textbf{(C) }n+m$ is even $\qquad\textbf{(D) }n+m$ is odd $\qquad \textbf{(E) }$ none of these are impossible

Find all solutions $x,y \in \mathbb{Z}$, $x,y \geq 0$, to the equation \[ 1 + 3^x = 2^y. \]

Find all triples $(a,b,c)$ of integers for which the equation \[x^3-a^2x^2+b^2x-ab+3c=0\] has three distinct integer roots $x_1,x_2,x_3$ which are pairwise coprime.

In the figure, $ABCD$ is an isosceles trapezoid with side lengths $AD = BC = 5, AB = 4,$ and $DC = 10$. The point $C$ is on $\overline{DF}$ and $B$ is the midpoint of hypotenuse $\overline{DE}$ in the right triangle $DEF$. Then $CF =$ [asy] size(200); defaultpen(fontsize(10)); pair D=origin, A=(3,4), B=(7,4), C=(10,0), E=(14,8), F=(14,0); draw(B--C--F--E--B--A--D--B^^C--D, linewidth(0.7)); dot(A^^B^^C^^D^^E^^F); pair point=(5,3); label("$A$", A, N); label("$B$", B, N); label("$C$", C, S); label("$D$", D, S); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); markscalefactor=0.05; draw(rightanglemark(E,F,D), linewidth(0.7));[/asy] $\text{(A)} \ 3.25 \qquad \text{(B)} \ 3.5 \qquad \text{(C)} \ 3.75 \qquad \text{(D)} \ 4.0 \qquad \text{(E)} \ 4.25$

How many pairs of real numbers $(x,y)$ are there such that $x^4+y^4 + 2x^2y + 2xy^2+ 2 = x^2 + y^2 + 2x + 2y$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 2 $