Show that, if $a,b,...,n$ are distinct natural numbers, none divisible by any primes greater than $3$, then $$\frac{1}{a}+\frac{1}{b}+...+ \frac{1}{n}< 3$$

Show that there is no positive integer $n$ such that $n + k^2$ is a perfect square for at least $n$ positive integer values of $k$.

For which real values of $ m$ are the simultaneous equations \begin{align*} y &= mx + 3 \\ y &= (2m - 1)x + 4 \end{align*} satisfied by at least one pair of real numbers $ (x,y)$? $ \textbf{(A)}\ \text{all } m \qquad \textbf{(B)}\ \text{all } m \neq 0 \qquad \textbf{(C)}\ \text{all } m \neq 1/2 \qquad \textbf{(D)}\ \text{all } m \neq 1 \qquad$ $ \textbf{(E)}\ \text{no values of } m$

Let $C$ be a circle, and let $A, B, P$ be three points on $C$. Let $L_A$ and $L_B$ be the tangent lines to $C$ passing through $A$ and $B$, respectively. Let $a$ and $b$ be the distances from $P$ to $L_A$ and $L_B$, respectively, and let $c$ be the distance from $P$ to the chord of $C$ determined by $A$ and $B$. Prove that $c^2 = a \cdot b$.

Divide the plane into an infinite square grid by drawing all the lines $x=m$ and $y=n$ for $m,n \in \mathbb Z$. Next, if a square's upper-right corner has both coordinates even, color it black; otherwise, color it white (in this way, exactly $1/4$ of the squares are black and no two black squares are adjacent). Let $r$ and $s$ be odd integers, and let $(x,y)$ be a point in the interior of any white square such that $rx-sy$ is irrational. Shoot a laser out of this point with slope $r/s$; lasers pass through white squares and reflect off black squares. Prove that the path of this laser will form a closed loop.

\[\int_e^{e^2} (\log(x))^{\log(x)}(2+\log(\log(x)))\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

If $T(n)$ is the numbers of triangles with integers sizes(not congruent with each other) with it´s perimeter is equal to $n$, prove that: $$T(2012)<T(2015)$$ $$T(2013)=T(2016)$$

Let $S$ be a set of integers such that [list][*] there exist $a, b \in S$ with $\gcd(a, b)=\gcd(a-2,b-2)=1$, [*] if $x,y\in S$, then $x^2 -y\in S$.[/list] Prove that $S=\mathbb{Z}$.

Suppose that $a,b,c$ are real numbers such that $a < b < c$ and $a^3-3a+1=b^3-3b+1=c^3-3c+1=0$. Then $\frac1{a^2+b}+\frac1{b^2+c}+\frac1{c^2+a}$ can be written as $\frac pq$ for relatively prime positive integers $p$ and $q$. Find $100p+q$. [i]Proposed by Michael Ren[/i]

A floor is tiled with equilateral triangles of side length $1$, as shown. If you drop a needle of length $2$ somewhere on the floor , what is the largest number of triangles it could end up intersecting? (Only count the triangles whose interiors are met by the needle --- touching along edges or at corners doesn't qualify.) [img]https://cdn.artofproblemsolving.com/attachments/5/6/e7555c22ffe890b46a3ebdbda2169d23e43700.png[/img]

Let $a$ and $b$ be positive integers such that the number $b^2 + (b +1)^2 +...+ (b + a)^2-3$ is multiple of $5$ and $a + b$ is odd. Calculate the digit of the units of the number $a + b$ written in decimal notation.

In the plane points $A,B,C$ are marked in blue and points $P,Q$ are marked in red (no 3 marked points lie on a line, and no 4 marked points lie on a circle). A circle is called [b]separating[/b] if all points of one color are inside it, and all points of the other color are outside of it. Denote by $O$ the circumcenter of $ABC$ and by $R$ the circumradius of $ABC$. Prove that [b]exactly one[/b] of the following holds: [list] [*] There exists a separating circle; [*] There exists a point $X$ on the segment $PQ$ which also lies inside the triangle $ABC$, for which $PX\cdot XQ = R^2-OX^2$.

Let $k$ be an even positive integer and define a sequence $<x_n>$ by \[ x_1= 1 , x_{n+1} = k^{x_n} +1. \] Show that $x_n ^2$ divides $x_{n-1}x_{n+1}$ for each $n \geq 2.$

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

Let $ABCD$ be a unit square. Let $Q_1$ be the midpoint of $\overline{CD}$. For $i=1,2,\dots,$ let $P_i$ be the intersection of $\overline{AQ_i}$ and $\overline{BD}$, and let $Q_{i+1}$ be the foot of the perpendicular from $P_i$ to $\overline{CD}$. What is $$\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_i P_i \, ?$$ $\textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{1}{3} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ 1$

On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order. "Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes. "Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes. "Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes. How many pieces of candy in all did the principal give to the children who always tell the truth? $\textbf{(A) }7\qquad\textbf{(B) }12\qquad\textbf{(C) }21\qquad\textbf{(D) }27\qquad\textbf{(E) }31$

Let $x$ be in the interval $\left(0, \frac{\pi}{2}\right)$ such that $\sin x - \cos x = \frac12$ . Then $\sin^3 x + \cos^3 x = \frac{m\sqrt{p}}{n}$ , where $m, n$, and $p$ are relatively prime positive integers, and $p$ is not divisible by the square of any prime. Find $m + n + p$.

Alice and Bob are playing a guessing game. Bob is thinking of a number n of the form $2^a3^b$, where a and b are positive integers between $ 1$ and $2020$, inclusive. Each turn, Alice guess a number m, and Bob will tell her either $\gcd (m, n)$ or $lcm (m, n)$ (letting her know that he is saying that $gcd$ or $lcm$), as well as whether any of the respective powers match up in their prime factorization. In particular, if $m = n$, Bob will let Alice know this, and the game is over. Determine the smallest number $k$ so that Alice is always able to find $n$ within $k$ guesses, regardless of Bob’s number or choice of revealing either the $lcm$, or the $gcd$ .

Let $p(x)$ be a polynomial of degree strictly less than $100$ and such that it does not have $(x^3-x)$ as a factor. If $$\frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)}$$ for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$.

$a$, $b$, $c$ are real numbers. Show that $\min((a-b)^2,(b-c)^2,(c-a)^2)\leq \frac{a^2+b^2+c^2}{2}$

From a point $ M $ on the surface of a sphere, three mutually perpendicular chords $ MA $, $ MB $, $ MC $ are drawn. Prove that the segment joining the point $ M $ with the center of the sphere intersects the plane of the triangle $ ABC $ at the center of gravity of this triangle.

Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$

Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

Given a set $P$ of $n>100$ points on the plane such that no three of them are collinear, and a set $S$ of $20n$ distinct segments, each joining a pair of points from $P$. Prove that there exists a line not passing through a point from $P$ and intersecting at least $200$ segments from $S$.

A line through the vertex $A{}$ of the triangle $ABC{}$ which doesn't coincide with $AB{}$ or $AC{}$ intersectes the altitudes from $B{}$ and $C{}$ at $D{}$ and $E{}$ respectively. Let $F{}$ be the reflection of $D{}$ in $AB{}$ and $G{}$ be the reflection of $E{}$ in $AC{}.$ Prove that the circles $ABF{}$ and $ACG{}$ are tangent.