Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$, then Todd must say the next two numbers ($2$ and $3$), then Tucker must say the next three numbers ($4$, $5$, $6$), then Tadd must say the next four numbers ($7$, $8$, $9$, $10$), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number $10,000$ is reached. What is the $2019$th number said by Tadd? $ \textbf{(A)}\ 5743 \qquad\textbf{(B)}\ 5885 \qquad\textbf{(C)}\ 5979 \qquad\textbf{(D)}\ 6001 \qquad\textbf{(E)}\ 6011 $

One hundred concentric circles with radii $1, 2, 3, \dots, 100$ are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Evaluate $\int_0^1 \frac{x^2+5}{(x+1)^2(x-2)}dx.$ [i]2011 Doshisya University entrance exam/Science and Technology[/i]

A polynomial $p$ with real coefficients satisfies $p(x+1)-p(x)=x^{100}$ for all $x \in \mathbb{R}.$ Prove that $p(1-t) \ge p(t)$ for $0 \le t \le 1/2.$

Do there exist polynomials $f(x)$, $g(x)$ with real coefficients and a positive integer $k$ satisfying the following condition? (Here, the equation $x^2 = 0$ is considered to have $1$ distinct real roots. The equation $0 = 0$ has infinitely many distinct real roots.) For any real numbers $a, b$ with $(a,b) \neq (0,0)$, the number of distinct real roots of $a f(x) + b g(x) = 0$ is $k$.

Let $ABC$ be a scalene triangle and $\omega$ is your incircle. The sides $BC,CA$ and $AB$ are tangents to $\omega$ in $X,Y,Z$ respectively. Let $M$ be the midpoint of $BC$ and $D$ is the intersection point of $BC$ with the angle bisector of $\angle BAC$. Prove that $\angle BAX=\angle MAC$ if and only if $YZ$ passes by the midpoint of $AD$.

Find the radius of the inscribed circle of a triangle with sides of length $40$, $42$, and $58$.

Pete wrote down $21$ pairwise distinct positive integers, each not greater than $1,000,000$. For every pair $(a, b)$ of numbers written down by Pete, Nick wrote the number $$F(a;b)=a+b -\gcd(a;b)$$ on his piece of paper. Prove that one of Nick’s numbers differs from all of Pete’s numbers.

Let $P(x) = x^2 + ax + b$ be a quadratic polynomial. For how many pairs $(a, b)$ of positive integers where $a, b < 1000$ do the quadratics $P(x+1)$ and $P(x) + 1$ have at least one root in common?

Let $d(n)$ be the sum of positive integers divisors of number $n$ and $\phi(n)$ the quantity of integers in the interval $[0,n]$ such that these integers are coprime with $n$. For instance $d(6)=12$ and $\phi(7)=6$. Determine if the set of the integers $n$ such that, $d(n)\cdot \phi (n)$ is a perfect square, is finite or infinite set.

Let $f(z) = a_m z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$ be a polynomial of degree $n \geq 3$ with real coefficients.Suppose all roots of $f(z) =0$ lie in the half plane ${\ z \in \mathbb{C} : Re(z) < 0 \}}$. Prove that $a_k a_{k+3} < a_{k+1}a_{k+2}$ for $k = 0,1,2,3,.... n-3$

Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end? ${ \textbf{(A)}\ 62 \qquad\textbf{(B)}\ 82 \qquad\textbf{(C)}\ 83\qquad\textbf{(D}}\ 102\qquad\textbf{(E)}\ 103 $

$ABCD$ is a regular tetrahedron. The points $P,Q,R$ and $S$ lie outside this tetrahedron in such a way that $ABCP$, $ABDQ$, $ACDR$ and $BCDS$ are regular tetrahedra. Prove that the volume of the tetrahedron $PQRS$ is less than the sum of the volumes of $ABCP$,$ABDQ$,$ACDR$, $BCDS$ and $ABCD$.

Triangle $ABC$ has a right angle at $C$. If $\sin A = \frac{2}{3}$, then $\tan B$ is $ \textbf{(A)}\ \frac{3}{5}\qquad\textbf{(B)}\ \frac{\sqrt 5}{3}\qquad\textbf{(C)}\ \frac{2}{\sqrt 5}\qquad\textbf{(D)}\ \frac{\sqrt 5}{2}\qquad\textbf{(E)}\ \frac{5}{3} $

consider $n\geq 6$ points $x_1,x_2,\dots,x_n$ on the plane such that no three of them are colinear. We call graph with vertices $x_1,x_2,\dots,x_n$ a "road network" if it is connected, each edge is a line segment, and no two edges intersect each other at points other than the vertices. Prove that there are three road networks $G_1,G_2,G_3$ such that $G_i$ and $G_j$ don't have a common edge for $1\leq i,j\leq 3$. Proposed by Morteza Saghafian

Let $A$ be a subset of the irrational numbers such that the sum of any two distinct elements of it be a rational number. Prove that $A$ has two elements at most.

Triangle $\vartriangle ABC$ has $\angle ABC = \angle BCA = 45^o$ and $AB = 1$. Let $D$ be on $\overline{AC}$ such that $\angle ABD =30^o$. Let $\overleftrightarrow{BD}$ and the line through $A$ parallel to $\overleftrightarrow{BC}$ intersect at $E$. Compute the area of $\vartriangle ADE$.

The incircle of triangle $\triangle ABC$ is the unique inscribed circle that is internally tangent to the sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$. How many non-congruent right triangles with integer side lengths have incircles of radius $2015$?

Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that: $a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area. $b) a\cdot AP=b\cdot BP=c\cdot PC.$

Prove that for evey positive integer n, there exits a positive integer k such that $ 2^n | 19^k \minus{} 97$

Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$. Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.

For all sets $A$ of complex numbers, let $P(A)$ be the product of the elements of $A$. Let $S_z = \{1, 2, 9, 99, 999, \frac{1}{z},\frac{1}{z^2}\}$, let $T_z$ be the set of nonempty subsets of $S_z$ (including $S_z$), and let $f(z) = 1 + \sum_{s\in T_z} P(s)$. Suppose $f(z) = 6125000$ for some complex number $z$. Compute the product of all possible values of $z$.

What is the largest integer $n$ with no repeated digits that is relatively prime to $6$? Note that two numbers are considered relatively prime if they share no common factors besides $1$. [i]Proposed by Jacob Stavrianos[/i]

Let $ABC$ be a triangle with $AB=34,BC=25,$ and $CA=39$. Let $O,H,$ and $ \omega$ be the circumcenter, orthocenter, and circumcircle of $\triangle ABC$, respectively. Let line $AH$ meet $\omega$ a second time at $A_1$ and let the reflection of $H$ over the perpendicular bisector of $BC$ be $H_1$. Suppose the line through $O$ perpendicular to $A_1O$ meets $\omega$ at two points $Q$ and $R$ with $Q$ on minor arc $AC$ and $R$ on minor arc $AB$. Denote $\mathcal H$ as the hyperbola passing through $A,B,C,H,H_1$, and suppose $HO$ meets $\mathcal H$ again at $P$. Let $X,Y$ be points with $XH \parallel AR \parallel YP, XP \parallel AQ \parallel YH$. Let $P_1,P_2$ be points on the tangent to $\mathcal H$ at $P$ with $XP_1 \parallel OH \parallel YP_2$ and let $P_3,P_4$ be points on the tangent to $\mathcal H$ at $H$ with $XP_3 \parallel OH \parallel YP_4$. If $P_1P_4$ and $P_2P_3$ meet at $N$, and $ON$ may be written in the form $\frac{a}{b}$ where $a,b$ are positive coprime integers, find $100a+b$. [i]Proposed by Vincent Huang[/i]

Find all the functions $f:\mathbb{N}\to\mathbb{R}$ that satisfy \[ f(x+y)=f(x)+f(y) \] for all $x,y\in\mathbb{N}$ satisfying $10^6-\frac{1}{10^6} < \frac{x}{y} < 10^6+\frac{1}{10^6}$. Note: $\mathbb{N}$ denotes the set of positive integers and $\mathbb{R}$ denotes the set of real numbers.