What is the largest integer with distinct digits such that no two of its digits sum to a perfect square?

Let $ABC$ be a triangle with $BC>CA>AB$ and let $G$ be the centroid of the triangle. Prove that \[ \angle GCA+\angle GBC<\angle BAC<\angle GAC+\angle GBA . \] [i]Dinu Serbanescu[/i]

Let $r_1$, $r_2$, $r_3$ be the distinct real roots of $x^3-2019x^2-2020x+2021=0$. Prove that $r_1^3+r_2^3+r_3^3$ is an integer multiple of $3$.

Joshua likes to play with numbers and patterns. Joshua's favorite number is $6$ because it is the units digit of his birth year, $1996$. Part of the reason Joshua likes the number $6$ so much is that the powers of $6$ all have the same units digit as they grow from $6^1$: \begin{align*}6^1&=6,\\6^2&=36,\\6^3&=216,\\6^4&=1296,\\6^5&=7776,\\6^6&=46656,\\\vdots\end{align*} However, not all units digits remain constant when exponentiated in this way. One day Joshua asks Michael if there are simple patterns for the units digits when each one-digit integer is exponentiated in the manner above. Michael responds, "You tell me!" Joshua gives a disappointed look, but then Michael suggests that Joshua play around with some numbers and see what he can discover. "See if you can find the units digit of $2008^{2008}$," Michael challenges. After a little while, Joshua finds an answer which Michael confirms is correct. What is Joshua's correct answer (the units digit of $2008^{2008}$)?

Find the smallest value that the expression takes $x^4 + y^4 - x^2y - xy^2$, for positive numbers $x$ and $y$ satisfying $x + y \le 1$.

Claire takes a square piece of paper and folds it in half four times without unfolding, making an isosceles right triangle each time. After unfolding the paper to form a square again, the creases on the paper would look like

Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that \[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \] Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.

There are $19$ balls in a box, numbered $1$ through $19$. When we go out get that box without looking five different balls, which number has the largest probability of being the difference between the highest and lowest number drawn? Justify you reply .

Let $\theta$ and $p(p<1)$ ) be nonnegative real numbers. Suppose that $f:X\to Y$ is mapping with $f(0)=0$ and $$\left |\left| 2f\left (\frac{x+y}{2}\right )-f(x)-f(y) \right |\right|_Y \leq \theta\left (\left |\left |x\right |\right |_X^p +\left |\left |y\right |\right |_X^p \right )$$ for all $x$, $y\in \mathbb{Z}$ with $x\perp y$ where $X$ is an orthogonality space and $Y$ is a real Banach space. Prove that there exists a unique orthogonally Jensen additive mapping $T:X\to Y$, namely a mapping $T$ that satisfies the so-called orthogonally Jensen additive functional equation $$2f\left (\frac{x+y}{2}\right )=f(x)+f(y)$$for all $x$, $y\in \mathbb{X}$ with $x\perp y$, satisfying the property $$\left |\left|f(x)-T(x) \right |\right|_Y \leq \frac{2^p\theta}{2-2^p}\left |\left |x\right |\right |_X^p$$ for all $x\in X$

For $f(x)=\frac{1}{x}\ (x>0)$, prove the following inequality. \[f\left(t+\frac 12 \right)\leq \int_t^{t+1} f(x)\ dx\leq \frac 16\left\{f(t)+4f\left(t+\frac 12\right)+f(t+1)\right\}\]

While Peter was driving from home to work, he noticed that after driving 21 miles, the distance he had left to drive was 30 percent of the total distance from home to work. How many miles was his complete trip home to work?

Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.

Determine the number of all positive integers which cannot be written in the form $80k + 3m$, where $k,m \in N = \{0,1,2,...,\}$

There is secret society with $2011$ members. Every member has bank account with integer balance ( can be negative). Sometimes some member give one dollar to every his friend. It is known, that after some such moves members can redistribute their money arbitrarily. Prove, that there are exactly $2010$ pairs of friends.

Let $S$ be the set of positive integers $x$ for which there exist positive integers $y$ and $m$ such that $y^2-2^m=x^2$. (a) Find all of the elements of $S$. (b) Find all $x$ such that both $x$ and $x+1$ are in $S$.

A sequence $\{a_n\}$ is defined by: $a_1 = 1, a_{n+1} = a_n + \dfrac{1}{\sqrt{a_n}}$ for $n = 1, 2, 3, \ldots$. Find all real numbers $q$ such that the sequence $\{u_n\}$ defined by $u_n = a_n^q$, $n = 1, 2, 3, \ldots$ has nonzero finite limit when $n$ goes to infinity. THERE MIGHT BE A TYPO!

How many sequences of integers $1 \le a_1 \le a_2\le ... \le a_{11 }\le 2015$ that satisfy $a_i \equiv i^2$ (mod $12$) for all $1 \le i \le 11$ are there? (Le Anh Vinh)

$$\lim_{n \to \infty} nr\sqrt[2]{1-\cos \frac{2\pi}{n}}=?$$

Sarah stands at $(0,0)$ and Rachel stands at $(6,8)$ in the Euclidena plane. Sarah can only move $1$ unit in the positive $x$ or $y$ direction, and Rachel can only move $1$ unit in the negative $x$ or $y$ direction. Each second, Sarah and Rachel see each other, independently pick a direction to move, and move to their new position. Sarah catches Rachel if Sarah and Rachel are every at the same point. Rachel wins if she is able to get $(0,0)$ without being caught; otherwise, Sarah wins. Given that both of them play optimally to maximize their probability of winning, what is the probability that Rachel wins?

Denote $\textrm{SL}_2 (\mathbb{Z})$ and $\textrm{SL}_3 (\mathbb{Z})$ the sets of matrices with $2$ rows and $2$ columns, respectively with $3$ rows and $3$ columns, with integer entries and their determinant equal to $1$. $\textbf{(a)}$ Let $N$ be a positive integer and let $g$ be a matrix with $3$ rows and $3$ columns, with rational entries. Suppose that for each positive divisor $M$ of $N$ there exists a rational number $q_M$, a positive divisor $f (M)$ of $N$ and a matrix $\gamma_M \in \textrm{SL}_3 (\mathbb{Z})$ such that \[ g = q_M \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & f (M) \end{array}\right) \gamma_M \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & M^{} \end{array}\right) . \] Moreover, if $q_1 = 1$, prove that $\det (g) = N$ and $g$ has the following shape: \[ g = \left(\begin{array}{ccc} a_{11} & a_{12} & Na_{13}\\ a_{21} & a_{22} & Na_{23}\\ Na_{31} & Na_{32} & Na_{33} \end{array}\right), \] where $a_{ij}$ are all integers, $i, j \in \{ 1, 2, 3 \} .$ $\textbf{(b)}$ Provide an example of a matrix $g$ with $2$ rows and $2$ columns which satisfies the following properties: $\bullet$ For each positive divisor $M$ of $6$ there exists a rational number $q_M$, a positive divisor $f (M)$ of $6$ and a matrix $\gamma_M \in \textrm{SL}_2 (\mathbb{Z})$ such that \[ g = q_M \left(\begin{array}{cc} 1 & 0\\ 0 & f (M) \end{array}\right) \gamma_M \left(\begin{array}{cc} 1 & 0\\ 0 & M^{} \end{array}\right) \] and $q_1 = 1$. $\bullet$ $g$ does not have its determinant equal to $6$ and is not of the shape \[ g = \left(\begin{array}{cc} a_{22} & 6 a_{23}\\ 6 a_{32} & 6 a_{33} \end{array}\right), \] where $a_{ij}$ are all positive integers, $i, j \in \{ 2, 3 \}$. [i](Radu Toma)[/i]

Let $g(n)$ be defined as follows: \[ g(1) = 0, g(2) = 1 \] and \[ g(n+2) = g(n) + g(n+1) + 1, n \geq 1. \] Prove that if $n > 5$ is a prime, then $n$ divides $g(n) \cdot (g(n) + 1).$

Define the sequences $(a_n),(b_n)$ by \begin{align*} & a_n, b_n > 0, \forall n\in\mathbb{N_+} \\ & a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}} \\ & b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}} \end{align*} 1) If $a_{100}b_{100} = a_{101}b_{101}$, find the value of $a_1-b_1$; 2) If $a_{100} = b_{99}$, determine which is larger between $a_{100}+b_{100}$ and $a_{101}+b_{101}$.

A1,A2,...,Am are subsets of X and we have |Ai|=mk (m,k natural numbers) prove that we can separate X into k sets such that every set has at least one member of each Ai.

Let $a,b,c,x,y,$ and $z$ be complex numbers such that \[a=\dfrac{b+c}{x-2},\qquad b=\dfrac{c+a}{y-2},\qquad c=\dfrac{a+b}{z-2}.\] If $xy+yz+xz=67$ and $x+y+z=2010$, find the value of $xyz$.

A circle centered at $ O$ has radius $ 1$ and contains the point $ A$. Segment $ AB$ is tangent to the circle at $ A$ and $ \angle{AOB} \equal{} \theta$. If point $ C$ lies on $ \overline{OA}$ and $ \overline{BC}$ bisects $ \angle{ABO}$, then $ OC \equal{}$ [asy]import olympiad; unitsize(2cm); defaultpen(fontsize(8pt)+linewidth(.8pt)); labelmargin=0.2; dotfactor=3; pair O=(0,0); pair A=(1,0); pair B=(1,1.5); pair D=bisectorpoint(A,B,O); pair C=extension(B,D,O,A); draw(Circle(O,1)); draw(O--A--B--cycle); draw(B--C); label("$O$",O,SW); dot(O); label("$\theta$",(0.1,0.05),ENE); dot(C); label("$C$",C,S); dot(A); label("$A$",A,E); dot(B); label("$B$",B,E);[/asy] $ \textbf{(A)}\ \sec^2\theta \minus{} \tan\theta \qquad \textbf{(B)}\ \frac {1}{2} \qquad \textbf{(C)}\ \frac {\cos^2\theta}{1 \plus{} \sin\theta} \qquad \textbf{(D)}\ \frac {1}{1 \plus{} \sin\theta} \qquad \textbf{(E)}\ \frac {\sin\theta}{\cos^2\theta}$