How many 6-tuples $ (a_1,a_2,a_3,a_4,a_5,a_6)$ are there such that each of $ a_1,a_2,a_3,a_4,a_5,a_6$ is from the set $ \{1,2,3,4\}$ and the six expressions \[ a_j^2 \minus{} a_ja_{j \plus{} 1} \plus{} a_{j \plus{} 1}^2\] for $ j \equal{} 1,2,3,4,5,6$ (where $ a_7$ is to be taken as $ a_1$) are all equal to one another?

Prove that the squares with sides $\frac{1}{1}, \frac{1}{2}, \frac{1}{3},\ldots$ may be put into the square with side $\frac{3}{2} $ in such a way that no two of them have any interior point in common.

A right triangle $ABC$ has the right angle at vertex $A$. Circle $c$ passes through vertices $A$ and $B$ of the triangle $ABC$ and intersects the sides $AC$ and $BC$ correspondingly at points $D$ and $E$. The line segment $CD$ has the same length as the diameter of the circle $c$. Prove that the triangle $ABE$ is isosceles.

Let $a, b, c$ be real numbers, $a \neq 0$. If the equation $2ax^2 + bx + c = 0$ has real root on the interval $[-1, 1]$. Prove that $$\min \{c, a + c + 1\} \leq \max \{|b - a + 1|, |b + a - 1|\},$$ and determine the necessary and sufficient conditions of $a, b, c$ for the equality case to be achieved.

$ AB \equal{} AC$, $ \angle BAC \equal{} 80^\circ$. Let $ E$ be a point inside $ \triangle ABC$ such that $ AE \equal{} EC$ and $ \angle EAC \equal{} 10^\circ$. What is the measure of $ \angle EBC$? $\textbf{(A)}\ 10^\circ \qquad\textbf{(B)}\ 15^\circ \qquad\textbf{(C)}\ 20^\circ \qquad\textbf{(D)}\ 25^\circ \qquad\textbf{(E)}\ 30^\circ$

If you have $65$ points in a plane, we will make the lines that passes by any two points in this plane and we obtain exactly $2015$ distinct lines, prove that least $4$ points are collinears!!

The function $f(x) = x^2+ax+b$ has two distinct zeros. If $f(x^2+2x-1)=0$ has four distinct zeros $x_1<x_2<x_3<x_4$ that form an arithmetic sequence, compute the range of $a-b$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 11)[/i]

In the equation below, $A$ and $B$ are consecutive positive integers, and $A$, $B$, and $A+B$ represent number bases: \[132_A + 43_B = 69_{A+B.}\] What is $A + B$? $ \textbf{(A)}\ 9\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 17 $

Your team receives up to $100$ points total for the team round. To play this minigame for up to $10$ bonus points, you must decide how to construct an optimal army with number of soldiers equal to the points you receive. Construct an army of $100$ soldiers with $5$ flanks; thus your army is the union of battalions $B_1$, $B_2$, $B_3$, $B_4$, and $B_5$. You choose the size of each battalion such that $|B_1|+|B_2|+|B_3|+|B_4|+|B_5|=100$. The size of each batallion must be integral and non-negative. Then, suppose you receive $n$ points for the Team Round. We will then "supply" your army as follows: if $n>B_1$, we fill in battalion $1$ so that it has $|B_1|$ soldiers; then repeat for the next battalion with $n-|B_1|$ soldiers. If at some point there are not enough soldiers to fill the battalion, the remainder will be put in that battalion and subsequent battalions will be empty. (Ex: suppose you tell us to form battalions of size $\{20,30,20,20,10\}$, and your team scores $73$ points. Then your battalions will actually be $\{20,30,20,3,0\}$.) Your team's army will then "fight" another's. The $B_i$ of both teams will be compared with the other $B_i$, and the winner of the overall war is the army who wins the majority of the battalion fights. The winner receives $1$ victory point, and in case of ties, both teams receive $\tfrac12$ victory points. Every team's army will fight everyone else's and the team war score will be the sum of the victory points won from wars. The teams with ranking $x$ where $7k\leq x\leq 7(k+1)$ will earn $10-k$ bonus points. For example: Team Princeton decides to allocate its army into battalions with size $|B_1|$, $|B_2|$, $|B_3|$, $|B_4|$, $|B_5|$ $=$ $20$, $20$, $20$, $20$, $20$. Team MIT allocates its army into battalions with size $|B_1|$, $|B_2|$, $|B_3|$, $|B_4|$, $|B_5|$ $=$ $10$, $10$, $10$, $10$, $60$. Now suppose Princeton scores $80$ points on the Team Round, and MIT scores $90$ points. Then after supplying, the armies will actually look like $\{20, 20, 20, 20, 0\}$ for Princeton and $\{10, 10, 10, 10, 50\}$ for MIT. Then note that in a war, Princeton beats MIT in the first four battalion battles while MIT only wins the last battalion battle; therefore Princeton wins the war, and Princeton would win $1$ victory point.

How many ordered pairs $(x, y)$ of real numbers $x$ and $y$ are there such that $-100 \pi \le x \le 100 \pi$, $-100 \pi \le y \le 100 \pi$, $x + y = 20.19$, and $\tan x + \tan y = 20.19$?

Gwendoline rolls a pair of six-sided dice and records the product of the two values rolled. Gwendoline repeatedly rolls the two dice and records the product of the two values until one of the values she records appears for a third time. What is the maximum number of times Gwendoline will need to roll the two dice?

The four points of a circle are in the following order: $A, B, C, D$. Extensions of chord $AB$ beyond point $B$ and of chord $CD$ beyond point $C$ intersect at point $E$, with $\angle AED= 60^o$. If $\angle ABD =3 \angle BAC$ , prove that $AD$ is the diameter of the circle.

A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and \[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\]. Find $f_n(x)$.

Point $P$ is chosen inside triangle $ABC$ so that $\angle APC+\angle ABC=180^o$ and $BC=AP.$ On the side $AB$, a point $K$ is chosen such that $AK = KB + PC$. Prove that $CK \perp AB$.

Let $(C)$ and $(L)$ be a circle and a line. $P_{1},\dots,P_{2n+1}$ are odd number of points on $(L)$. $A_{1}$ is an arbitrary point on $(C)$. $A_{k+1}$ is the intersection point of $A_{k}P_{k}$ and $(C)$ ($1\leq k\leq 2n+1$). Prove that $A_{1}A_{2n+2}$ passes through a constant point while $A_{1}$ varies on $(C)$.

A point $X$ exactly $\sqrt{2}-\frac{\sqrt{6}}{3}$ away from the origin is chosen randomly. A point $Y$ less than $4$ away from the origin is chosen randomly. The probability that a point $Z$ less than $2$ away from the origin exists such that $\triangle XYZ$ is an equilateral triangle can be expressed as $\frac{a\pi + b}{c \pi}$ for some positive integers $a, b, c$ with $a$ and $c$ relatively prime. Find $a+b+c$.

Let $f$ be a function which takes in $0, 1, 2$ and returns $0, 1, $ or $2$. The values need not be distinct: for instance we could have $f(0) = 1, f(1) = 1, f(2) = 2$. How many such functions are there which satisfy \[f(2) + f(f(0)) + f(f(f(1))) = 5?\]

Let $a_1,a_2,\dots,a_{2017}$ be reals satisfied $a_1=a_{2017}$, $|a_i+a_{i+2}-2a_{i+1}|\le 1$ for all $i=1,2,\dots,2015$. Find the maximum value of $\max_{1\le i<j\le 2017}|a_i-a_j|$.

Susan is presented with six boxes $B_1, \dots, B_6$, each of which is initially empty, and two identical coins of denomination $2^k$ for each $k = 0, \dots, 5$. Compute the number of ways for Susan to place the coins in the boxes such that each box $B_k$ contains coins of total value $2^k$. [i]Proposed by Ankan Bhattacharya[/i]

Let $A$ be a subgroup of the symmetric group $S_n$, and $G$ be a normal subgroup of $A$. Show that if $G$ is transitive, then $|A\colon G|\le 5^{n-1}$ (translated by Miklós Maróti)

Prove that for no integer $ n$ is $ n^7 \plus{} 7$ a perfect square.

Prove the inequality $x_1y_1+x_2y_2+x_2y_1+2x_2y_2\le 1996$ if $x_1^2+2x_1x_2+2x_2^2\le 998$ and $y_1^2+2y_1y_2+2y_2^2\le 3992$.

For some real numbers $ a$ and $ b$, the equation \[ 8x^3 \plus{} 4ax^2 \plus{} 2bx \plus{} a \equal{} 0 \]has three distinct positive roots. If the sum of the base-$ 2$ logarithms of the roots is $ 5$, what is the value of $ a$? $ \textbf{(A)}\minus{}\!256 \qquad \textbf{(B)}\minus{}\!64 \qquad \textbf{(C)}\minus{}\!8 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 256$

Find all $f:R-R$ such that $f(x^2+y+f(y))=2y+f(x)^2$

A set of lines in the plane is in [i]general position[/i] if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its [i]finite regions[/i]. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ lines blue in such a way that none of its finite regions has a completely blue boundary. [i]Note[/i]: Results with $\sqrt{n}$ replaced by $c\sqrt{n}$ will be awarded points depending on the value of the constant $c$.