The operations below can be applied on any expression of the form \(ax^2+bx+c\). $(\text{I})$ If \(c \neq 0\), replace \(a\) by \(4a-\frac{3}{c}\) and \(c\) by \(\frac{c}{4}\). $(\text{II})$ If \(a \neq 0\), replace \(a\) by \(-\frac{a}{2}\) and \(c\) by \(-2c+\frac{3}{a}\). $(\text{III}_t)$ Replace \(x\) by \(x-t\), where \(t\) is an integer. (Different values of \(t\) can be used.) Is it possible to transform \(x^2-x-6\) into each of the following by applying some sequence of the above operations? $(\text{a})$ \(5x^2+5x-1\) $(\text{b})$ \(x^2+6x+2\)

Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.

Let $\phi(n)$ be the number of positive integers less than $n$ that are relatively prime to $n$, where $n$ is a positive integer. Find all pairs of positive integers $(m,n)$ such that \[2^n + (n-\phi(n)-1)! = n^m+1.\]

Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ such that \[f(f(x) - y) = f(xy) + f(x)f(-y)\] for any two real numbers $x, y$. [i]Proposed by Pablo Valeriano[/i]

Let $x$ and $y$ be integers with $x + y \ne 0$. Find all pairs $(x, y)$ such that $$\frac{x^2 + y^2}{x + y}= 10.$$ (Walther Janous)

Triangle $\vartriangle ABC$ has side lengths $AB = 39$, $BC = 16$, and $CA = 25$. What is the volume of the solid formed by rotating $\vartriangle ABC$ about line $BC$?

Let $p,q$ prime numbers such that $$p+q \mid p^3-q^3$$ Show that $p=q$.

Two sequences $b_i$, $c_i$, $0 \le i \le 100$ contain positive integers, except $c_0=0$ and $b_{100}=0$. Some towns in Graphland are connected with roads, and each road connects exactly two towns and is precisely $1$ km long. Towns, which are connected by a road or a sequence of roads, are called [i]neighbours[/i]. The length of the shortest path between two towns $X$ and $Y$ is denoted as [i]distance[/i]. It is known that the greatest [i]distance[/i] between two towns in Graphland is $100$ km. Also the following property holds for every pair $X$ and $Y$ of towns (not necessarily distinct): if the [i]distance[/i] between $X$ and $Y$ is exactly $k$ km, then $Y$ has exactly $b_k$ [i]neighbours[/i] that are at the [i]distance[/i] $k+1$ from $X$, and exactly $c_k$ [i]neighbours[/i] that are at the [i]distance[/i] $k-1$ from $X$. Prove that $$\frac{b_0b_1 \cdot \cdot \cdot b_{99}}{c_1c_2 \cdot \cdot \cdot c_{100}}$$ is a positive integer.

A paper equilateral triangle $ABC$ has side length $12$. The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance $9$ from point $B$. The length of the line segment along which the triangle is folded can be written as $\frac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$. [asy] import cse5; size(12cm); pen tpen = defaultpen + 1.337; real a = 39/5.0; real b = 39/7.0; pair B = MP("B", (0,0), dir(200)); pair A = MP("A", (9,0), dir(-80)); pair C = MP("C", (12,0), dir(-20)); pair K = (6,10.392); pair M = (a*B+(12-a)*K) / 12; pair N = (b*C+(12-b)*K) / 12; draw(B--M--N--C--cycle, tpen); draw(M--A--N--cycle); fill(M--A--N--cycle, mediumgrey); pair shift = (-20.13, 0); pair B1 = MP("B", B+shift, dir(200)); pair A1 = MP("A", K+shift, dir(90)); pair C1 = MP("C", C+shift, dir(-20)); draw(A1--B1--C1--cycle, tpen);[/asy]

Let $M$ be the set of the positive rational numbers less than 1, which can be expressed with a 10-distinct digits period in decimal representation. a) Find the arithmetic mean of all the elements in $M$; b) Prove that there exists a positive integer $n$, $1<n<10^{10}$, such that $n\cdot a - a$ is a non-negative integer, for all $a\in M$.

A function from the positive integers to the positive integers satisfies these properties 1. $f(ab)=f(a)f(b)$ for any two coprime positive integers $a,b$. 2. $f(p+q)=f(p)+f(q)$ for any two primes $p,q$. Prove that $f(2)=2, f(3)=3, f(1999)=1999$.

Compute the side length of the largest cube contained in the region \[ \{(x, y, z) : x^2+y^2+z^2 \le 25 \text{ and } x, y \ge 0 \} \] of three-dimensional space.

Let $ r$ be a positive rational. Prove that $\frac{8r + 21}{3r + 8}$ is a better approximation to $\sqrt7$ that $ r$.

[b]p1.[/b] Find the smallest positive integer which is $1$ more than multiple of $3$, $2$ more than a multiple of $4$, and $4$ more than a multiple of $7$. [b]p2.[/b] Let $p = 4$, and let $a =\sqrt1$, $b =\sqrt2$, $c =\sqrt3$, $...$. Compute the value of $(p-a)(p-b) ... (p-z)$. [b]p3.[/b] There are $6$ points on the circumference of a circle. How many convex polygons are there having vertices on these points? [b]p4.[/b] David and I each have a sheet of computer paper, mine evenly spaced by $19$ parallel lines into $20$ sections, and his evenly spaced by $29$ parallel lines into $30$ sections. If our two sheets are overlayed, how many pairs of lines are perfectly incident? [b]p5.[/b] A pyramid is created by stacking equilateral triangles of balls, each layer having one fewer ball per side than the triangle immediately beneath it. How many balls are used if the pyramid’s base has $5$ balls to a side? [b]p6.[/b] Call a positive integer $n$ good if it has $3$ digits which add to $4$ and if it can be written in the form $n = k^2$, where $k$ is also a positive integer. Compute the average of all good numbers. [b]p7.[/b] John’s birthday cake is a scrumptious cylinder of radius $6$ inches and height $3$ inches. If his friends cut the cake into $8$ equal sectors, what is the total surface area of a piece of birthday cake? [b]p8.[/b] Evaluate $\sum^{10}_{i=1}\sum^{10}_{j=1} ij$. [b]p9.[/b] If three numbers $a$, $b$, and $c$ are randomly selected from the interval $[-2, 2]$, what is the probability that $a^2 + b^2 + c^2 \ge 4$? [b]p10.[/b] Evaluate $\sum^{\infty}_{x=2} \frac{2}{x^2 - 1}.$ [b]p11.[/b] Consider $4x^2 - kx - 1 = 0$. If the roots of this polynomial are $\sin \theta$ and $\cos \theta$, compute $|k|$. [b]p12.[/b] Given that $65537 = 2^{16} + 1$ is a prime number, compute the number of primes of the form $2^n + 1$ (for $n \ge 0$) between $1$ and $10^6$. [b]p13.[/b] Compute $\sin^{-1}(36/85) + \cos^{-1}(4/5) + \cos^{-1}(15/17).$ [b]p14.[/b] Find the number of integers $n$, $1\le n \le 2003$, such that $n^{2003} - 1$ is a multiple of $10$. [b]p15.[/b] Find the number of integers $n,$ $1 \le n \le 120$, such that $n^2$ leaves remainder $1$ when divided by $120$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In an All-Area track meet, $216$ sprinters enter a $100-$meter dash competition. The track has $6$ lanes, so only $6$ sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion sprinter? $\textbf{(A)}\mbox{ }36\qquad\textbf{(B)}\mbox{ }42\qquad\textbf{(C)}\mbox{ }43\qquad\textbf{(D)}\mbox{ }60\qquad\textbf{(E)}\mbox{ }72$

Let $n$ be a positive integer, and let $A$ and $B$ be $n\times n$ matrices with complex entries such that $A^2=B^2$. Show that there exists an $n\times n$ invertible matrix $S$ with complex entries that satisfies $S(AB-BA)=(BA-AB)S$.

Find all real numbers $ t $ not less than $1 $ that satisfy the following requirements: for any $a,b\in [-1,t]$ , there always exists $c,d \in [-1,t ]$ such that $ (a+c)(b+d)=1.$

A [i]sum decomposition[/i] of the number 100 is given by a positive integer $n$ and $n$ positive integers $x_1<x_2<\cdots <x_n$ such that $x_1 + x_2 + \cdots + x_n = 100$. Determine the largest possible value of the product $x_1x_2\cdots x_n$, and $n$ , as $x_1, x_2,\dots, x_n$ vary among all sum decompositions of the number $100$.

Let $x,y$ and $a_0, a_1, a_2, \cdots $ be integers satisfying $a_0 = a_1 = 0$, and $$a_{n+2} = xa_{n+1}+ya_n+1$$for all integers $n \geq 0$. Let $p$ be any prime number. Show that $\gcd(a_p,a_{p+1})$ is either equal to $1$ or greater than $\sqrt{p}$.

Let $a, b,c$ be distinct positive integers, and suppose that $p = ab+bc+ca$ is a prime number. $(a)$ Show that $a^2,b^,c^2$ give distinct remainders after division by $p$. (b) Show that $a^3,b^3,c^3$ give distinct remainders after division by $p$.

Find all polynomials \(P\) with real coefficients satisfying $$P(\frac{1}{1+x})=\frac{1}{1+P(x)}$$ for all real numbers \(x\neq -1\)

A triangle with vertices at $(1003,0)$, $(1004,3)$, and $(1005,1)$ in the $xy$-plane is revolved all the way around the $y$-axis. Find the volume of the solid thus obtained.

The diagonals of convex quadrilateral $ABCD$ are perpendicular. Points $A' , B' , C' , D' $ are the circumcenters of triangles $ABD, BCA, CDB, DAC$ respectively. Prove that lines $AA' , BB' , CC' , DD' $ concur. (A. Zaslavsky)

Maryam labels each vertex of a tetrahedron with the sum of the lengths of the three edges meeting at that vertex. She then observes that the labels at the four vertices of the tetrahedron are all equal. For each vertex of the tetrahedron, prove that the lengths of the three edges meeting at that vertex are the three side lengths of a triangle.

Triangle $ABC$ has $AB = 13$, $BC = 14$, and $CA = 15$. The internal angle bisector of $\angle ABC$ intersects side $CA$ at $X$. The circumcircles of triangles $AXB$ and $BXC$ intersect sides $BC$ and $AB$ at $M$ and $N$, respectively. The value of $MN^2$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m+n$ is divided by $1000$. [i]Proposed by [b] Th3Numb3rThr33[/b][/i]