The vertical diameter of a circle is moved a centimetres to the right, and the horizontal diameter of this circle is moved $b$ centimetres up. These two lines divide the circle into four pieces. Consider the sum of the areas of the largest and the smallest pieces, and the sum of the areas of the other two pieces. Find the difference between these two sums. (G Galperin, NB Vassiliev)

Compute $\displaystyle\lim_{x\to 0}\dfrac{e^{x\cos x}-1-x}{\sin(x^2)}.$

Let $ \omega$ be the circumcircle of $ ABC$ and let $ D$ be a fixed point on $ BC$, $ D\neq B$, $ D\neq C$. Let $ X$ be a variable point on $ (BC)$, $ X\neq D$. Let $ Y$ be the second intersection point of $ AX$ and $ \omega$. Prove that the circumcircle of $ XYD$ passes through a fixed point.

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

In a book with page numbers from $ 1$ to $ 100$ some pages are torn off. The sum of the numbers on the remaining pages is $ 4949$. How many pages are torn off?

Compute the greatest common divisor of $4^8 - 1$ and $8^{12} - 1$.

Suppose, $x, y, z \in \mathbb{R}_+$ such that $xy+yz+zx=1$. Prove that, \[x(1-y^2)(1-z^2)+y(1-z^2)(1-x^2)+z(1-x^2)(1-y^2)\leq \frac{4\sqrt{3}}{9}\]

In an isosceles right triangle $ABC$ with a right angle $C$, point $M$ is the midpoint of leg $AC$. At the perpendicular bisector of $AC$, point $D$ was chosen such that $\angle CDM = 30^o$, and $D$ and $B$ lie on different sides of $AC$. Find the angle $\angle ABD$. (Volodymyr Petruk)

Determine all sequences $ a_1,a_2,a_3,...$ of $ 1$ and $ \minus{}1$ such that $ a_{mn}\equal{}a_ma_n$ for all $ m,n$ and among any three successive terms $ a_n,a_{n\plus{}1},a_{n\plus{}2}$ both $ 1$ and $ \minus{}1$ occur.

Let $P_{1}, P_{2}, \ldots, P_{2021}$ be 2021 points in the quarter plane $\{(x, y): x \geq 0, y \geq 0\}$. The centroid of these 2021 points lies at the point $(1,1)$. Show that there are two distinct points $P_{i}, P_{j}$ such that the distance from $P_{i}$ to $P_{j}$ is no more than $\sqrt{2} / 20$.

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? $\textbf{(A)}\mbox{ }36\qquad\textbf{(B)}\mbox{ }42\qquad\textbf{(C)}\mbox{ }43\qquad\textbf{(D)}\mbox{ }60\qquad\textbf{(E)}\mbox{ }72$