[b]Problem:[/b]For a positive integer $ n$,let $ V(n; b)$ be the number of decompositions of $ n$ into a product of one or more positive integers greater than $ b$. For example,$ 36 \equal{} 6.6 \equal{}4.9 \equal{} 3.12 \equal{} 3 .3. 4$, so that $ V(36; 2) \equal{} 5$.Prove that for all positive integers $ n$; b it holds that $ V(n;b)<\frac{n}{b}$. :)

A rectangle $ABCD$ and a point $P$ are given. Lines passing through $A$ and $B$ and perpendicular to $PC$ and $PD$ respectively, meet at a point $Q$. Prove that $PQ \perp AB$.

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

Let $ABCD$ be a convex quadrilateral with $AB$ is not parallel to $CD$. Circle $\omega_1$ with center $O_1$ passes through $A$ and $B$, and touches segment $CD$ at $P$. Circle $\omega_2$ with center $O_2$ passes through $C$ and $D$, and touches segment $AB$ at $Q$. Let $E$ and $F$ be the intersection of circles $\omega_1$ and $\omega_2$. Prove that $EF$ bisects segment $PQ$ if and only if $BC$ is parallel to $AD$.

In a triangle $ABC$, let $D$ and $E$ trisect $BC$, so $BD = DE = EC$. Let $F$ be the point on $AB$ such that $\frac{AF}{F B}= 2$, and $G$ on $AC$ such that $\frac{AG}{GC} =\frac12$ . Let $P$ be the intersection of $DG$ and $EF$, and extend $AP$ to intersect $BC$ at a point $X$. Find $\frac{BX}{XC}$

Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n - 1$, and $n-2$ divides $2^n - 2$.

What are the last two digits of $9^{8^{.^{.^{.^2}}}}$ ?

The sum of several (not necessarily different) positive integers not exceeding $10$ is equal to $S$. Find all possible values of $S$ such that these numbers can always be partitioned into two groups with the sum of the numbers in each group not exceeding $70$. [i](I. Voronovich)[/i]

On a plane with a fixed coordinate system, there is a convex polygon whose all vertices have integer coordinates. Prove that twice the area of this polygon is an integer.

What is the maximum value of the difference between the largest real root and the smallest real root of the equation system \[\begin{array}{rcl} ax^2 + bx+ c &=& 0 \\ bx^2 + cx+ a &=& 0 \\ cx^2 + ax+ b &=& 0 \end{array}\], where at least one of the reals $a,b,c$ is non-zero? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \sqrt 2 \qquad\textbf{(D)}\ 3\sqrt 2 \qquad\textbf{(E)}\ \text{There is no upper-bound} $

Evaluate \[\int_0^1 \frac{1-x^2}{(1+x^2)\sqrt{1+x^4}}\ dx\]

What is the largest difference that can be formed by subtracting two numbers chosen from the set $\{ -16,-4,0,2,4,12 \}$? $\text{(A)}\ 10 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 16 \qquad \text{(D)}\ 28 \qquad \text{(E)}\ 48$

[b]p1.[/b] Consider the following function $f(x) = \left(\frac12 \right)^x - \left(\frac12 \right)^{x+1}$. Evaluate the infinite sum $f(1) + f(2) + f(3) + f(4) +...$ [b]p2.[/b] Find the area of the shape bounded by the following relations $$y \le |x| -2$$ $$y \ge |x| - 4$$ $$y \le 0$$ where |x| denotes the absolute value of $x$. [b]p3.[/b] An equilateral triangle with side length $6$ is inscribed inside a circle. What is the diameter of the largest circle that can fit in the circle but outside of the triangle? [b]p4.[/b] Given $\sin x - \tan x = \sin x \tan x$, solve for $x$ in the interval $(0, 2\pi)$, exclusive. [b]p5.[/b] How many rectangles are there in a $6$ by $6$ square grid? [b]p6.[/b] Find the lateral surface area of a cone with radius $3$ and height $4$. [b]p7.[/b] From $9$ positive integer scores on a $10$-point quiz, the mean is $ 8$, the median is $ 8$, and the mode is $7$. Determine the maximum number of perfect scores possible on this test. [b]p8.[/b] If $i =\sqrt{-1}$, evaluate the following continued fraction: $$2i +\frac{1}{2i +\frac{1}{2i+ \frac{1}{2i+...}}}$$ [b]p9.[/b] The cubic polynomial $x^3-px^2+px-6$ has roots $p, q$, and $r$. What is $(1-p)(1-q)(1-r)$? [b]p10.[/b] (Variant on a Classic.) Gilnor is a merchant from Cutlass, a town where $10\%$ of the merchants are thieves. The police utilize a lie detector that is $90\%$ accurate to see if Gilnor is one of the thieves. According to the device, Gilnor is a thief. What is the probability that he really is one? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A unit square is decorated with snippets of the graph of $y = x^2$ as follows: We consider the graph of $y=x^2$ restricted to the domain $0 \le x \le 6$. We cut up the first quadrant (the positive quadrant) into unit squares with lattice vertices. We translate each square so that they are stacked, one on top of the other. We merge all of these squares. How many regions is the unit square divided into by all the overlaid snippets of the graph of the parabola?

How many sequences of nonnegative integers $a_1,a_2,\ldots, a_n$ ($n\ge1$) are there such that $a_1\cdot a_n > 0$, $a_1+a_2+\cdots + a_n = 10$, and $\prod_{i=1}^{n-1}(a_i+a_{i+1}) > 0$? [i]Ray Li.[/i] [hide="Clarifications"][list=1][*]If you find the wording of the problem confusing, you can use the following, equivalent wording: "How many finite sequences of nonnegative integers are there such that (i) the sum of the elements is 10; (ii) the first and last elements are both positive; and (iii) among every pair of adjacent integers in the sequence, at least one is positive."[/list][/hide]

Are there such $x,y$ that $\lg{(x+y)}=\lg x \lg y$ and $\lg{(x-y)}=\frac{\lg x}{\lg y}$ ?

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Find a finite sequence of 16 numbers such that: (a) it reads same from left to right as from right to left. (b) the sum of any 7 consecutive terms is $ \minus{}1$, (c) the sum of any 11 consecutive terms is $ \plus{}1$.

Let equilateral triangle $ABC$ with side length $6$ be inscribed in a circle and let $P$ be on arc $AC$ such that $AP \cdot P C = 10$. Find the length of $BP$.

Let $ABC$ be an acute triangle, with $\angle A > 60^\circ$, and let $H$ be it's orthocenter. Let $M$ and $N$ be points on $AB$ and $AC$, respectively, such that $\angle HMB = \angle HNC = 60^\circ$. Also, let $O$ be the circuncenter of $HMN$ and $D$ be a point on the semiplane determined by $BC$ that contains $A$ in such a way that $DBC$ is equilateral. Prove that $H$, $O$ and $D$ are collinear.

$A, B, C$ are collinear with $B$ betweeen $A$ and $C$. $K_{1}$ is the circle with diameter $AB$, and $K_{2}$ is the circle with diameter $BC$. Another circle touches $AC$ at $B$ and meets $K_{1}$ again at $P$ and $K_{2}$ again at $Q$. The line $PQ$ meets $K_{1}$ again at $R$ and $K_{2}$ again at $S$. Show that the lines $AR$ and $CS$ meet on the perpendicular to $AC$ at $B$.

[u]Down the Infinite Corridor[/u] Consider an isosceles triangle $T$ with base $10$ and height $12$. Define a sequence $\omega_1$, $\omega_2$,$...$of circles such that $\omega_1$ is the incircle of $T$ and $\omega_{i+1}$ is tangent to $\omega_i$ and both legs of the isosceles triangle for $i > 1$. [b]p1.[/b] Find the radius of $\omega_1$. [b]p2.[/b] Find the ratio of the radius of $\omega_{i+1}$ to the radius of $\omega_i$. [b]p3.[/b] Find the total area contained in all the circles.

Find all pairs $(x, y)$ of real numbers satisfying the system : $\begin{cases} x + y = 3 \\ x^4 - y^4 = 8x - y \end{cases}$

How many integers less than $2502$ are equal to the square of a prime number?

On the plane with a rectangular coordinate system, a set of infinitely many rectangles is given. Every rectangle has the origin as one of its vertices. The sides of all rectangles are parallel to the coordinate axes, and all sides have integer lengths. Prove that there are at least two rectangles in the set, one of which completely covers the other.