Two circles that are not equal are tangent externally at point $R$. Suppose point $P$ is the intersection of the external common tangents of the two circles. Let $A$ and $B$ are two points on different circles so that $RA$ is perpendicular to $RB$. Show that the line $AB$ passes through $P$.

Let $p(n)$ denote the product of decimal digits of a positive integer $n$. Computer the sum $p(1)+p(2)+\ldots+p(2001)$.

Let $\omega (n)$ denote the number of prime divisors of the integer $n>1$. Find the least integer $k$ such that the inequality $2^{\omega (n) } \leq k \cdot n^{\frac 1 4}$ holds for all $n > 1.$ Pierre.

Two cubical dice each have removable numbers $ 1$ through $ 6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is $ 7$? $ \textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{8} \qquad \textbf{(C)}\ \frac{1}{6} \qquad \textbf{(D)}\ \frac{2}{11} \qquad \textbf{(E)}\ \frac{1}{5}$

For any positive integer number $k$, the factorial $k!$ is defined as a product of all integers between $1$ and $k$ inclusive: $k!=k\times{(k-1)}\times\dots\times{1}$. Let $s(n)$ denote the sum of the first $n$ factorials, i.e. $$s(n)=\underbrace{n\times{(n-1)}\times\dots\times{1}}_{n!}+\underbrace{(n-1)\times{(n-2)}\times\dots\times{1}}_{(n-1)!}+\cdots +\underbrace{2\times{1}}_{2!}+\underbrace{1}_{1!}$$ Find the remainder when $s(2024)$ is divided by $8$

Alice thinks of an integer $1 \le n \le 2048$. Bob asks $k$ true or false questions about Alice's integer; Alice then answers each of the questions, but she may lie on at most one question. What is the minimum value of $k$ for which Bob can guarantee he knows Alice's integer after she answers?

You roll three fair six-sided dice. Given that the highest number you rolled is a $5$, the expected value of the sum of the three dice can be written as $\tfrac ab$ in simplest form. Find $a+b$.

A number from $1$ to $144$ is guessed. You are allowed to select a subset of the set of numbers from $ 1$ to $144$ and ask whether the guessed number belongs to it. For the answer “yes” you have to pay $2$ rubles, for the answer “no” - $1$ ruble. What is the smallest amount of money needed to surely guess that?

Alex thinks of a two-digit integer (any integer between $10$ and $99$). Greg is trying to guess it. If the number Greg names is correct, or if one of its digits is equal to the corresponding digit of Alex’s number and the other digit differs by one from the corresponding digit of Alex’s number, then Alex says “hot”; otherwise, he says “cold”. (For example, if Alex’s number was $65$, then by naming any of $64, 65, 66, 55$ or $75$ Greg will be answered “hot”, otherwise he will be answered “cold”.) [list][b](a)[/b] Prove that there is no strategy which guarantees that Greg will guess Alex’s number in no more than 18 attempts. [b](b)[/b] Find a strategy for Greg to find out Alex’s number (regardless of what the chosen number was) using no more than $24$ attempts. [b](c)[/b] Is there a $22$ attempt winning strategy for Greg?[/list]

Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube? $ \textbf{(A)}\ \frac{1}{4} \qquad \textbf{(B)}\ \frac{3}{8} \qquad \textbf{(C)}\ \frac{4}{7} \qquad \textbf{(D)}\ \frac{5}{7} \qquad \textbf{(E)}\ \frac{3}{4}$

Assume that the following generating function equation is correct, prove the following statement: $\Pi_{i=1}^{\infty} (1+x^{3i})\Pi_{j=1}^{\infty} (1-x^{6j+3})=1$ Statement: The number of partitions of $n$ to numbers not of the form $6k+1$ or $6k-1$ is equal to the number of partitions of $n$ in which each summand appears at least twice. (10 points) [i]Proposed by Morteza Saghafian[/i]

Let $p(z)$ be a polynomial of degree $n>0$ with complex coefficients. Prove that there are at least $n+1$ complex numbers $z$ for which $p(z)\in \{0,1\}$.

Consider the figure (attached), where every small triangle is equilateral with side length 1. Compute the area of the polygon $ AEKS $. (Fun fact: this problem was originally going to ask for the area of $ DANK $, as in "dank memes!")

Let $D$ be a point on side $[BC]$ of $\triangle ABC$ such that $|AB|=3, |CD|=1$ and $|AC|=|BD|=\sqrt{5}$. If the $B$-altitude of $\triangle ABC$ meets $AD$ at $E$, what is $|CE|$? $ \textbf{(A)}\ \dfrac{2}{\sqrt{5}} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \dfrac{2}{\sqrt{3}} \qquad\textbf{(D)}\ \dfrac{\sqrt{5}}{2} \qquad\textbf{(E)}\ \dfrac{3}{2} $

[b]a)[/b] Show that the expression $ x^3-5x^2+8x-4 $ is nonegative, for every $ x\in [1,\infty ) . $ [b]b)[/b] Determine $ \min_{a,b\in [1,\infty )} \left( ab(a+b-10) +8(a+b) \right) . $

Let $ABCD$ be a parallelogram which is not a rectangle and $E$ be the point in its plane such that $AE \perp AB$ and $CE \perp CB$. Prove that $\angle DEA = \angle CEB$.

In triangle $ ABC$ we have $ \angle{ACB} \equal{} 2\angle{ABC}$ and there exists the point $ D$ inside the triangle such that $ AD \equal{} AC$ and $ DB \equal{} DC$. Prove that $ \angle{BAC} \equal{} 3\angle{BAD}$.

A convex pentagon $P $ is divided by all its diagonals into ten triangles and one smaller pentagon $P'$. Let $N$ be the sum of areas of five triangles adjacent to the sides of $P$ decreased by the area of $P'$. The same operations are performed with the pentagon $P'$, let $N'$ be the similar difference calculated for this pentagon. Prove that $N > N'$. (A.Belov)

Let an integer $k > 1$ be given. For each integer $n > 1$, we put \[f(n) = k \cdot n \cdot \left(1-\frac{1}{p_1}\right) \cdot \left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_r}\right)\] where $p_1, p_2, \ldots, p_r$ are all distinct prime divisors of $n$. Find all values $k$ for which the sequence $\{x_m\}$ defined by $x_0 = a$ and $x_{m+1} = f(x_m), m = 0, 1, 2, 3, \ldots$ is bounded for all integers $a > 1$.

The vertices of a cube are labeled with the integers $1$ through $8,$ with each used exactly once. Let $s$ be the maximum sum of the labels of two edge-adjacent vertices. Compute the minimum possible value of $s$ over all such labelings.

What is the smallest positive integer $t$ such that there exist integers $x_{1},x_{2}, \cdots, x_{t}$ with \[{x_{1}}^{3}+{x_{2}}^{3}+\cdots+{x_{t}}^{3}=2002^{2002}\;\;?\]

[b]p1.[/b] Let $n$ be the number so that $1 - 2 + 3 - 4 + ... - (n - 1) + n = 2012$. What is $4^{2012}$ (mod $n$)? [b]p2. [/b]Consider three unit squares placed side by side. Label the top left vertex $P$ and the bottom four vertices $A,B,C,D$ respectively. Find $\angle PBA + \angle PCA + \angle PDA$. [b]p3.[/b] Given $f(x) = \frac{3}{x-1}$ , then express $\frac{9(x^2-2x+1)}{x^2-8x+16}$ entirely in terms of $f(x)$. In other words, $x$ should not be in your answer, only $f(x)$. [b]p4.[/b] Right triangle with right angle $B$ and integer side lengths has $BD$ as the altitude. $E$ and $F$ are the incenters of triangles $ADB$ and $BDC$ respectively. Line $EF$ is extended and intersects $BC$ at $G$, and $AB$ at $H$. If $AB = 15$ and $BC = 8$, find the area of triangle $BGH$. [b]p5.[/b] Let $a_1, a_2, ..., a_n$ be a sequence of real numbers. Call a $k$-inversion $(0 < k\le n)$ of a sequence to be indices $i_1, i_2, .. , i_k$ such that $i_1 < i_2 < .. < i_k$ but $a_{i1} > a_{i2} > ...> a_{ik}$ . Calculate the expected number of $6$-inversions in a random permutation of the set $\{1, 2, ... , 10\}$. [b]p6.[/b] Chell is given a strip of squares labeled $1, .. , 6$ all placed side to side. For each $k \in {1, ..., 6}$, she then chooses one square at random in $\{1, ..., k\}$ and places a Weighted Storage Cube there. After she has placed all $6$ cubes, she computes her score as follows: For each square, she takes the number of cubes in the pile and then takes the square (i.e. if there were 3 cubes in a square, her score for that square would be $9$). Her overall score is the sum of the scores of each square. What is the expected value of her score? PS. You had better use hide for answers.

$\{a_n\}$ is an infinite, strictly increasing sequence of positive integers and $a_{a_n}\leq a_n+a_{n+3}$ for all $n\geq 1$. Prove that, there are infinitely many triples $(k,l,m)$ of positive integers such that $k<l<m$ and $a_k+a_m=2a_l$

A sphere is touched by all the four sides of a (space) quadrilateral. Prove that all the four touching points are in the same plane.

As shown in the figure, two circles $\Gamma_1$ and $\Gamma_2$ on the plane intersect at two points $A$ and $B$. The two rays passing through $A$, $\ell_1$ and $\ell_2$ intersect $\Gamma_1$ at points $D$ and $E$ respectively, and $\Gamma_2$ at points $F$ and $C$ respectively (where $E$ and $F$ lie on line segments $AC$ and $AD$ respectively, and neither of them coincides with the endpoints). It is known that the three lines $AB$, $CF$ and $DE$ have a common point, the circumscribed circle of $\vartriangle AEF$ intersects $AB$ at point $G$, the straight line $EG$ intersects the circle $\Gamma_1$ at point $P$, the straight line $FG$ intersects the circle $\Gamma_2$ at point $Q$. Let the symmetric points of $C$ and $D$ wrt the straight line $AB$ be $C'$ and $D'$ respectively. If $PD'$ and $QC'$ intersect at point$ J$, prove that $J$ lies on the straight line $AB$. [img]https://cdn.artofproblemsolving.com/attachments/3/7/eb3acdbad52750a6879b4b6955dfdb7de19ed3.png[/img]