I'm thinking of two whole numbers. Their product is 24 and their sum is 11. What is the larger number? $ \text{(A)}\ 3\qquad\text{(B)}\ 4\qquad\text{(C)}\ 6\qquad\text{(D)}\ 8\qquad\text{(E)}\ 12 $

A ball was launched on a rectangular billiard table at an angle of $45^o$ to one of the sides. Reflected from all sides (the angle of incidence is equal to the angle of reflection), he returned to his original position . It is known that one of the sides of the table has a length of one meter. Find the length of the second side. [img]https://cdn.artofproblemsolving.com/attachments/3/d/e0310ea910c7e3272396cd034421d1f3e88228.png[/img]

The writer Arthur has $n \ge1$ co-authors who write books with him. Each book has a list of authors including Arthur himself. No two books have the same set of authors. At a party with all his co-author, each co-author writes on a note how many books they remember having written with Arthur. Inspecting the numbers on the notes, they discover that the numbers written down are the first $n$ Fibonacci numbers (defined by $F_1 = F_2 = 1$ and $F_{k+2}= F_{k+1} + F_k$). For which $n$ is it possible that none of the co-authors had a lapse of memory?

A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities $$P(x, y, z)=P(x, y, xy-z)=P(x, zx-y, z)=P(yz-x, y, z).$$ Prove that there exists a polynomial $F(t)$ in one variable such that $$P(x,y,z)=F(x^2+y^2+z^2-xyz).$$

Let $ABCD$ be a cyclic quadrilateral, and angle bisectors of $\angle BAD$ and $\angle BCD$ meet at point $I$. Show that if $\angle BIC = \angle IDC$, then $I$ is the incenter of triangle $ABD$.

Given a prime $p$, prove that the sum $\sum_{k=1}^{\lfloor \frac{q}{p} \rfloor}{k^{p-1}}$ is not divisible by $q$ for all but finitely many primes $q$.

Prove that there exist infinitely many pairs $(a, b)$ of relatively prime positive integers such that \[\frac{a^{2}-5}{b}\;\; \text{and}\;\; \frac{b^{2}-5}{a}\] are both positive integers.

Let $ABCD$ be a square and $E$ a point on side $CD$ such that $\angle DAE = 30^{\circ}$. The bisector of angle $\angle AEC$ intersects line $BD$ at point $F$. Lines $FC$ and $AE$ intersect at $S$. Find $\angle SDC$. [i]Proposed by Ana Boiangiu[/i]

A convicted person will be released when he reaches the top of a $100$-step staircase. But he cannot advance as he pleases, since he is obliged to go up one step each day of the odd-numbered months and go down one step each day of the even-numbered months. If it begins on January $ 1$, $2001$, what day will it be released?

Let $a, b, c > 0$ be real numbers with $abc = 1$. Show $$1 + ab + bc + ca \ge \min \left\{ \frac{(a + b)^2}{ab} , \frac{(b+c)^2}{bc} , \frac{(c + a)^2}{ca}\right\}.$$ When does equality holds?

Compute the sum of all positive integers $n$ for which there exists a real number $x$ satisfying $$\left(x +\frac{n}{x} \right)^n= 2^{20}.$$

Given two integers $ m,n$ satisfying $ 4 < m < n.$ Let $ A_{1}A_{2}\cdots A_{2n \plus{} 1}$ be a regular $ 2n\plus{}1$ polygon. Denote by $ P$ the set of its vertices. Find the number of convex $ m$ polygon whose vertices belongs to $ P$ and exactly has two acute angles.

Draw on the plane a set of points whose coordinates $(x,y)$ satisfy the equation $x^3 + y^3 = x^2y^2 + xy$.

The integers from 1 to 49 are written in a $7\times 7$ table, such that for any $k\in\{1,2,\ldots,7\}$ the product of the numbers in the $k$-th row equals the product of the numbers in the $(8-k)$-th row. [list=a] [*]Prove that there exists a row such that the sum of the numbers written on it is a prime number. [*]Give an example of such a table. [/list] [i]Cristi Săvescu[/i]

Let $ a$ be a positive integer. Consider the sequence $ (a_n)$ defined as $ a_0\equal{}a$ and $ a_n\equal{}a_{n\minus{}1}\plus{}40^{n!}$ for $ n > 0$. Prove that the sequence $ (a_n)$ has infinitely many numbers divisible by $ 2009$.

Find the largest positive integer $k$ such that we can find a set $A \subseteq \{1,2, \ldots, 100 \}$ with $k$ elements such that, for any $a,b \in A$, $a$ divides $b$ if and only if $s(a)$ divides $s(b)$, where $s(k)$ denotes the sum of the digits of $k$.

Let $ n$ be a positive integer and $ [ \ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied: $ (1)$ $ n$ is not a perfect square; $ (2)$ $ a^{3}$ divides $ n^{2}$.

5-Two circles $ S_1$ and $ S_2$ with equal radius and intersecting at two points are given in the plane.A line $ l$ intersects $ S_1$ at $ B,D$ and $ S_2$ at $ A,C$(the order of the points on the line are as follows:$ A,B,C,D$).Two circles $ W_1$ and $ W_2$ are drawn such that both of them are tangent externally at $ S_1$ and internally at $ S_2$ and also tangent to $ l$ at both sides.Suppose $ W_1$ and $ W_2$ are tangent.Then PROVE $ AB \equal{} CD$.

Let be given a triangle $ ABC$. Points $ P$ on side $ AC$ and $ Y$ on the production of $ CB$ beyond $ B$ are chosen so that $ Y$ subtends equal angles with $ AP$ and $ PC$. Similarly, $ Q$ on side $ BC$ and $ X$ on the production of $ AC$ beyond $ C$ are such that $ X$ subtends equal angles with $ BQ$ and $ QC$. Lines $ YP$ and $ XB$ meet at $ R$, $ XQ$ and $ YA$ meet at $ S$, and $ XB$ and $ YA$ meet at $ D$. Prove that $ PQRS$ is a parallelogram if and only if $ ACBD$ is a cyclic quadrilateral.

Square corners, $5$ units on a side, are removed from a $20$ unit by $30$ unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is [asy] fill((0,0)--(20,0)--(20,5)--(0,5)--cycle,lightgray); fill((20,0)--(20+5*sqrt(2),5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)--cycle,lightgray); draw((0,0)--(20,0)--(20,5)--(0,5)--cycle); draw((0,5)--(5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)); draw((20+5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5*sqrt(2))--(20,0)); draw((5*sqrt(2),5+5*sqrt(2))--(5*sqrt(2),5*sqrt(2))--(5,5),dashed); draw((5*sqrt(2),5*sqrt(2))--(15+5*sqrt(2),5*sqrt(2)),dashed); [/asy] $\text{(A)}\ 300 \qquad \text{(B)}\ 500 \qquad \text{(C)}\ 550 \qquad \text{(D)}\ 600 \qquad \text{(E)}\ 1000$

Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$

You are given a piece of paper. You can cut the paper into $8$ or $12$ pieces. Then you can do so for any of the new pieces or let them uncut and so on. Can you get exactly $60$ pieces¿ Show that you can get every number of pieces greater than $60$.

What is the largest value for $m$ for which I can find nonnegative integers $a_1, a_2, \ldots, a_m < 2024$ such that for all indices $i>j,$ $17$ divides $\tbinom{a_i}{a_j}$?

It is now between $10:00$ and $11:00$ o'clock, and six minutes form now, the minute hand of the watch will be exactly opposite the place where the hour hand was three minutes ago. What is the exact time now? $\textbf{(A) }10:05\dfrac{5}{11}\qquad\textbf{(B) }10:07\dfrac{1}{2}\qquad\textbf{(C) }10:10\qquad\textbf{(D) }10:15\qquad$ $\textbf{(E) }10:17\dfrac{1}{2}$

Two cars travel around a track at equal and constant speeds, each completing a lap every hour. From a common starting point, the first starts at time $t=0$ and the second at an arbitrary later time $t=T>0.$ Prove that there is a total period of exactly one hour during the motion in which the first has completed twice as many laps as the second.