In quadrilateral $ ABCD$, it is given that $ \angle A \equal{} 120^\circ$, angles $ B$ and $ D$ are right angles, $ AB \equal{} 13$, and $ AD \equal{} 46$. Then $ AC \equal{}$ $ \textbf{(A)}\ 60 \qquad \textbf{(B)}\ 62 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 65 \qquad \textbf{(E)}\ 72$

Find all integers $n > 2$ for which $(2n)! = (n-2)!n!(n+2)!$ .

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

Let be three natural numbers $ k,m,n $ an $ m\times n $ matrix $ A, $ an $ n\times m $ matrix $ B, $ and $ k $ complex numbers $ a_0,a_1,\ldots ,a_k $ such that the following conditions hold. $ \text{(i)}\quad m\ge n\ge 2 $ $ \text{(ii)}\quad a_0I_m+a_1AB+a_2(AB)^2+\cdots +a_k(AB)^k=O_m $ $ \text{(iii)}\quad a_0I_m+a_1BA+a_2(BA)^2+\cdots +a_k(BA)^k\neq O_n $ Prove that $ a_0=0. $

The diameter $\overline{AB}$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and the line $ED$ is perpendicular to the line $AD$. Segment $\overline{AE}$ intersects the circle at point $C$ between $A$ and $E$. What is the area of $\triangle ABC$? $\textbf{(A) \ } \frac{120}{37}\qquad \textbf{(B) \ } \frac{140}{39}\qquad \textbf{(C) \ } \frac{145}{39}\qquad \textbf{(D) \ } \frac{140}{37}\qquad \textbf{(E) \ } \frac{120}{31}$

Find the largest prime number less than $2008$ that is a divisor of some integer in the infinite sequence \[\left\lfloor\dfrac{2008}1\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^2}2\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^3}3\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^4}4\right\rfloor,\,\,\,\,\,\,\,\,\,\ldots.\]

$x,y$ are naturals. $GCM(x^7,y^4)*GCM(x^8,y^5)=xy$ Prove that $xy$ is cube

Let $n$ be an odd positive integer such that both $\phi(n)$ and $\phi (n+1)$ are powers of two. Prove $n+1$ is power of two or $n=5$.

Let the diagonals of the square $ABCD$ intersect at $S$ and let $P$ be the midpoint of $AB$. Let $M$ be the intersection of $AC$ and $PD$ and $N$ the intersection of $BD$ and $PC$. A circle is incribed in the quadrilateral $PMSN$. Prove that the radius of the circle is $MP- MS$.

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that \[ \frac{1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|. \] When does equality hold?

There are real numbers $a,b$ and $c$ and a positive number $\lambda$ such that $f(x)=x^3+ax^2+bx+c$ has three real roots $x_1, x_2$ and $x_3$ satisfying $(1) x_2-x_1=\lambda$ $(2) x_3>\frac{1}{2}(x_1+x_2)$. Find the maximum value of $\frac{2a^3+27c-9ab}{\lambda^3}$

Let $n \in \mathbb{N}$ and $S_{n}$ the set of all permutations of $\{1,2,3,...,n\}$. For every permutation $\sigma \in S_{n}$ denote $I(\sigma) := \{ i: \sigma (i) \le i \}$. Compute the sum $\sum_ {\sigma \in S_{n}} \frac{1}{|I(\sigma )|} \sum_ {i \in I(\sigma)} (i+ \sigma(i))$.

Let $(a,\ b)$ be a point on the curve $y=\frac{x}{1+x}\ (x\geq 0).$ Denote $U$ the volume of the figure enclosed by the curve , the $x$ axis and the line $x=a$, revolved around the the $x$ axis and denote $V$ the volume of the figure enclosed by the curve , the $y$ axis and th line $y=b$, revolved around the $y$ axis. What's the relation of $U$ and $V?$ 1978 Chuo university entrance exam/Science and Technology

In the plane, $2022$ points are chosen such that no three points lie on the same line. Each of the points is coloured red or blue such that each triangle formed by three distinct red points contains at least one blue point. What is the largest possible number of red points? [i]Proposed by Art Waeterschoot, Belgium[/i]

The real numbers x and у satisfy the equations $$\begin{cases} \sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\ \\ \sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt2 \end{cases}$$ Find the numerical value of the ratio $y/x$.

Given in the plane the square $ABCD$, the square $A_1B_1C_1D_1$, smaller than the first, and a quadrilateral $PQRS$ that satisfy the following conditions $\bullet$ $ABCD$ and $A_1B_1C_1D_1$ have a common center and respectively parallel sides. $\bullet$$P$, $Q$, $R$, $S$ belong one to each side of the square $ABCD$. $\bullet$ $A_1$, $B_1$, $C_1$, $D_1$ belong one to each side of the quadrilateral $PQRS$. Prove that $PQRS$ is a square.

[b]p1.[/b] Solve the system of equations $$xy = 2x + 3y$$ $$yz = 2y + 3z$$ $$zx =2z+3x$$ [b]p2.[/b] For any integer $k$ greater than $1$ and any positive integer $n$ , prove that $n^k$ is the sum of $n$ consecutive odd integers. [b]p3.[/b] Determine all pairs of real numbers, $x_1$, $x_2$ with $|x_1|\le 1$ and $|x_2|\le 1$ which satisfy the inequality: $|x^2-1|\le |x-x_1||x-x_2|$ for all $x$ such that $|x| \ge 1$. [b]p4.[/b] Find the smallest positive integer having exactly $100$ different positive divisors. (The number $1$ counts as a divisor). [b]p5.[/b] $ABC$ is an equilateral triangle of side $3$ inches. $DB = AE = 1$ in. and $F$ is the point of intersection of segments $\overline{CD}$ and $\overline{BE}$ . Prove that $\overline{AF} \perp \overline{CD}$. [img]https://cdn.artofproblemsolving.com/attachments/f/a/568732d418f2b1aa8a4e8f53366df9fbc74bdb.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A piece of cardboard has the shape of a pentagon $ABCDE$ in which $BCDE$ is a square and $ABE$ is an isosceles triangle with a right angle at $A$. Prove that the pentagon can be divided in two different ways in three parts that can be rearranged in order to recompose a right isosceles triangle.

A point $(a,b)$ in the plane is called [i]sparkling[/i] if it also lies on the line $ax+by=1$. Find the maximum possible distance between two sparkling points. [i]Proposed by Evan Chen[/i]

The [i]giraffe[/i] is a chess piece that moves $4$ squares in one direction and then a box in a perpendicular direction. What is the smallest value of $n$ such that the giraffe that starts from a corner on an $n \times n$ board can visit all the squares of said board?

Let G, H be two finite groups, and let $\varphi, \psi: G \to H$ be two surjective but non-injective homomorphisms. Prove that there exists an element of G that is not the identity element of G but whose images under $\varphi$ and $\psi$ are the same.

Point $L$ is marked on the side $AB$ of a triangle $ABC$. The incircle of the triangle $ABC$ meets the segment $CL$ at points $P$ and $Q$ .Is it possible that the equalities $CP = PQ = QL$ hold if $CL$ is a) the median? b) the bisector? c) the altitude? d) the segment joining vertex $C$ with the point $L$ of tangency of the excircle of the triangie $ABC$ with $AB$ ? (I. Gorodnin)

Find all integer solutions $(x,y)$ of the equation $y^2=x^3-p^2x,$ where $p$ is a prime such that $p\equiv 3 \mod 4.$

$N$ is the set of positive integers. Does there exist a function $f: N \to N$ such that $f(n+1) = f( f(n) ) + f( f(n+2) )$ for all $n$?