Square $ EFGH$ is inside the square $ ABCD$ so that each side of $ EFGH$ can be extended to pass through a vertex of $ ABCD$. Square $ ABCD$ has side length $ \sqrt {50}$ and $ BE \equal{} 1$. What is the area of the inner square $ EFGH$? [asy]unitsize(4cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair D=(0,0), C=(1,0), B=(1,1), A=(0,1); pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0]; pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H); draw(A--B--C--D--cycle); draw(D--F); draw(C--E); draw(B--H); draw(A--G); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E,NNW); label("$F$",F,ENE); label("$G$",G,SSE); label("$H$",H,WSW);[/asy]$ \textbf{(A)}\ 25\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 40\qquad \textbf{(E)}\ 42$

Prove the following inequality. \[\frac{e-1}{n+1}\leqq\int^e_1(\log x)^n dx\leqq\frac{(n+1)e+1}{(n+1)(n+2)}\ (n=1,2,\cdot\cdot\cdot) \] 1994 Kyoto University entrance exam/Science

In triangle $ ABC$, sides $ a,b$ and $ c$ are opposite angles $ A,B$ and $ C$ respectively. $ AD$ bisects angle $ A$ and meets $ BC$ at $ D$. Then if $ x \equal{} \overline{CD}$ and $ y \equal{} \overline{BD}$ the correct proportion is: $ \textbf{(A)}\ \frac {x}{a} \equal{} \frac {a}{b \plus{} c} \qquad\textbf{(B)}\ \frac {x}{b} \equal{} \frac {a}{a \plus{} c} \qquad\textbf{(C)}\ \frac {y}{c} \equal{} \frac {c}{b \plus{} c} \\ \textbf{(D)}\ \frac {y}{c} \equal{} \frac {a}{b \plus{} c} \qquad\textbf{(E)}\ \frac {x}{y} \equal{} \frac {c}{b}$

Let $n$ be a positive integer. A panel of dimenisions $2n\times2n$ is divided in $4n^2$ squares with dimensions $1\times1$. What is the highest possible number of diagonals that can be drawn in $1\times1$ squares, such that each two diagonals have no common points.

Decide, whether every positive rational number can present in the form $\frac{a^2 + b^3}{c^5 + d^7}$, where $a, b, c, d$ are positive integers.

Sets $A,B$ satisfy that $A\cup B=\{a_1,a_2,a_3\}$. If $A\neq B$, then $(A,B)$ is different from $(B,A)$. The number of such sets $(A,B)$ is $\text{(A)}8\qquad\text{(B)}9\qquad\text{(C)}26\qquad\text{(D)}27$

Is there a power of $2$ that when written in the decimal system has all its digits different from zero and it is possible to reorder them to form another power of $2$?

You are given a deck of $n \cdot m$ different cards where $n$ and $m$ are fixed numbers both between $10^{99}$ and $10^{100}$. You perform an [i]$m$-perfect shuffle[/i] for some times. In a single $m$-perfect shuffle, you divide the deck into $m$ piles with $n$ consecutive cards in each pile. You take one card from each pile, in order of the piles, for $n$ times to form the new deck. (The $m$-perfect shuffle is deterministic) For example, if the cards are labeled 12345678 where $n=4$ and $m=2$, you divide the deck into 1234 and 5678, and after one $2$-perfect shuffle you get 15263748. In another example, if the cards are labeled 123456789 where $n=3$ and $m=3$, you divide the deck into 123, 456, and 789, and after one $3$-perfect shuffle you get 147258369. Find an algorithm that, in at most $k$ steps, outputs the smallest positive number of $m$-perfect shuffle after which the deck is exactly the same as the original deck. In each step, you can do one arithmetic operation in $\{+, -, *, /, \bmod\}$, do one comparison, break out of a loop, or store one number to a specific location of an array. You can use the following precomputed numbers of steps in your solution: [list] [*] Checking if $a$ divides $b$ for any two integers $a$ and $b$ takes 2 steps because you need to compute $b \bmod a$ then compare with $0$. [*]A loop over $k$ iterations takes $2k$ steps because you need to increment the loop index by $1$ $k$ times and check the loop guard $k$ times. [*]Simulating one "$m$-perfect shuffle" takes $7nm$ steps because there is one loop index increment, four arithmetic operations, and one store in each iteration of the loop. [/list] [b]Scoring:[/b] An algorithm that completes in at most $k$ steps will be awarded: [list] [*] 1 pt for $k > 10^{10^{10^{10}}}$ [*] 10 pts for $k = 10^{10^{10^{10}}}$ [*] 20 pts for $k = 10^{420}$ [*] 30 pts for $k = 10^{360}$ [*] 50 pts for $k = 10^{240}$ [*] 70 pts for $k = 10^{202}$ [*] 80 pts for $k = 10^{201}$ [*] 95 pts for $k = 10^{120}$ [*] 98 pts for $k = 10^{102}$ [*] 100 pts for $k = 10^{101}$ [/list] [i]Proposed by Mingkuan Xu[/i]

Let $p$ be a prime number. Prove that the determinant of the matrix \[ \begin{bmatrix}x & y & z\\ x^p & y^p & z^p \\ x^{p^2} & y^{p^2} & z^{p^2} \end{bmatrix} \] is congruent modulo $p$ to a product of polynomials of the form $ax+by+cz$, where $a$, $b$, and $c$ are integers. (We say two integer polynomials are congruent modulo $p$ if corresponding coefficients are congruent modulo $p$.)

There is a unique function $f: \mathbb{N} \to \mathbb{R}$ such that $f(1) > 0$ and such that \[\sum_{d \mid n} f(d) f\left(\frac{n}{d}\right) = 1\] for all $n \ge 1$. What is $f(2018^{2019})$?

Find the smallest $x \in\mathbb{N}$ for which $\frac{7x^{25}-10}{83}$ is an integer.

Do there exist positive integers $k$ and $n$ such that for any finite graph $G$ with diameter $k+1$ there exists a set $S$ of at most $n$ vertices such that for any $v\in V(G)\setminus S$, there exists a vertex $u\in S$ of distance at most $k$ from $v$? [i]David Yang.[/i]

For a point $O$ inside a triangle $ABC$, denote by $A_1,B_1, C_1,$ the respective intersection points of $AO, BO, CO$ with the corresponding sides. Let \[n_1 =\frac{AO}{A_1O}, n_2 = \frac{BO}{B_1O}, n_3 = \frac{CO}{C_1O}.\] What possible values of $n_1, n_2, n_3$ can all be positive integers?

Real numbers $a,b,c$ with $a\neq b$ verify $$a^2(b+c)=b^2(c+a)=2023.$$ Find the numerical value of $E=c^2(a+b)$.

Let S be a set of positive integers such that: min { lcm (x, y) : x, y ∈ S, $x \neq y$ } $\ge$ 2 + max S. Prove that $\displaystyle\sum\limits_{x \in S} \frac{1}{x} \le \frac{3}{2} $.

Let $\sigma(n)=\sum_{d|n} d$, the sum of positive divisors of an integer $n>0$. [list] [b](a)[/b] Show that $\sigma(mn)=\sigma(m)\sigma(n)$ for positive integers $m$ and $n$ with $gcd(m,n)=1$ [b](b)[/b] Find all positive integers $n$ such that $\sigma(n)$ is a power of $2$.[/list]