What is the value of $$(\sqrt{2}-1)^4+\frac{1}{(\sqrt{2}-1)^4}?$$ $\textbf{(A) }32-16\sqrt{2}\qquad\textbf{(B) }30\qquad\textbf{(C) }34\qquad\textbf{(D) }15+15\sqrt{2}\qquad\textbf{(E) }16+16\sqrt{2}$

If $x,y,z$ are real numbers satisfying relations \[x+y+z = 1 \quad \textrm{and} \quad \arctan x + \arctan y + \arctan z = \frac{\pi}{4},\] prove that $x^{2n+1} + y^{2n+1} + z^{2n+1} = 1$ holds for all positive integers $n$.

Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers ${{O} _ {1}}$ and ${{O} _ {2}}$ intersect at points $A$ and $B$, respectively. The line ${{O} _ {1}} {{O} _ {2}}$ intersects ${{w} _ {1}}$ at the point $Q$, which does not lie inside the circle ${{w} _ {2}}$, and ${{w} _ {2}}$ at the point $X$ lying inside the circle ${{w} _ {1} }$. Around the triangle ${{O} _ {1}} AX$ circumscribe a circle ${{w} _ {3}}$ intersecting the circle ${{w} _ {1}}$ for the second time in point $T$. The line $QT$ intersects the circle ${{w} _ {3}}$ at the point $K$, and the line $QB$ intersects ${{w} _ {2}}$ the second time at the point $H$. Prove that a) points $T, \, \, X, \, \, B$ lie on one line; b) points $K, \, \, X, \, \, H$ lie on one line. (Vadym Mitrofanov)

At a store, when a length is reported as $x$ inches that means the length is at least $x-0.5$ inches and at most $x+0.5$ inches. Suppose the dimensions of a rectangular tile are reported as $2$ inches by $3$ inches. In square inches, what is the minimum area for the rectangle? $ \textbf{(A)}\ 3.75 \qquad \textbf{(B)}\ 4.5 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8.75 $

How many subsets, which does not contain two consecutive numbers, are there of the set $\{1,2,\dots ,20\}$ with $8$ elements? $\textbf{(A)}\ {{13}\choose{8}} \qquad\textbf{(B)}\ {{13}\choose{9}} \qquad\textbf{(C)}\ {{14}\choose{8}} \qquad\textbf{(D)}\ {{14}\choose{9}} \qquad\textbf{(E)}\ {{20}\choose{15}}$

Let $e > 0$ be a given real number. Find the least value of $f(e)$ (in terms of $e$ only) such that the inequality $a^{3}+ b^{3}+ c^{3}+ d^{3} \leq e^{2}(a^{2}+b^{2}+c^{2}+d^{2}) + f(e)(a^{4}+b^{4}+c^{4}+d^{4})$ holds for all real numbers $a, b, c, d$.

Let n numbers $x_1, x_2, \ldots, x_n$ be chosen in such a way that $1 \geq x_1 \geq x_2 \geq \cdots \geq x_n \geq 0$. Prove that \[(1 + x_1 + x_2 + \cdots + x_n)^\alpha \leq 1 + x_1^\alpha+ 2^{\alpha-1}x_2^\alpha+ \cdots+ n^{\alpha-1}x_n^\alpha\] if $0 \leq \alpha \leq 1$.

The natural numbers $a$ and $b$ satisfy the inequalities $a > b > 1$ . It is also known that the equation $\frac{a^x - 1}{a - 1}=\frac{b^y - 1}{b - 1}$ has at least two solutions in natural numbers, when $x > 1$ and $y > 1$. Prove that the numbers $a$ and $b$ are coprime (their greatest common divisor is $1$).

Consider the function $ f: \mathbb{N}_0\to\mathbb{N}_0$, where $ \mathbb{N}_0$ is the set of all non-negative integers, defined by the following conditions : $ (i)$ $ f(0) \equal{} 0$; $ (ii)$ $ f(2n) \equal{} 2f(n)$ and $ (iii)$ $ f(2n \plus{} 1) \equal{} n \plus{} 2f(n)$ for all $ n\geq 0$. $ (a)$ Determine the three sets $ L \equal{} \{ n | f(n) < f(n \plus{} 1) \}$, $ E \equal{} \{n | f(n) \equal{} f(n \plus{} 1) \}$, and $ G \equal{} \{n | f(n) > f(n \plus{} 1) \}$. $ (b)$ For each $ k \geq 0$, find a formula for $ a_k \equal{} \max\{f(n) : 0 \leq n \leq 2^k\}$ in terms of $ k$.

Let $ABC$ be a triangle, let $I$ be its incentre and let $D$ be the orthogonal projection of $I$ on $BC.$ The circle $\odot(ABC)$ crosses the line $AI$ again at $M,$ and the line $DM$ again at $N.$ Prove that the lines $AN$ and $IN$ are perpendicular. [i]Freddie Illingworth & Dominic Yeo[/i]

Find all triplets $(p,a,b)$ of positive integers such that $$p=b\sqrt{\frac{a-8b}{a+8b}}$$ is prime

Let $M$ be the set of the positive rational numbers less than 1, which can be expressed with a 10-distinct digits period in decimal representation. a) Find the arithmetic mean of all the elements in $M$; b) Prove that there exists a positive integer $n$, $1<n<10^{10}$, such that $n\cdot a - a$ is a non-negative integer, for all $a\in M$.

On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots? $ \textbf{(A)}\ \frac {5}{11} \qquad \textbf{(B)} \ \frac {10}{21} \qquad \textbf{(C)}\ \frac {1}{2} \qquad \textbf{(D)} \ \frac {11}{21} \qquad \textbf{(E)}\ \frac {6}{11}$

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is $100$, and one of the numbers is $28$. What is the other number? $\textbf{(A) } 8 \qquad\textbf{(B) } 11 \qquad\textbf{(C) } 14 \qquad\textbf{(D) } 15 \qquad\textbf{(E) } 18 $

Find all pairs of positive integers $m$ and $n$ such that $$(m + 1)! + (n + 1)! = m^2n.$$

$201$ positive integers are written on a line, such that both the first one and the last one are equal to $19999$. Each one of the remaining numbers is less than the average of its neighbouring numbers, and the differences between each one of the remaining numbers and the average of its neighbouring numbers are all equal to a unique integer. Find the second-to-last term on the line.

In a tennis tournament, any two of the $n$ participants played a match (the winner of a match gets $1$ point, the loser gets $0$). The scores after the tournament were $r_1\le r_2\le\ldots\le r_n$. A match between two players is called wrong if after it the winner has a score less than or equal to that of the loser. Consider the set $I=\{i|r_1\ge i\}$. Prove that the number of wrong matches is not less than $\sum_{i\in I}(r_i-i+1)$, and show that this value is realizable