Prove that the area of the circle inscribed in a regular hexagon is greater than $90\%$ of the area of the hexagon.

A hiking club wants to hike around a lake along an exactly circular route. On the shoreline they determine two points, which are the most distant from each other, and start to walk along the circle, which has these two points as the endpoints of its diameter. Can they be sure that, independent of the shape of the lake, they do not have to swim across the lake on any part of their route?

Is it true that of the four heights of an arbitrary tetrahedron, three can be selected from which a triangle can be made?

Prove that there exists a positive non-prime integer such that if any three of its neighboring digits are replaced with any given triple of the digits, the number remains non-prime. Does there exist a $1997$-digit such number?

Bishops on a chessboard move along the diagonals ( that is , on lines parallel to the two main diagonals ) . Prove that the maximum number of non-attacking bishops on an $n*n$ chessboard is $2n-2$. (Two bishops are said to be attacking if they are on a common diagonal).

Determine all polynomials $p$ with real coefficients for which $p(0)=0$ and $$f(f(n))+n=4f(n)\qquad\text{for all }n\in\mathbb N,$$where $f(n)=\lfloor p(n)\rfloor$.

Find all functions $f : R \to R$ such that $$2f(x)f(x + y) -f(x^2) =\frac{x}{2}(f(2x) + 4f(f(y)))$$ for all $x, y \in R$.

Suppose that $f:\{1, 2,\ldots ,1600\}\rightarrow\{1, 2,\ldots ,1600\}$ satisfies $f(1)=1$ and \[f^{2005}(x)=x\quad\text{for}\ x=1,2,\ldots ,1600. \] $(a)$ Prove that $f$ has a fixed point different from $1$. $(b)$ Find all $n>1600$ such that any $f:\{1,\ldots ,n\}\rightarrow\{1,\ldots ,n\}$ satisfying the above condition has at least two fixed points.

Let $n\geq 1$ be an integer. Find all rings $(A,+,\cdot)$ such that all $x\in A\setminus\{0\}$ satisfy $x^{2^{n}+1}=1$.

The King decided to reduce his Council consisting of thousand wizards. He placed them in a line and placed hats with numbers from $1$ to $1001$ on their heads not necessarily in this order (one hat was hidden). Each wizard can see the numbers on the hats of all those before him but not on himself or on anyone who stayed behind him. By King's command, starting from the end of the line each wizard calls one integer from $1$ to $1001$ so that every wizard in the line can hear it. No number can be repeated twice. In the end each wizard who fails to call the number on his hat is removed from the Council. The wizards knew the conditions of testing and could work out their strategy prior to it. (a) Can the wizards work out a strategy which guarantees that more than $500$ of them remain in the Council? (b) Can the wizards work out a strategy which guarantees that at least $999$ of them remain in the Council?

A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$ and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$ and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\angle{XIY}=90^{\circ}$.

Show that in a convex region in the plane whose boundary contains at most a finite number of straight line segments and whose area is greater than $\frac{\pi}{4}$ there is at least one pair of points a unit distance apart.

Let $\{a_{n}\}$ be a sequence which satisfy $a_{1}=5$ and $a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \qquad \forall n\geq2$ [b](a)[/b] Find the general fomular for $a_{n}$ [b](b)[/b] Prove that $\{a_{n}\}$ is decreasing sequences

In Ace Attorney, Phoenix Wright is rolling a standard fair $20$-sided die. He can roll this die up to three times. After each roll, Phoenix can yell "Objection!'' to roll again, or "Hold It!'' to stop and keep his current number. If Phoenix plays optimally to maximize his final number, find the expected value of this number.

The sum of the numbers $a,b$ and $c$ is zero, and their product is negative. Prove that the number $\frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}$ is positive.

$n$ parallel segments of lengths $a_1 \le a_2 \le a_3 \le ... \le a_n$ were painted to mark an airport atrium. However, the architect decided that the $n$ segments should have equal length. If the cost per meter of extending the lines is equal to the cost of reducing them, how long should the lines be in order to minimize costs?

Prove that two congruent regular hexagons can be cut up into (altogether) $6$ parts such that these $6$ parts can be composed to form an equilateral triangle (without gaps or overlaps).

A function $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, where $\mathbb{Z}_+$ is the set of positive integers, is non-decreasing and satisfies $f(mn) = f(m)f(n)$ for all relatively prime positive integers $m$ and $n$. Prove that $f(8)f(13) \ge (f(10))^2$.

Find all the ways of placing the integers $1,2,3,\cdots,16$ in the boxes below, such that each integer appears in exactly one box, and the sum of every pair of neighboring integers is a perfect square. [asy] import graph; real r=10; size(r*cm); picture square1; draw(square1, (0,0)--(0,1)--(1,1)--(1,0)--cycle); add(scale(r/31*cm)*square1,(0,0)); picture square2; draw(square2, (-1,0.5)--(0,0.5)--(0,1)--(1,1)--(1,0)--(0,0)--(0,0.5)); for(int i=1; i<16; ++i) { add(scale(r/31*cm)*square2,(i,0)); }[/asy]

One of the receipts for a math tournament showed that $72$ identical trophies were purchased for $\$$-$99.9$-, where the first and last digits were illegible. How much did each trophy cost?

Given a point $P$ on a triangular piece of paper $ABC,$ consider the creases that are formed in the paper when $A, B,$ and $C$ are folded onto $P.$ Let us call $P$ a fold point of $\triangle ABC$ if these creases, which number three unless $P$ is one of the vertices, do not intersect. Suppose that $AB=36, AC=72,$ and $\angle B=90^\circ.$ Then the area of the set of all fold points of $\triangle ABC$ can be written in the form $q\pi-r\sqrt{s},$ where $q, r,$ and $s$ are positive integers and $s$ is not divisible by the square of any prime. What is $q+r+s$?

Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $$\frac{1}{2y}\int_{x-y}^{x+y}f(t)\, dt=f(x)\qquad\forall~x\in\mathbb{R}~\&~y>0$$ Show that there exist $a,b\in\mathbb{R}$ such that $f(x)=ax+b$ for all $x\in\mathbb{R}$.

Let $z=x+yi$ be a complex number with $x$ and $y$ rational and with $|z|=1.$ Prove that the number $|z^{2n} -1|$ is rational for every integer $n$.

There are $90$ cards and two different digits are written on each one: $01$, $02$, $03$, $04$, $05$, $06$, $07$, $08$, $09$, $10$, $12$, and so on up to $98$. A set of cards is [i]correct [/i]if it does not contain any cards whose first digit is the same as the second digit of another card in the set. We call the [i]value [/i]of a set of cards the sum of the numbers written on each card. For example, the four cards $04$, $35$, $78$ and $98$ form a correct set and their value is $215$, since$ 04+35+78+98=215$. Find a correct set that has the largest possible value. Explain why it is impossible to achieve a correct set of higher value.

A four-digit number $n$ is said to be [i]literally 1434[/i] if, when every digit is replaced by its remainder when divided by $5$, the result is $1434$. For example, $1984$ is [i]literally 1434[/i] because $1$ mod $5$ is $1$, $9$ mod $5$ is $4$, $8$ mod $5$ is $3$, and $4$ mod $5$ is $4$. Find the sum of all four-digit positive integers that are [i]literally 1434[/i]. [i]Proposed by Evin Liang[/i] [hide=Solution] [i]Solution.[/i] $\boxed{67384}$ The possible numbers are $\overline{abcd}$ where $a$ is $1$ or $6$, $b$ is $4$ or $9$, $c$ is $3$ or $8$, and $d$ is $4$ or $9$. There are $16$ such numbers and the average is $\dfrac{8423}{2}$, so the total in this case is $\boxed{67384}$. [/hide]