Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be such that $$f(x+f(y))^2\geq f(x)\left(f(x+f(y))+f(y)\right)$$ for all $x,y\in\mathbb{R}^+$. Show that $f$ is [i]unbounded[/i], i.e. for each $M\in\mathbb{R}^+$, there exists $x\in\mathbb{R}^+$ such that $f(x)>M$.

The circle circumscribed about an acute triangle $ABC$ and the vertex $C$ are fixed. Orthocenter $H$ moves in a circle with center at point $C$. Find the locus of the midpoints of the segments connecting the feet of altitudes drawn from vertices $A$ and $B$.

There are 1997 pieces of medicine. Three bottles $A, B, C$ can contain at most 1997, 97, 19 pieces of medicine respectively. At first, all 1997 pieces are placed in bottle $A$, and the three bottles are closed. Each piece of medicine can be split into 100 part. When a bottle is opened, all pieces of medicine in that bottle lose a part each. A man wishes to consume all the medicine. However, he can only open each of the bottles at most once each day, consume one piece of medicine, move some pieces between the bottles, and close them. At least how many parts will be lost by the time he finishes consuming all the medicine?

Let $ABC$ be a triangle with $AB<AC$. Let $\omega$ be a circle passing through $B, C$ and assume that $A$ is inside $\omega$. Suppose $X, Y$ lie on $\omega$ such that $\angle BXA=\angle AYC$. Suppose also that $X$ and $C$ lie on opposite sides of the line $AB$ and that $Y$ and $B$ lie on opposite sides of the line $AC$. Show that, as $X, Y$ vary on $\omega$, the line $XY$ passes through a fixed point. [i]Proposed by Aaron Thomas, UK[/i]

Messages are coded using sequences consisting of zeroes and ones only. Only sequences with at most two consecutive ones or zeroes are allowed. (For instance the sequence $011001$ is allowed, but $011101$ is not.) Determine the number of sequences consisting of exactly $12$ numbers.

Find all pairs of positive integers $(x,n) $ that are solutions of the equation $3 \cdot 2^x +4 =n^2$.

Let $r, s, t, u$ be the distinct roots of the polynomial $x^4 + 2x^3 + 3x^2 + 3x + 5$. For $n \ge 1$, define $s_n = r^n + s^n + t^n + u^n$ and $t_n = s_1 + s_2 + ...+ s_n$. Compute $t_4 + 2t_3 + 3t_2 + 3t_1 + 5$.

Let $a_0$ be an irrational number such that $0 < a_0 < \frac 12$ . Define $a_n = \min \{2a_{n-1},1 - 2a_{n-1}\}$ for $n \geq 1$. [list][b](a)[/b] Prove that $a_n < \frac{3}{16}$ for some $n$. [b](b)[/b] Can it happen that $a_n > \frac{7}{40}$ for all $n$?[/list]

Let $ M,M'$ be two conjugates point in triangle $ ABC$ (in the sense that $ \angle MAB\equal{}\angle M'AC,\dots$). Let $ P,Q,R,P',Q',R'$ be foots of perpendiculars from $ M$ and $ M'$ to $ BC,CA,AB$. Let $ E\equal{}QR\cap Q'R'$, $ F\equal{}RP\cap R'P'$ and $ G\equal{}PQ\cap P'Q'$. Prove that the lines $ AG, BF, CE$ are parallel.

A circle of radius $2$ is inscribed in an isosceles trapezoid with the area of $28$. Find the length of the side of the trapezoid.

For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases} $$ Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$. [i]Proposed by Stephan Wagner, South Africa[/i]

Let $ S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\}$ be a $ (k \plus{} l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called [i]nice[/i] if \[ \left |\frac {1}{k}\sum_{x_i\in A} x_i \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl}\] Prove that the number of nice subsets is at least $ \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}$. [i]Proposed by Andrey Badzyan, Russia[/i]

Let $a, b, c$ be positive real numbers with $abc = 1$. Prove that $1 + \frac{3}{a+b+c}\ge \frac{6}{ab+bc+ca}$

[b]p1.[/b] A right triangle has hypotenuse of length $12$ cm. The height corresponding to the right angle has length $7$ cm. Is this possible? [img]https://cdn.artofproblemsolving.com/attachments/0/e/3a0c82dc59097b814a68e1063a8570358222a6.png[/img] [b]p2.[/b] Prove that from any $5$ integers one can choose $3$ such that their sum is divisible by $3$. [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a knight on an arbitrary square. Then the second player can put another knight on a free square that is not controlled by the first knight. Then the first player can put a new knight on a free square that is not controlled by the knights on the board. Then the second player can do the same, etc. A player who cannot put a new knight on the board loses the game. Who has a winning strategy? [b]p4.[/b] Consider a regular octagon $ABCDEGH$ (i.e., all sides of the octagon are equal and all angles of the octagon are equal). Show that the area of the rectangle $ABEF$ is one half of the area of the octagon. [img]https://cdn.artofproblemsolving.com/attachments/d/1/674034f0b045c0bcde3d03172b01aae337fba7.png[/img] [b]p5.[/b] Can you find a positive whole number such that after deleting the first digit and the zeros following it (if they are) the number becomes $24$ times smaller? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A large golden square land lot of dimension $100 \times 100$ m was subdivided into $100$ square lots, each measured $10\times10$ m. A king of landfill had his men dump wastes onto some of the lots. There was a practice that if a particular lot was not dumped and twoof its adjacents had waste materials, then the lot would be filled with wastes the next day by the people. One day if all the lotswere filled with wastes, the king would claim his ownership ofthe whole land lot. At least how many lots should have the kind had his men dump wastes onto? Vu Ha Van, Mathematics Faculty, Yale University, USA.

Prove: (a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$ (b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$

Siva has the following expression, which is missing operations: $$\frac12 \,\, \_ \,\,\frac14 \,\, \_ \,\, \frac18 \,\, \_ \,\,\frac{1}{16} \,\, \_ \,\,\frac{1}{32}.$$ For each blank, he flips a fair coin: if it comes up heads, he fills it with a plus, and if it comes up tails, he fills it with a minus. Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coinflips and operations. If the probability that the positive difference between his computed values is greater than $\frac12$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, then find $a + b$.

Let $A_1A_2A_3A_4$ be a tetrahedron. We construct four mutually tangent spheres $S_1,S_2,S_3,S_4$ with centers $A_1,A_2,A_3,A_4$ respectively. Suppose that there exists a point $Q$ such that we can construct two spheres centered at $Q$ satisfying the following conditions: i) One sphere with radius $r$ is tangent to $S_1,S_2,S_3,S_4$; ii) One sphere with radius $R$ is tangent to every edges of tetrahedron $A_1A_2A_3A_4$. Prove that $A_1A_2A_3A_4$ is a regular tetrahedron.

Let $n\geq 2$ be an integer. Find the smallest real value $\rho (n)$ such that for any $x_i>0$, $i=1,2,\ldots,n$ with $x_1 x_2 \cdots x_n = 1$, the inequality \[ \sum_{i=1}^n \frac 1{x_i} \leq \sum_{i=1}^n x_i^r \] is true for all $r\geq \rho (n)$.

$g$ and $n$ are natural numbers such that $gcd(g^2-g,n)=1$ and $A=\{g^i|i \in \mathbb N\}$ and $B=\{x\equiv (n)|x\in A\}$(by $x\equiv (n)$ we mean a number from the set $\{0,1,...,n-1\}$ which is congruent with $x$ modulo $n$). if for $0\le i\le g-1$ $a_i=|[\frac{ni}{g},\frac{n(i+1)}{g})\cap B|$ prove that $g-1|\sum_{i=0}^{g-1}ia_i$.( the symbol $|$ $|$ means the number of elements of the set)($\frac{100}{6}$ points) the exam time was 4 hours

Let $A$ be the area of the region in the first quadrant bounded by the line $y = \frac{x}{2}$, the x-axis, and the ellipse $\frac{x^2}{9} + y^2 = 1$. Find the positive number $m$ such that $A$ is equal to the area of the region in the first quadrant bounded by the line $y = mx,$ the y-axis, and the ellipse $\frac{x^2}{9} + y^2 = 1.$

Prove that there exists a function $f\colon [\omega_1]^2 \rightarrow \omega _1$ such that (i) $f(\alpha, \beta)< \mathrm{min}(\alpha, \beta)$ whenever $\mathrm{min}(\alpha,\beta)>0$; and (ii) if $\alpha_0<\alpha_1<\ldots<\alpha_i<\ldots<\omega_1$ then $\sup\left\{ a_i \colon i<\omega \right\} =\sup \left\{ f(\alpha_i, \alpha_j)\colon i,j<\omega\right\}$.

A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square? [i]Proposed by Evan Chen[/i]

Let $n\ge 2$ and $d\ge 1$ be integers with $d\mid n$, and let $x_1,x_2,\ldots x_n$ be real numbers such that $x_1+x_2+\cdots + x_n=0$. Show that there are at least $\binom{n-1}{d-1}$ choices of $d$ indices $1\le i_1<i_2<\cdots <i_d\le n $ such that $x_{i_{1}}+x_{i_{2}}+\cdots +x_{i_{d}}\ge 0$.

The incircle of a right-angled triangle $ABC$ touches hypotenus $AB$ at $P$, $BC$ and $AC$ at $R$ and $Q$ respectively. $C_1$ and $C_2$ are reflections of $C$ in $PQ$ and $PR$. Find the angle $C_1IC_2$, where $I$ is the incenter of $ABC$.