Found problems: 1671
Given a sequence $(x_1,x_2,\ldots, x_n)$ of 0's and 1's, let $A$ be the number of triples $(x_i,x_j,x_k)$ with $i<j<k$ such that $(x_i,x_j,x_k)$ equals $(0,1,0)$ or $(1,0,1)$. For $1\leq i \leq n$, let $d_i$ denote the number of $j$ for which either $j < i$ and $x_j = x_i$ or else $j > i$ and $x_j\neq x_i$.
(a) Prove that \[A = \binom n3 - \sum_{i=1}^n\binom{d_i}2.\] (Of course, $\textstyle\binom ab = \tfrac{a!}{b!(a-b)!}$.) [5 points]
(b) Given an odd number $n$, what is the maximum possible value of $A$? [15 points]
Prove that for all positive real numbers $a, b, c$ with $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} =1$ the following inequality holds:
$$3(ab+bc+ca)+\frac{9}{a+b+c} \le \frac{9abc}{a+b+c} + 2(a^2+b^2+c^2)+1$$
Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m,n\in \mathbb{Z}$: \[f(m+f(n))=f(m)-n.\]
Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that
\[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\]
and
\[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\]
[i]Calvin Deng.[/i]
Find all $f:R-R$ such that $f(x^2+y+f(y))=2y+f(x)^2$
Prove for positive $a,b,c$ that
$$ (a^2+\frac{b^2}{c^2})(b^2+\frac{c^2}{a^2})(c^2+\frac{a^2}{b^2}) \geq abc (a+\frac{1}{a})(b+\frac{1}{b})(c+\frac{1}{c})$$
Do there exist 100 positive distinct integers such that a cube of one of them equals the sum of the cubes of all the others?
[i]Mikhail Evdokimov[/i]
Determine all functions $f:\mathbb R\to\mathbb R$ such that equality
$$f(x + y + yf(x)) = f(x) + f(y) + xf(y)$$
holds for all real numbers $x$, $y$.
Proposed by Athanasios Kontogeorgis
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions
\[f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},\]
and $f(-1) \neq 0$.
A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\omega$. Suppose that \[
(AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2.
\] Prove that $ABCD$ is an isosceles trapezoid.
Let $a,b,c,d,x,y$ denote the lengths of the sides $AB, BC,CD,DA$ and the diagonals $AC,BD$ of a cyclic quadrilateral $ABCD$ respectively.
Prove that $$(\frac{1}{a}+\frac{1}{c})^2+(\frac{1}{b}+\frac{1}{d})^2 \geq 8 ( \frac{1}{x^2}+\frac{1}{y^2})$$
Prove that the only function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following is $f(x)=x$.
(i) $\forall x \not= 0$, $f(x) = x^2f(\frac{1}{x})$.
(ii) $\forall x, y$, $f(x+y) = f(x)+f(y)$.
(iii) $f(1)=1$.
Let $x_1, x_2, \dots ,x_n$ ($n\ge 2$) be positive real numbers satisfying $\sum^{n}_{i=1}x_i=1$. Prove that:\[\sum^{n}_{i=1}\dfrac{x_i}{\sqrt{1-x_i}}\ge \dfrac{\sum_{i=1}^{n}\sqrt{x_i}}{\sqrt{n-1}}.\]
In a group of $m$ girls and $n$ boys, any two persons either know each other or do not know each other. For any two boys and any two girls, there are at least one boy and one girl among them,who do not know each other. Prove that the number of unordered pairs of (boy, girl) who know each other does not exceed $m+\frac{n(n-1)}{2}$.
Prove that
$$\sqrt{x-1}+\sqrt{2x+9}+\sqrt{19-3x}<9$$
for all real $x$ for which the left-hand side is well defined.
Let $P$ be a point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12$. If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$, then the minimum possible value of $PX^2 + PY^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Andrew Wen[/i]
Let $\overline{AH_1}, \overline{BH_2}$, and $\overline{CH_3}$ be the altitudes of an acute scalene triangle $ABC$. The incircle of triangle $ABC$ is tangent to $\overline{BC}, \overline{CA},$ and $\overline{AB}$ at $T_1, T_2,$ and $T_3$, respectively. For $k = 1, 2, 3$, let $P_i$ be the point on line $H_iH_{i+1}$ (where $H_4 = H_1$) such that $H_iT_iP_i$ is an acute isosceles triangle with $H_iT_i = H_iP_i$. Prove that the circumcircles of triangles $T_1P_1T_2$, $T_2P_2T_3$, $T_3P_3T_1$ pass through a common point.
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
Let $x$, $y$, $z$ be positive real numbers satisfying $x+y+z\ge 6$. Find, with proof, the minimum value of \[ x^2+y^2+z^2+\frac{x}{y^2+z+1}+\frac{y}{z^2+x+1}+\frac{z}{x^2+y+1}. \]
Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\to0$ as $x\to\infty$.
Let $f$ be a function defined on the positive integers, taking positive integral values, such that
$f(a)f(b) = f(ab)$ for all positive integers $a$ and $b$,
$f(a) < f(b)$ if $a < b$,
$f(3) \geq 7$.
Find the smallest possible value of $f(3)$.
If $x$, $y$, $z$ are positive numbers satisfying
\[x+\frac{y}{z}=y+\frac{z}{x}=z+\frac{x}{y}=2.\]
Find all the possible values of $x+y+z$.
Let $x, y, z$ be positive real numbers satisfying $x + y + z =\frac{2}{x+y}+\frac{2}{y+z}+\frac{2}{z+x}$. Prove that $$\frac{3}{2}\leq\frac{3}{\sqrt[3]{xyz}(\sqrt[3]{xyz}+1)}\leq\frac{1}{x(y+1)}+\frac{1}{y(z+1)}+\frac{1}{z(x+1)}.$$
Find the largest possible value of $M$ for which $\frac{x}{1+\frac{yz}{x}}+\frac{y}{1+\frac{zx}{y}}+\frac{z}{1+\frac{xy}{z}}\geq M$ for all $x,y,z>0$ with $xy+yz+zx=1$
$\triangle ABC$ has semiperimeter $s$ and area $F$ . A square $P QRS$ with side length $x$ is inscribed in $ABC$ with $P$ and $Q$ on $BC$, $R$ on $AC$, and $S$ on $AB$. Similarly, $y$ and $z$ are the sides of squares two vertices of which lie on $AC$ and $AB$, respectively. Prove that
\[\frac 1x +\frac 1y + \frac 1z \le \frac{s(2+\sqrt3)}{2F}\]