If $kx + 12 = 3k$, for how many integer values of $k$ is $x$ a positive integer?
Four towns are located at $A(0,0)$, $B(2,12)$, $C(12,8)$, and $D(7,2)$. A warehouse is built at point $P$ so the sum of the distances $PA + PB + PC + PD$ is minimized. We must find the coordinates of point $P$.
Consider all integer values of $a$ and $b$ for which $a < 2$ and $b \ge -2$. We are asked to find the minimum value of $b -a$.
Let $z$ be a complex number, and $|z|=1$. Find the maximal value of $u=|z^3-3z+2|$.
Let positive real number $x$, $y$, and $z$ satisfy $x+y+z=1$. Find the minimal value of $u=\sqrt{x^2 + y^2 + xy} + \sqrt{y^2 +z^2 +yz} +\sqrt{z^2 +x^2 + xz}$
Let $A=x\cos^2{\theta} + y\sin^2{\theta}$, $B=x\sin^2{\theta}+y\sin^2{\theta}$, where $x$, $y$, $A$, and $B$ are all real numbers. Prove $x^2 + y^2 \ge A^2 + B^2$
Let $x$ and $y$ be two positive real numbers. Find the maximum value of $\frac{(3x+4y)^2}{x^2 + y^2}$.
Let $a_1, a_2, \cdots, a_n$ be $n > 2$ real numbers. Show that it is possible to select $\epsilon_1, \epsilon_2, \cdots, \epsilon_n \in \{1, -1\}$ such that $$(\sum_{i=1}^na_i)^2 + (\sum_{i=1}^n\epsilon_ia_i)^2 \le (n+1)(\sum_{i=1}^na_i^2)$$
Let $x$ be a real number between 0 and 1. Find the maximum value of $x(1-x^4)$.
Let $a_1, a_2, \cdots, a_n$ be a sequence of positive numbers, and $b_1, b_2, \cdots, b_n$ be any permutation of the first sequence. Prove that $$\frac{a_1}{b_1}+\frac{a_2}{b_2}+\cdots \frac{a_n}{b_n}\ge n$$
For all real numbers $a, b, c$, prove that $$\frac{a^2+b^2}{a+b}+\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}\ge a+b+c$$
For all positive real numbers, prove the following inequalities:
a) $x^5 + y^5 + z^5 \ge x^4y + y^4z + z^4x$
b) $x^5 + y^5 + z^5 \ge x^2y^2z + y^2z^2x + z^2x^2y$
c) $x^3y^2 +y^3z^2 + z^3x^2 \ge x^2y^2z+y^2z^2x+z^2x^2y$
Let $x, y, z$ be positive real numbers, prove that $$\Large(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\Large)(\sqrt{x}+\sqrt{y}+\sqrt{z})^4\ge 9\sqrt{3}$$
Let $a, b, c$ be positive real numbers such that $a^2 + b^2 +c^2 +(a+b+c)^2 \le 4$. Prove that $$\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\ge 3$$
Let $A=(2,0)$, $B=(0,2)$, $C=(-2,0)$, and $D=(0, -2)$. Compute the greatest possible value of the product $PA\cdot PB\cdot PC\cdot PD$, where $P$ is a point on the circle $x^2 + y^2=9$.
Show that if $k \ge 4$, then $lcm(1; 3;\cdots; 2k- 3; 2k- 1) > (2k + 1)^2$ where $lcm$ stands for least common multiple.
Find all positive integers $n$ and $k_i$ $(1\le i \le n)$ such that $$k_1 + k_2 + \cdots + k_n = 5n-4$$ and $$\frac{1}{k_1} + \frac{1}{k_2} + \cdots + \frac{1}{k_n}=1$$
Let $\alpha, \beta \in (0, \frac{\pi}{2})$. Show that $\alpha + \beta = \frac{\pi}{2}$ if and only if $$\frac{\sin^4 \alpha}{\cos^2 \beta} + \frac{\cos^4\alpha}{\sin^2\beta} = 1$$
Let $ a,\, b,\, c$ be side lengths of a triangle and $ a+b+c = 3$. Find the minimum of \[ a^{2}+b^{2}+c^{2}+\frac{4abc}{3}\]
Solve in positive integers the equation $x^2y + y^2z +z^2x = 3xyz$
Let $a$ and $b$ be non-negative real numbers such that $a + b = 2$. Show that: $$\frac{1}{a^2+1}+\frac{1}{b^2 +1} \le \frac{2}{ab+1}$$
On the number line, consider the point $x$ that corresponds to the value 10. Consider 24 distinct integer points $y_1, y_2 \cdots y_{24}$ on the number line such that for all $k$ such that $1\le k\le 12$, we have that $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of $$\sum_{n=1}^{24}(\mid y_n-1 \mid + \mid y_n+1\mid)$$
Let $x_1$ and $x_2$ be the two real roots of the equation $x^2 - 2mx + (m^2+2m+3)=0$. Find the minimal value of $x_1^2 + x_2^2$.
If $-1 < a < b < 0$, then which relationship below holds?
$(A)\quad a < a^3 < ab^2 < ab \qquad (B)\quad a < ab^2 < ab < a^3 \qquad (C)\quad a< ab < ab^2 < a^3 \qquad (D) a^3 < ab^2 < a < ab$
If real numbers $a$ and $b$ satisfy $a^2 + b^2=1$, find the minimal value of $a^4 + ab+b^4$.