Let P be a point in an acute triangle $ABC$, and $d_A, d_B, d_C$ be the distance from P to vertices of the triangle respectively. If the distance from P to the three edges are $d_1, d_2, d_3$ respectively, prove that \[d_A+d_B+d_C\geq 2(d_1+d_2+d_3)\]

10.2 Prove for all $x>0$ and $n\in\mathbb{N}$ the following inequality \[1+x^{n+1}\geq \frac{(2x)^n}{(1+x)^{n-1}}.\] ([i]A. Khrabrov[/i])

For each continuous function $f: [0,1]\to\mathbb{R},$ let $I(f)=\int_{0}^{1}x^{2}f(x)\,dx$ and $J(f)=\int_{0}^{1}x\left(f(x)\right)^{2}\,dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$

Let $n$ and $k$ be positive integers. Prove that for $a_1, \dots, a_n \in [1,2^k]$ one has \[ \sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}. \]

Find the least value of $x>0$ such that for all positive real numbers $a,b,c,d$ satisfying $abcd=1$, the inequality $a^x+b^x+c^x+d^x\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$ is true.

Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2+(a+b+c)^2\leq4$. Prove that \[\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\geq 3.\]

Let $n \geq 4$ be a natural and let $x_1,\ldots,x_n$ be non-negative reals such that $x_1 + \cdots + x_n = 1$. Determine the maximum value of $x_1x_2x_3 + x_2x_3x_4 + \cdots + x_nx_1x_2$.

p1. A regular $6$-face dice is thrown three times. Calculate the probability that the number of dice points on all three throws is $ 12$? p2. Given two positive real numbers $x$ and $y$ with $xy = 1$. Determine the minimum value of $\frac{1}{x^4}+\frac{1}{4y^4}.$ p3. Known a square network which is continuous and arranged in forming corners as in the following picture. Determine the value of the angle marked with the letter $x$. [img]https://cdn.artofproblemsolving.com/attachments/6/3/aee36501b00c4aaeacd398f184574bd66ac899.png[/img] p4. Find the smallest natural number $n$ such that the sum of the measures of the angles of the $n$-gon, with $n > 6$ is less than $n^2$ degrees. p5. There are a few magic cards. By stating on which card a number is there, without looking at the card at all, someone can precisely guess the number. If the number is on Card $A$ and $B$, then the number in question is $1 + 2$ (sum of corner number top left) cards $A$ and $B$. If the numbers are in $A$, $B$, and $C$, the number what is meant is $1 + 2 + 4$ or equal to $7$ (which is obtained by adding the numbers in the upper left corner of each card $A$,$B$, and $C$). [img]https://cdn.artofproblemsolving.com/attachments/e/5/9e80d4f3bba36a999c819c28c417792fbeff18.png[/img] a. How can this be explained? b. Suppose we are going to make cards containing numbers from $1$ to with $15$ based on the rules above. Try making the cards. [hide=original wording for p5, as the wording isn't that clear]Ada suatu kartu ajaib. Dengan menyebutkan di kartu yang mana suatu bilan gan berada, tanpa melihat kartu sama sekali, seseorang dengan tepat bisa menebak bilangan yang dimaksud. Kalau bilangan tersebut ada pada Kartu A dan B, maka bilangan yang dimaksud adalah 1 + 2 (jumlah bilangan pojok kiri atas) kartu A dan B. Kalau bilangan tersebut ada di A, B, dan C, bilangan yang dimaksud adalah 1 + 2 + 4 atau sama dengan 7 (yang diperoleh dengan menambahkan bilangan-bilangan di pojok kiri atas masing-masing kartu A, B, dan C) a. Bagaimana hal ini bisa dijelaskan? b. Andai kita akan membuat kartu-kartu yang memuat bilangan dari 1 sampai dengan 15 berdasarkan aturan di atas. Coba buatkan kartu-kartunya[/hide]

Let $a,b,c$ be positive reals such that $a+b+c=3$. Prove the inequality \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geq \frac{3}{2}.\]

Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that \[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]

On the circle $O,$ six points $A,B,C,D,E,F$ are on the circle counterclockwise. $BD$ is the diameter of the circle and it is perpendicular to $CF.$ Also, lines $CF, BE, AD$ is concurrent. Let $M$ be the foot of altitude from $B$ to $AC$ and let $N$ be the foot of altitude from $D$ to $CE.$ Prove that the area of $\triangle MNC$ is less than half the area of $\square ACEF.$

Let a ,b and c be positive real numbers such that $a^2+b^2+c^2=3$. Prove that for whatever positive real numbers x y and z, the inequality below holds. $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge \sqrt{xy}+\sqrt{yz}+\sqrt{zx}$ At first I noticed $\sqrt{xy}+\sqrt{yz}+\sqrt{zx}\le \sqrt{x+y+z}\sqrt{x+y+z}=x+y+z$, so perhaps the next move is to prove $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge x+y+z$, but I don't see how to do that, the best thing that I can do with the LHS of this inequality is to prove it by AM-GM in the way that $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge 3\left(\frac{xyz}{abc}\right)^{\frac{1}{3}}\ge 3(xyz)^{\frac{1}{3}}$, but this isn't going to be helpful...

Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=10$. What is the maximum value of $A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A$? $\text{(A)}\ 49 \qquad\text{(B)}\ 59 \qquad\text{(C)}\ 69 \qquad\text{(D)}\ 79\qquad\text{(E)}\ 89$

Let $ABC$ be a triangle with $\angle A=\tfrac{135}{2}^\circ$ and $\overline{BC}=15$. Square $WXYZ$ is drawn inside $ABC$ such that $W$ is on $AB$, $X$ is on $AC$, $Z$ is on $BC$, and triangle $ZBW$ is similar to triangle $ABC$, but $WZ$ is not parallel to $AC$. Over all possible triangles $ABC$, find the maximum area of $WXYZ$.

Let $a>1$ be a real number. Prove that for all $n\in\mathbb{N}*$ that : $\frac{a^n-1}{n}\ge \sqrt{a}^{n+1}-\sqrt{a}^{n-1}$

Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$