Prove or disprove that $\forall a,b,c,d \in \mathbb{R}^+$ we have the following inequality: \[3 \leq \frac{4a+b}{a+4b} + \frac{4b+c}{b+4c} + \frac{4c+a}{c+4a} < \frac{33}{4}\]

Point $D$ is chosen on the Euler line of triangle $ABC$ and it is inside of the triangle. Points $E,F$ are were the line $BD,CD$ intersect with $AC,AB$ respectively. Point $X$ is on the line $AD$ such that $\angle EXF =180 - \angle A$, also $A,X$ are on the same side of $EF$. If $P$ is the second intersection of circumcircles of $CXF,BXE$ then prove the lines $XP,EF$ meet on the altitude of $A$ Proposed by [i]Alireza Danaie[/i]

Let $H$ be the orthocenter of an acute-angled triangle $ABC$; $A_1, B_1, C_1$ be the touching points of the incircle with $BC, CA, AB$ respectively; $E_A, E_B, E_C$ be the midpoints of $AH, BH, CH$ respectively. The circle centered at $E_A$ and passing through $A$ meets for the second time the bisector of angle $A$ at $A_2$; points $B_2, C_2$ are defined similarly. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.

Sitting at a desk, Alice writes a nonnegative integer $N$ on a piece of paper, with $N \le 10^{10}$. Interestingly, Celia, sitting opposite Alice at the desk, is able to properly read the number upside-down and gets the same number $N$, without any leading zeros. (Note that the digits 2, 3, 4, 5, and 7 will not be read properly when turned upside-down.) Find the number of possible values of $N$. [i]Proposed by Yannick Yao[/i]

Find all positive integers $(m, n)$ that satisfy $$m^2 =\sqrt{n} +\sqrt{2n + 1}.$$

A palindrome number is a positive integer that reads the same forward and backward. For example, $1221$ and $8$ are palindrome numbers whereas $69$ and $157$ are not. $A$ and $B$ are $4$-digit palindrome numbers. $C$ is a $3$-digit palindrome number. Given that $A-B=C$, what is the value of $C$?

How many positive integers have digits whose product is 20 and sum is 23? [i]Individual #1[/i]

Let $A$, $B$, $C$, $D$, $E$, $F$ be six points on a circle such that $AE\parallel BD$ and $BC\parallel DF$. Let $X$ be the reflection of the point $D$ in the line $CE$. Prove that the distance from the point $X$ to the line $EF$ equals to the distance from the point $B$ to the line $AC$.

The random variables $X, Y$ can each take a finite number of integer values. They are not necessarily independent. Express $P(\min(X,Y)=k)$ in terms of $p_1=P(X=k)$, $p_2=P(Y=k)$ and $p_3=P(\max(X,Y)=k)$.

Let $a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that \[\left\vert a+b\sqrt{2}+c\sqrt{3}\right\vert > \frac{1}{10^{21}}.\]

We are given an acute $\triangle ABC$ with circumcenter $O$ such that $BC<AB$. The bisector of $\angle ACB$ meets the circumcircle of $\triangle ABC$ at a second point $D$. The perpendicular bisector of $AC$ meets the circumcircle of $\triangle BOD$ for the second time at $E$. The line $DE$ meets the circumcircle of $\triangle ABC$ for the second time at $F$. Prove that the lines $CF$, $OE$ and $AB$ are concurrent. [i]Authored by Petar Filipovski[/i]

Let $ABCDEF$ be a convex hexagon satisfying $\overline{AB} \parallel \overline{DE}$, $\overline{BC} \parallel \overline{EF}$, $\overline{CD} \parallel \overline{FA}$, and \[ AB \cdot DE = BC \cdot EF = CD \cdot FA. \] Let $X$, $Y$, and $Z$ be the midpoints of $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$. Prove that the circumcenter of $\triangle ACE$, the circumcenter of $\triangle BDF$, and the orthocenter of $\triangle XYZ$ are collinear.

Fill in each square with a number from $1$ to $5$; some numbers have been given. If two squares $A$ and $B$ have equal numbers, then $A$ and $B$ cannot share a side, and there also cannot exist a third square $C$ sharing a side with both $A$ and $B$. There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy]unitsize(30); int a[][] = { {9, 9, 0, 0, 0, 9, 0, 0, 0}, {4, 0, 0, 2, 0, 0, 0, 2, 0}, {0, 1, 0, 0, 0, 2, 0, 0, 5}, {0, 0, 0, 9, 0, 0, 0, 9, 9} }; for (int i = 0; i < a.length; ++i) { for (int j = 0; j < a[0].length; ++j) { if (a[i][j] < 9) draw(shift(j, -i-1) * unitsquare); label((a[i][j] >= 1 && a[i][j] <= 5) ? string(a[i][j]) : "", (j+.5, -i-.5), fontsize(24pt)); } }[/asy]

Let $c$ be a fixed positive integer, and let ${a_n}^{\inf}_{n=1}$ be a sequence of positive integers such that $a_n < a_{n+1} < a_n+c$ for every positive integer $n$. Let $s$ denote the infinite string of digits obtained by writing the terms in the sequence consecutively from left to right, starting from the first term. For every positive integer $k$, let $s_k$ denote the number whose decimal representation is identical to the $k$ most left digits of $s$. Prove that for every positive integer $m$ there exists a positive integer $k$ such that $s_k$ is divisible by $m$.

Determine all real numbers $a, b, c, d$ such that the polynomial $f(x) = ax^3 +bx^2 + cx + d$ satisfies simultaneously the folloving conditions $\begin {cases} |f(x)| \le 1 \,for \, |x| \le 1 \\ f(2) = 26 \end {cases}$

Let $x$ be a positive real number. Define \[ A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad\text{and}\quad C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}. \] Given that $A^3+B^3+C^3 + 8ABC = 2014$, compute $ABC$. [i]Proposed by Evan Chen[/i]

Let $ABC$ be a triangle. Take $n$ point lying on the side $AB$ (different from $A$ and $B$) and connect all of them by straight lines to the vertex $C$. Similarly, take $n$ points on the side $AC$ and connect them to $B$. Into how many regions is the triangle $ABC$ partitioned by these lines? Further, take $n$ points on the side $BC$ also and join them with $A$. Assume that no three straight lines meet at a point other than $A,B$ and $C$. Into how many regions is the triangle $ABC$ partitioned now?

In the diagram to the right, squares are drawn on the side of the triangle with side lengths $5$, $6$, and $7$ as shown below. The corners of adjacent squares are then connected. What is the area of the resulting hexagon?

Positive integers $ a,b,$ and $ c$ are randomly and independently selected with replacement from the set $ \{ 1,2,3,\dots,2010 \}.$ What is the probability that $ abc \plus{} ab \plus{} a$ is divisible by $ 3$? $ \textbf{(A)}\ \dfrac{1}{3} \qquad\textbf{(B)}\ \dfrac{29}{81} \qquad\textbf{(C)}\ \dfrac{31}{81} \qquad\textbf{(D)}\ \dfrac{11}{27} \qquad\textbf{(E)}\ \dfrac{13}{27}$

The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $165$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?

Given $n$ numbers different from $0$, ($n \in \mathbb{N}$) which are arranged randomly. We do the following operation: Choose some consecutive numbers in the given order and change their sign (i.e. $x \rightarrow -x$). What is the minimum number of operations needed, in order to make all the numbers positive for any given initial configuration of the $n$ numbers?

Find all positive integers $\overline{xyz}$ ($x$, $y$ and $z$ are digits) such that $\overline{xyz} = x+y+z+xy+yz+zx+xyz$

[u]Round 5[/u] [b]p13.[/b] Find all ordered pairs of real numbers $(x, y)$ satisfying the following equations: $$\begin{cases} \dfrac{1}{xy} + \dfrac{y}{x}= 2 \\ \dfrac{1}{xy^2} + \dfrac{y^2}{x} = 7 \end{cases}$$ [b]p14.[/b] An egg plant is a hollow prism of negligible thickness, with height $2$ and an equilateral triangle base. Inside the egg plant, there is enough space for four spherical eggs of radius $1$. What is the minimum possible volume of the egg plant? [b]p15.[/b] How many ways are there for Farmer James to color each square of a $2\times 6$ grid with one of the three colors eggshell, cream, and cornsilk, so that no two adjacent squares are the same color? [u]Round 6[/u] [b]p16.[/b] In a triangle $ABC$, $\angle A = 45^o$, and let $D$ be the foot of the perpendicular from $A$ to segment $BC$. $BD = 2$ and $DC = 4$. Let $E$ be the intersection of the line $AD$ and the perpendicular line from $B$ to line $AC$. Find the length of $AE$. [b]p17.[/b] Find the largest positive integer $n$ such that there exists a unique positive integer $m$ satisfying $$\frac{1}{10} \le \frac{m}{n} \le \frac19$$ [b]p18.[/b] How many ordered pairs $(A,B)$ of positive integers are there such that $A+B = 10000$ and the number $A^2 + AB + B$ has all distinct digits in base $10$? [u]Round 7[/u] [b]p19.[/b] Pentagon $JAMES$ satisfies $JA = AM = ME = ES = 2$. Find the maximum possible area of $JAMES$. [b]p20.[/b] $P(x)$ is a monic polynomial (a polynomial with leading coecient $1$) of degree $4$, such that $P(2^n+1) =8^n + 1$ when $n = 1, 2, 3, 4$. Find the value of $P(1)$. [b]p21[/b]. PEAcock and Zombie Hen Hao are at the starting point of a circular track, and start running in the same direction at the same time. PEAcock runs at a constant speed that is $2018$ times faster than Zombie Hen Hao's constant speed. At some point in time, Farmer James takes a photograph of his two favorite chickens, and he notes that they are at different points along the track. Later on, Farmer James takes a second photograph, and to his amazement, PEAcock and Zombie Hen Hao have now swapped locations from the first photograph! How many distinct possibilities are there for PEAcock and Zombie Hen Hao's positions in Farmer James's first photograph? (Assume PEAcock and Zombie Hen Hao have negligible size.) [u]Round 8[/u] [b]p22.[/b] How many ways are there to scramble the letters in $EGGSEATER$ such that no two consecutive letters are the same? [b]p23.[/b] Let $JAMES$ be a regular pentagon. Let $X$ be on segment $JA$ such that $\frac{JX}{XA} = \frac{XA}{JA}$ . There exists a unique point $P$ on segment $AE$ such that $XM = XP$. Find the ratio $\frac{AE}{PE}$ . [b]p24.[/b] Find the minimum value of the function $$f(x) = \left|x - \frac{1}{x} \right|+ \left|x - \frac{2}{x} \right| + \left|x - \frac{3}{x} \right|+... + \left|x - \frac{9}{x} \right|+ \left|x - \frac{10}{x} \right|$$ over all nonzero real numbers $x$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949191p26406082]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Real numbers $a \neq 0, b, c$ are given. Prove that there is a polynomial $P(x)$ with real coefficients such that the polynomial $x^2+1$ divides the polynomial $aP(x)^2+bP(x)+c$. [i]Proposed by A. Golovanov[/i]

Let the sequence $\{a_i\}$ of $n$ positive reals denote the lengths of the sides of an arbitrary $n$-gon. Let $s=\sum_{i=1}^{n}{a_i}$. Prove that $2\ge \sum_{i=1}^{n}{\frac{a_i}{s-a_i}}\ge \frac{n}{n-1}$.