If ${a, b}$ and $c$ are positive real numbers, prove that \begin{align*} a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 &\ge{ abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right)}. \end{align*} [i](Montenegro).[/i]

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying relation : $$f(xf(y)-f(x))=2f(x)+xy$$ $\forall x,y \in \mathbb{R}$

Solve the equation (for positive $x$) $$x^x=\frac{1}{\sqrt2}$$

Determine all continuous functions $f:\mathbb{R}\to\mathbb{R}$ for which $f(1)=e$ and \[f(x+y)=e^{3xy}\cdot f(x)f(y),\]for all real numbers $x{}$ and $y{}$.

Let $f(w) = w^3 - rw^2 + sw - \frac{4\sqrt2}{27}$ denote a polynomial, where $r^2 =\left(\frac{8\sqrt2+10}{7}\right) s$. The roots of $f$ correspond to the sides of a right triangle. Compute the smallest possible area of this triangle

A sequence $a_1,a_2,\cdots$ of positive integers contains each positive integer exactly once. Moreover for every pair of distinct positive integer $m$ and $n$, $\frac{1}{1998} < \frac{|a_n- a_m|}{|n-m|} < 1998$, show that $|a_n - n | <2000000$ for all $n$.

Find all functions $f:\mathbb R\to \mathbb R$ such that \begin{align*} \left (x \left (f(x)-\dfrac{f(y)+f(z)}{2} \right) +y \left (f(y)-\dfrac{f(z)+f(x)}{2} \right ) +z\left (f(z)- \dfrac{f(x)+f(y)}{2} \right) \right )f(x+y+z)= \\ f(x^3)+f(y^3)+f(z^3)-3f(xyz) \end{align*} for all $x,y,z\in \mathbb R.$

Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality \[af^2 + bfg +cg^2 \geq 0\] holds if and only if the following conditions are fulfilled: \[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]

Show, for all positive integers $n = 1,2 , \dots ,$ that $14$ divides $ 3 ^ { 4 n + 2 } + 5 ^ { 2 n + 1 }$.

If $a+b=1$, $ \in \mathbb{R}$ and $ab \ne 0$, prove that $$\frac{a}{b^3-1}+\frac{b}{a^3-1}=\frac{2(ab-2)}{a^2b^2+3}$$

Consider real numbers $a, b ,c \ge0$ with $a+b+c=2$. Prove that: $\frac{bc}{\sqrt[4]{3a^2+4}}+\frac{ca}{\sqrt[4]{3b^2+4}}+\frac{ab}{\sqrt[4]{3c^2+4}} \le \frac{2*\sqrt[4] {3}}{3}$

Find all integer-coefficient polynomials $f(x)$ satisfying the following conditions for every integer $n \geqslant 2$: [list] [*] $f(n) > 0$. [*] $f(n)$ divides $n^{f(n)} - 1$. [/list]

The sequence $\{a_n\}$ is defined as follows: $a_1$ is a positive rational number, $a_n= \frac{p_n}{q_n}$, ($n= 1,2,…$) is a positive integer, where $p_n$ and $q_n$ are positive integers that are relatively prime, then $a_{n+1} = \frac{p_n^2+2015}{p_nq_n}$ Is there a$_1>2015$, making the sequence $\{a_n\}$ a bounded sequence? Justify your conclusion.

Find the minimum value of the expression $f(a, b, c) = (a + b)^4 + (b + c)^4 + (c + a)^4 - \frac47 (a^4 + b^4 + c^4)$, as $a, b, c$ varies over the set of all real numbers

Given $x \ge 0$, prove that $$\frac{(x^2 + 1)^6}{2^7}+\frac12 \ge x^5 - x^3 + x$$

Solve the equation $x^2- 10[x] + 9 = 0$. ($[x]$ is the integer part of $x$, $[x]$ is equal to the largest integer not exceeding $x$. For example, $[3,33] = 3$, $[2] = 2$, $[- 3.01] = -4$).

Let $a, b, c$ be positive reals such that $a + b + c = 1$ and $P(x) = 3^{2005}x^{2007 }- 3^{2005}x^{2006} - x^2$. Prove that $P(a) + P(b) + P(c) \le -1$.

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

Given $4$ vectors $a,b,c,d$ in the plane, such that $a+b+c+d=0$. Prove the following inequality: $$|a|+|b|+|c|+|d| \ge |a+d|+|b+d|+|c+d|$$

Find all integers $m>1$ such that $m^3$ is a sum of $m$ squares of consecutive integers.

Find all continious $f: \mathbb R\longrightarrow\mathbb R$ that for any $x,y$ \[f(x)+f(y)+f(xy)=f(x+y+xy)\]

The sports plane flew along a diamond-shaped route in windy weather. He flew through the first three sides of the rhombus in $a $, $b$ and $c$ hours, respectively. How long did it take him to cover the fourth side of the diamond? (The speed of an aircraft is a vector equal to the sum of two vectors: the aircraft’s own speed and the wind speed. Wind speed is a constant vector. The aircraft’s own speed is a vector of constant length).

A triplet of polynomials $u,v,w \in \mathbb{R}[x,y,z]$ is called [i]smart[/i] if there exists polynomials $P,Q,R\in \mathbb{R}[x,y,z]$ such that the following polynomial identity holds :$$u^{2019}P +v^{2019 }Q+w^{2019} R=2019$$ a) Is the triplet of polynomials $$u=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y+z$$ [i]smart[/i]? b) Is the triplet of polynomials $$u=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y-z$$ [i]smart[/i]? [i]Proposed by Arturas Dubickas (Vilnius University). [/i]

The following light grid is given: \begin{tabular}{cccc} o & o & o & o \\ o & o & o & o \\ o & o & o & o \\ o & o & o & o \end{tabular} where `o` represents a switched-off light and `•` represents a switched-on light. Each time a light is pressed, it toggles its state (on/off) as well as the state of its four adjacent neighbors (left, right, above, below). The bottom edge lights are considered to be immediately above the top edge lights, and the same applies to the lateral edges.The right figure illustrates the effect of pressing a light in a corner. Pressing a certain combination of lights results in all lights turning on. Prove that all lights must have been pressed at least once.

[i]20 problems for 20 minutes. [/i] [b]p1.[/b] Evaluate $\frac{\sqrt2 \cdot \sqrt6}{\sqrt3}.$ [b]p2.[/b] If $6\%$ of a number is $1218$, what is $18\%$ of that number? [b]p3.[/b] What is the median of $\{42, 9, 8, 4, 5, 1,13666, 3\}$? [b]p4.[/b] Define the operation $\heartsuit$ so that $i\heartsuit u = 5i - 2u$. What is $3\heartsuit 4$? p5. How many $0.2$-inch by $1$-inch by $1$-inch gold bars can fit in a $15$-inch by $12$-inch by $9$-inch box? [b]p6.[/b] A tetrahedron is a triangular pyramid. What is the sum of the number of edges, faces, and vertices of a tetrahedron? [b]p7.[/b] Ron has three blue socks, four white socks, five green socks, and two black socks in a drawer. Ron takes socks out of his drawer blindly and at random. What is the least number of socks that Ron needs to take out to guarantee he will be able to make a pair of matching socks? [b]p8.[/b] One segment with length $6$ and some segments with lengths $10$, $8$, and $2$ form the three letters in the diagram shown below. Compute the sum of the perimeters of the three figures. [img]https://cdn.artofproblemsolving.com/attachments/1/0/9f7d6d42b1d68cd6554d7d5f8dd9f3181054fa.png[/img] [b]p9.[/b] How many integer solutions are there to the inequality $|x - 6| \le 4$? [b]p10.[/b] In a land for bad children, the flavors of ice cream are grass, dirt, earwax, hair, and dust-bunny. The cones are made out of granite, marble, or pumice, and can be topped by hot lava, chalk, or ink. How many ice cream cones can the evil confectioners in this ice-cream land make? (Every ice cream cone consists of one scoop of ice cream, one cone, and one topping.) [b]p11.[/b] Compute the sum of the prime divisors of $245 + 452 + 524$. [b]p12.[/b] In quadrilateral $SEAT$, $SE = 2$, $EA = 3$, $AT = 4$, $\angle EAT = \angle SET = 90^o$. What is the area of the quadrilateral? [b]p13.[/b] What is the angle, in degrees, formed by the hour and minute hands on a clock at $10:30$ AM? [b]p14.[/b] Three numbers are randomly chosen without replacement from the set $\{101, 102, 103,..., 200\}$. What is the probability that these three numbers are the side lengths of a triangle? [b]p15.[/b] John takes a $30$-mile bike ride over hilly terrain, where the road always either goes uphill or downhill, and is never flat. If he bikes a total of $20$ miles uphill, and he bikes at $6$ mph when he goes uphill, and $24$ mph when he goes downhill, what is his average speed, in mph, for the ride? [b]p16.[/b] How many distinct six-letter words (not necessarily in any language known to man) can be formed by rearranging the letters in $EXETER$? (You should include the word EXETER in your count.) [b]p17.[/b] A pie has been cut into eight slices of different sizes. Snow White steals a slice. Then, the seven dwarfs (Sneezy, Sleepy, Dopey, Doc, Happy, Bashful, Grumpy) take slices one by one according to the alphabetical order of their names, but each dwarf can only take a slice next to one that has already been taken. In how many ways can this pie be eaten by these eight persons? [b]p18.[/b] Assume that $n$ is a positive integer such that the remainder of $n$ is $1$ when divided by $3$, is $2$ when divided by $4$, is $3$ when divided by $5$, $...$ , and is $8$ when divided by $10$. What is the smallest possible value of $n$? [b]p19.[/b] Find the sum of all positive four-digit numbers that are perfect squares and that have remainder $1$ when divided by $100$. [b]p20.[/b] A coin of radius $1$ cm is tossed onto a plane surface that has been tiled by equilateral triangles with side length $20\sqrt3$ cm. What is the probability that the coin lands within one of the triangles? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].