Prove that: $\int_{-1}^1f^2(x)dx\ge \frac 1 2 (\int_{-1}^1f(x)dx)^2 +\frac 3 2(\int_{-1}^1xf(x)dx)^2$ Please give a proof without using even and odd functions. (the oficial proof uses those and seems to be un-natural) :D

Kostya had two sets of $17$ coins: in one set all the coins were real, and in the other set there were exactly $5$ fakes (all the coins look the same; all real coins weigh the same, all fake coins also weigh the same, but it is unknown lighter or heavier than real ones). Kostya gave away one of the sets friend, and subsequently forgot which of the two sets had stayed. With the help of two weighings, can Kostya on a cup scale without weights, find out which of the two did he give away the sets?

How many two-digit positive integers have at least one $ 7$ as a digit? $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 18\qquad \textbf{(C)}\ 19\qquad \textbf{(D)}\ 20\qquad \textbf{(E)}\ 30$

Solve in $ \mathbb{C}^3 $ the following chain of equalities: $$ x(x-y)(x-z)=y(y-x)(y-z)=z(z-x)(z-y)=3. $$

Let $r\geqslant 0$ be a real number and define $f(x)=1/(1+x^2)^r$. Prove that \[|f^{(k)}(x)|\leqslant\frac{2r\cdot(2r+1)\cdots(2r+k-1)}{(1+x^2)^{r+k/2}},\]for every natural number $k{}$. Here, $f^{(k)}(x)$ denotes the $k^{\text{th}}$ derivative of $f$.

Let $P_1, P_2, P_3, P_4$ be four points on a circle, let $I_1$ be incenter of the triangle of vertices $P_2P_3P_4$, $I_2$ the incenter of the triangle $P_1P_3P_4$, $I_3$ the incenter of the triangle $P_1P_2P_4$, $I_4$ the incenter of the triangle $P_2P_3P_1$. Prove that $I_1I_2I_3I_4$ is a rectangle.

$3n$ people take part in a chess tournament: $n$ girls and $2n$ boys. Each participant plays with each of the others exactly once. There were no ties and the number of games won by the girls is $\displaystyle\frac75$ the number of games won by the boys. How many people took part in the tournament?

An ordered quadruple of numbers is called [i]ten-esque[/i] if it is composed of 4 nonnegative integers whose sum is equal to $10$. Ana chooses a ten-esque quadruple $(a_1, a_2, a_3, a_4)$ and Banana tries to guess it. At each stage Banana offers a ten-esque quadtruple $(x_1,x_2,x_3,x_4)$ and Ana tells her the value of \[|a_1-x_1|+|a_2-x_2|+|a_3-x_3|+|a_4-x_4|\] How many guesses are needed for Banana to figure out the quadruple Ana chose?

The real-valued function $f$ is defined for positive integers, and the positive integer $a$ satisfies $f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), f(n + a) = \frac{f(n) - 1}{f(n) + 1}$ for all positive integers $n$. (i) Show that $f(n+ 4a) = f(n)$ for all positive integers $n$. (ii) Determine the smallest possible $a$.

Can there be a continuum set of continuum sets such that (i) the intersection of any two is finite, and (ii) every set that intersects all sets intersects any in an infinite set? note: a continuum set is a set that can be put into a 1-to-1 bijection with the reals.

$AB$ is a chord (not a diameter) of a circle with centre $O$. Let $T$ be a point on segment $OB$. The line through $T$ perpendicular to $OB$ meets $AB$ at $C$ and the circle at $D$ and $E$. Denote by $S$ the orthogonal projection of $T$ onto $AB$ . Prove that $AS \cdot BC = TE \cdot TD$.

Let $ABC$ be a triangle $D\in[BC]$ (different than $A$ and $B$).$E$ is the midpoint of $[CD]$. $F\in[AC]$ such that $\widehat{FEC}=90$ and $|AF|.|BC|=|AC|.|EC|.$ Circumcircle of $ADC$ intersect $[AB]$ at $G$ different than $A$.Prove that tangent to circumcircle of $AGF$ at $F$ is touch circumcircle of $BGE$ too.

The $100$th degree polynomial $P(x)$ satisfies $P(2^k) = k$ for $k = 0, 1, . . . 100$. Let $a$ denote the leading coefficient of $P(x)$. Find the unique integer $M$ such that $2^M < |a| < 2^{M+1}$. .

Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$. Let $P$ be the point on $\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$ $\textbf{(A) }\frac{42}5 \qquad \textbf{(B) }6\sqrt2 \qquad \textbf{(C) }\frac{84}5\qquad \textbf{(D) }12\sqrt2 \qquad \textbf{(E) }18$

Let $ x_1,x_2,..,x_n$ be non-negative numbers whose sum is $ 1$ . Show that there exist numbers $ a_1,a_2,\ldots ,a_n$ chosen from amongst $ 0,1,2,3,4$ such that $ a_1,a_2,\ldots ,a_n$ are different from $ 2,2,\ldots ,2$ and $ 2\leq a_1x_1\plus{}a_2x_2\plus{}\ldots\plus{}a_nx_n\leq 2\plus{}\frac{2}{3^n\minus{}1}$.

For a nonnegative integer $k$, let $f(k)$ be the number of ones in the base 3 representation of $k$. Find all complex numbers $z$ such that $$ \sum_{k=0}^{3^{1010}-1}(-2)^{f(k)}(z+k)^{2023}=0 $$

Prove that there are infinitely many positive integers $n$ such that $n \times n \times n$ can not be filled completely with 2 x 2 x 2 and 3 x 3 x 3 solid cubes.

Let $a \geq 0$ be a natural number. Determine all rational $x$, so that \[\sqrt{1+(a-1)\sqrt[3]x}=\sqrt{1+(a-1)\sqrt x}\] All occurring square roots, are not negative. [b]Note.[/b] It seems the set of natural numbers = $\mathbb N = \{0,1,2,\ldots\}$ in this problem.

100 people from 25 countries, four from each countries, stay on a circle. Prove that one may partition them onto 4 groups in such way that neither no two countrymans, nor two neighbours will be in the same group.

Denote by $s(n)$ the sum of all natural divisors of $n$ which are smaller than $n$. Does there exist a positive integer $a$ such that the equation $$s(n)=a+n$$ has infinitely many solutions in positive integers?

Which positive integers satisfy that the sum of the number’s last three digits added to the number itself yields $2029$?

Compute the number of two-digit numbers $\overline{ab}$ with nonzero digits $a$ and $b$ such that $a$ and $b$ are both factors of $\overline{ab}$. [i]Proposed by Nathan Xiong[/i]

Find non-negative integers $a, b, c, d$ such that $5^a + 6^b + 7^c + 11^d = 1999$.

[u]Round 5[/u] [b]5.1.[/b] In a triangle $\vartriangle ABC$ with $AB = 48$, let the angle bisectors of $\angle BAC$ and $\angle BCA$ meet at $I$. Given $\frac{[ABI]}{[BCI]}=\frac{24}{7}$ and $\frac{[ACI]}{[ABI]}=\frac{25}{24}$ , find the area of $\vartriangle ABC$. [b]5.2.[/b] At a dinner party, $9$ people are to be seated at a round table. If person $A$ cannot be seated next to person $B$ and person $C$ cannot be next to person $D$, how many ways can the $9$ people be seated? Rotations of the table are indistinguishable. [b]5.3.[/b] Let $f(x)$ be a monic cubic polynomial such that $f(1) = f(7) = f(10) = a$ and $f(2) = f(5) = f(11) = b$. Find $|a - b|$. [u]Round 6[/u] [b]6.1.[/b] If $N$ has $16$ positive integer divisors and the sum of all divisors of $N$ that are multiples of $3$ is $39$ times the sum of divisors of $N$ that are not multiples of $3$, what is the smallest value of $N$? [b]6.2.[/b] In the two parabolas $y = x^2/16$ and $x = y^2/16$, the single line tangent to both parabolas intersects the parabolas at $A$ and $B$. If the parabolas intersect each other at $C$ which is not the origin, find the area of $\vartriangle ABC$. [b]6.3.[/b] Five distinguishable noncollinear points are drawn. How many ways are there to draw segments connecting the points, such that there are exactly two disjoint groups of connected points? Note that a single point can be considered a connected group of points. [u]Round 7[/u] [b]7.1.[/b] Let $a, b$ be positive integers, and $1 = d_1 < d_2 < d_3 < ... < d_n = a$ be the divisors of $a$, and $1 = e_1 < e_2 <e_3 < ... < e_m = b$ be the divisors of b. Given $gcd(a, b) = d_2 = e_6$, find the smallest possible value of $a + b$. [b]7.2.[/b] Let $\vartriangle ABC$ be a triangle such that $AB = 2$ and $AC = 3$. Let X be the point on $BC$ such that $m \angle BAX =\frac13 m\angle BAC$. Given that $AX = 1$, the sum of all possible values of $CX^2$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Find $a + b$. [b]7.3.[/b] Bob has a playlist of $6$ different songs in some order, and he listens to his playlist repeatedly. Every time he finishes listening to the third song in the playlist, he randomly shuffles his playlist and listens to the playlist starting with the new first song. The expected number of times Bob shuffles his songs before he listens each one of his $6$ songs at least once can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Find $a+b$. [u]Round 8[/u] [b]8.1.[/b] $\underline{A}, \underline{B}, \underline{C}, \underline{D}, \underline{E}, \underline{F}, \underline{G}, \underline{H}, \underline{I}$, and $\underline{J}$ represent distinct digits ($0$ to $9$) in the equation $\underline{FBGA} - \underline{ABAC} = \underline{DCE}$ (where $\underline{ABAC}$ and $\underline{F BGA}$ are four-digit numbers, and $\underline{DCE }$ is a three-digit number). If $\underline{A} < \underline{B} < \underline{C} < \underline{D}$ and $\underline{ABCDEF GHIJ}$ is minimized, find $\underline{ABCD} + \underline{EF G} + \underline{HI} + \underline{J}$. [b]8.2.[/b] $\underline{A}, \underline{B}, \underline{C}, \underline{D}, \underline{E}$,,, and $\underline{F}$ represent distinct digits ($0$ to $9$) in the equations $\underline{ABC} \cdot \underline{C} = \underline{DEA}, \underline{ABC} \cdot \underline{D} = \underline{BAF E}$, and $ \underline{DEA} + \underline{BAF E}0 = \underline{BF ACA}$ (where $\underline{ABC}$ and $\underline{DEA}$ are three-digit numbers, $\underline{BAF E}$ is a four-digit number, and $\underline{BF ACA}$ is a five-digit number). Find $\underline{ABC} + \underline{DE} + \underline{F}$. [b]8.3.[/b] $\underline{A}, \underline{B}, \underline{C}, \underline{D}, \underline{E}, \underline{F}, \underline{G}$, and $\underline{H}$ represent distinct digits ($0$ to $9$) in the equations $\underline{ABC } \cdot \underline{D} = \underline{AF GE}$, $\underline{ABC } \cdot \underline{C} = \underline{GHC}$, $\underline{GHC} + \underline{HF F} = \underline{AEHC}$, and $\underline{AF GE}0 + \underline{AEHC} = \underline{AEABC}$ (where $\underline{ABC}$, $\underline{GHC}$ and $\underline{HF F}$ are three-digit numbers, $\underline{AF GE}$ is a four-digit number, and $\underline{AEABC}$ is a five-digit number). Find $\underline{ABCD} + \underline{EF GH}$. [u]Round 9[/u] Estimate the arithmetic mean of all answers to this question. Only integer answers between $0$ to $100, 000$ will count for credit and count toward the average. Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05|I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3129699p28347299]here [/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Inside a circle $c$ there are circles $c_1, c_2$ and $c_3$ which are tangent to $c$ at points $A, B$ and $C$ correspondingly, which are all different. Circles $c_2$ and $c_3$ have a common point $K$ in the segment $BC$, circles $c_3$ and $c_1$ have a common point $L$ in the segment $CA$, and circles $c_1$ and $c_2$ have a common point $M$ in the segment $AB$. Prove that the circles $c_1, c_2$ and $c_3$ intersect in the center of the circle $c$.