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 $c$.

Given a real number $x$, define the series \[ S(x) = \sum_{n=1}^{\infty} \{n! \cdot x\}, \] where $\{s\} = s - \lfloor s \rfloor$ is the fractional part of the number $s$. Determine if there exists an irrational number $x$ for which the series $S(x)$ converges.

We consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a sequence $(d_i)_{1\le i\le k}$ with $$1=d_1<d_2<\dots <d_k=n\quad (k\ge 6).$$ Find all positive integers $n$ such that $$n=d_5^2+d_6^2.$$

Let $ABCD$ be a convex quadrilateral whose diagonals intersect at $O$ at an angle $\theta$. Let us set $OA = a, OB = b, OC = c$, and $OD = d, c > a > 0$, and $d > b > 0.$ Show that if there exists a right circular cone with vertex $V$, with the properties: [b](1)[/b] its axis passes through $O$, and [b](2)[/b] its curved surface passes through $A,B,C$ and $D,$ then \[OV^2=\frac{d^2b^2(c + a)^2 - c^2a^2(d + b)^2}{ca(d - b)^2 - db(c - a)^2}.\] Show also that if $\frac{c+a}{d+b}$ lies between $\frac{ca}{db}$ and $\sqrt{\frac{ca}{db}},$ and $\frac{c-a}{d-b}=\frac{ca}{db},$ then for a suitable choice of $\theta$, a right circular cone exists with properties [b](1) [/b]and [b](2).[/b]

Today was realized the National Olimpiad in Cuba, this is the 3rd problem of the second day: Prof that we can color the lattice points in the plane with two color so that every rectangle with vertices in the lattice points and edges parallels to the co-ordinate axis that have area 2^n is not monocromatic [/img]

A three-digit bumber $\overline{abc}$ (in decimal representation) is such that (i) its hundreds digit is equal to the sum of the other two digits, and (ii)$b(c+1)=52-4a$. Find all such numbers.

Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.

Let $ABC$ be a non-isosceles triangle, let $AA_1$ be its angle bisector and $A_2$ be the touching point of the incircle with side $BC$. The points $B_1,B_2,C_1,C_2$ are defined similarly. Let $O$ and $I$ be the circumcenter and the incenter of triangle $ABC$. Prove that the radical center of the circumcircle of the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ lies on the line $OI$.

Given some points in the plane. For some pairs $A,B$ the vector $AB$ is chosen. For every point the number of the chosen vectors starting in that point equal to the number of the chosen vectors ending in that point. Prove that the sum of the chosen vectors equals to zero vector.

Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.

Let $ABCD$ be a parallelogram with $\angle ABC = 105^\circ$. Inside the parallelogram, there is a point $E$ such that triangle $BEC$ is equilateral and $\angle CED = 135^\circ$. Let $K$ be the midpoint of side $AB$. Determine the measure of angle $\angle BKC$.

In the following figure, the bigger wheel has circumference $12$m and the inscribed wheel has circumference $8 $m. $P_{1}$ denotes a point on the bigger wheel and $P_{2}$ denotes a point on the smaller wheel. Initially $P_{1}$ and $P_{2}$ coincide as in the figure. Now we roll the wheels on a smooth surface and the smaller wheel also rolls in the bigger wheel smoothly. What distance does the bigger wheel have to roll so that the points will be together again?

Find all sets X of points in a plane, not all collinear, such that: For any two distinct circumferences, each contains three points of X, its intersection points are points of X.

Let $ABC$ be a triangle with $\angle ABC=90^{\circ}$. The square $BDEF$ is inscribed in $\triangle ABC$, such that $D,E,F$ are in the sides $AB,CA,BC$ respectively. The inradius of $\triangle EFC$ and $\triangle EDA$ are $c$ and $b$, respectively. Four circles $\omega_1,\omega_2,\omega_3,\omega_4$ are drawn inside the square $BDEF$, such that the radius of $\omega_1$ and $\omega_3$ are both equal to $b$ and the radius of $\omega_2$ and $\omega_4$ are both equal to $a$. The circle $\omega_1$ is tangent to $ED$, the circle $\omega_3$ is tangent to $BF$, $\omega_2$ is tangent to $EF$ and $\omega_4$ is tangent to $BD$, each one of these circles are tangent to the two closest circles and the circles $\omega_1$ and $\omega_3$ are tangents. Determine the ratio $\frac{c}{a}$.

A positive integer is [i]bold[/i] iff it has $8$ positive divisors that sum up to $3240$. For example, $2006$ is bold because its $8$ positive divisors, $1$, $2$, $17$, $34$, $59$, $118$, $1003$ and $2006$, sum up to $3240$. Find the smallest positive bold number.

In a circle of diameter $1$ consider $65$ points, no three of them collinear. Prove that there exist three among these points which are the vertices of a triangle with area less than or equal to $\frac{1}{72}$.