let $f(x) = \sum_{n=0}^{\infty} 2^{-n} ||2^n x||$ , where ||x|| is the distance between x and the closest integer to x. Are the level sets $\{ x \in [0,1] : f(x)=y \}$ Lebesgue measurable for almost all $y \in f(R)$?

Let $S(n)$ be the sum of the digits of the positive integer $n$. Find all $n$ such that $S(n)(S(n)-1)=n-1$.

Figure shows a semicircle with diameter $AD$. The chords $AC$ and $BD$ meet at $P$. $Q$ is the foot of the perpendicular from $P$ to $AD$. find $\angle BCQ$ in terms of $\theta$ and $\phi$ . [img]https://cdn.artofproblemsolving.com/attachments/a/2/2781050e842b2dd01b72d246187f4ed434ff69.png[/img]

Let $ABC$ be a triangle with centroid $G$. Points $R$ and $S$ are chosen on rays $GB$ and $GC$, respectively, such that \[ \angle ABS=\angle ACR=180^\circ-\angle BGC.\] Prove that $\angle RAS+\angle BAC=\angle BGC$. [i]Merlijn Staps[/i]

Find the sum of all positive integers $n$ such that \[x^3+y^3+z^3=nx^2y^2z^2\] is satisfied by at least one ordered triplet of positive integers $(x,y,z)$.

A clock currently shows the time $10:10$. The obtuse angle between the hands measures $x$ degrees. What is the next time that the angle between the hands will be $x$ degrees? Round your answer to the nearest minute.

Determine all real numbers $x$ for which the following statement is true: the field $\mathbb C$ of complex numbers contains a proper subfield $F$ such that adjoining $x$ to $F$ we get $\mathbb C$. [M. Laczkovich]

A school has $n$ students and some super classes are provided for them. Each student can participate in any number of classes that he/she wants. Every class has at least two students participating in it. We know that if two different classes have at least two common students, then the number of the students in the first of these two classes is different from the number of the students in the second one. Prove that the number of classes is not greater that $\left(n-1\right)^2$.

Given a convex pentagon $ABCDE$ with $\angle ABC= \angle ADE$ and $\angle AEC= \angle ADB$ . Prove that $\angle BAC = \angle DAE$ .

There are cities in country, and some cities are connected by roads. Not more than $100$ roads go from every city. Set of roads is called as ideal if all roads in set have not common ends, and we can not add one more road in set without breaking this rule. Every day minister destroy one ideal set of roads. Prove, that he need not more than $199$ days to destroy all roads in country.

In how many ways can 1992 be expressed as the sum of one or more consecutive integers?

Let $(a_n)_{n=1}^\infty$ be a sequence of real numbers. We say that the sequence $(a_n)_{n=1}^\infty$ covers the set of positive integers if for any positive integer $m$ there exists a positive integer $k$ such that $\sum_{n=1}^\infty a_n^k=m$. a) Does there exist a sequence of real positive numbers which covers the set of positive integers? b) Does there exist a sequence of real numbers which covers the set of positive integers?

What's the largest number of integers from $1$ to $2022$ that you can choose so that no sum of any two different chosen integers is divisible by any difference of two different chosen integers? [i](Proposed by Oleksii Masalitin)[/i]

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

We number a hundred blank cards on both sides with the numbers $1$ to $100$. The cards are then stacked in order, with the card with the number $1$ on top. The order of the cards is changed step by step as follows: at the $1$st step the top card is turned around, and is put back on top of the stack (nothing changes, of course), at the $2$nd step the topmost $2$ cards are turned around, and put back on top of the stack, up to the $100$th step, in which the entire stack of $100$ cards is turned around. At the $101$st step, again only the top card is turned around, at the $102$nd step, the top most $2$ cards are turned around, and so on. Show that after a finite number of steps, the cards return to their original positions.

(1) Find the possible number of roots for the equation $|x + 1| + |x + 2| + |x + 3| = a$, where $x \in R$ and $a$ is parameter. (2) Let $\{ a_1, a_2, \ldots, a_n \}$ be an arithmetic progression, $n \in \mathbb{N}$, and satisfy the condition \[ \sum^{n}_{i=1}|a_i| = \sum^{n}_{i=1}|a_{i} + 1| = \sum^{n}_{i=1}|a_{i} - 2| = 507. \] Find the maximum value of $n$.

I am not a spammer, at least, this is the way I use to think about myself, and thus I will not open a new thread for the following problem from today's DeMO exam: Let Q(n) denote the sum of the digits of a positive integer n. Prove that $Q\left(Q\left(Q\left(2005^{2005}\right)\right)\right)=7$. [[b]EDIT:[/b] Since this post was split into a new thread, I comment: The problem is completely analogous to the problem posted at http://www.mathlinks.ro/Forum/viewtopic.php?t=31409 , with the only difference that you have to consider the number $2005^{2005}$ instead of $4444^{4444}$.] Darij

Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$th, $8$th, $12$th, $\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions $2019, 2020, 2021$? $\textbf{(A) } 7 \qquad\textbf{(B) } 9 \qquad\textbf{(C) } 10 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 12$

For a given natural number $n$, consider those sets $A\subseteq \mathbb{Z}_n$ for which the equation $xy=uv$ has no other solution in the residual classes $x,y,u,v\in A$ than the trivial solutions $x=u$, $y=v$ and $x=v$, $y=u$. Let $g(n)$ be the maximum of the size of such sets $A$. Prove that $$\limsup_{n\to\infty}\frac{g(n)}{\sqrt{n}}=1$$

Suppose that $k,m,n$ are positive integers with $k \le n$. Prove that: \[\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1\]

There are 6 squares in a row. Each one is labeled with the name of Ana or Beto and with a number from 1 to 6, using each number without repetition. Ana and Beto take turns painting each square according to the order of the numbers on the labels. Whoever paints the square will be the person whose name is on the label. When painting, the person can choose to paint the square either red or blue. Beto wins if at the end there are the same number of blue squares as red squares, and Ana wins otherwise. In how many of all the possible ways of labeling the squares can Beto ensure his victory? The following is an example of a labeling of the labels. [asy] size(12cm); draw((0,0)--(6,0)--(6,-1)--(0,-1)--cycle); for (int i=1; i<6; ++i) { draw((i,0)--(i,-1)); } for (int i=1; i<6; ++i) { draw((i,0)--(i,-1.25)); } draw((0,0)--(6,0)--(6,-1.25)--(0,-1.25)--cycle); for (int i=1; i<7; ++i) { draw((i-0.5,-1)--(i-0.5,-1.25)); } label("Ana", (0.25, -1.125)); label("Beto", (1.25, -1.125)); label("Ana", (2.25, -1.125)); label("Beto", (3.25, -1.125)); label("Ana", (4.25, -1.125)); label("Beto", (5.25, -1.125)); label("1", (0.75, -1.125)); label("3", (1.75, -1.125)); label("5", (2.75, -1.125)); label("2", (3.75, -1.125)); label("4", (4.75, -1.125)); label("6", (5.75, -1.125)); [/asy] First Ana paints the first square, then Beto paints the fourth square, then Beto paints the second square, and so on.

Travis Scott says "FEIN'' every $0.8$ seconds. Find the tens digit of the number of times he says "FEIN'' in $1$ minute.

On a $123 \times 123$ board, each square is painted red or blue according to the following conditions: a) Each square painted red that is not on the edge of the board has exactly $5$ blue squares among its $8$ neighboring squares. b) Each square painted blue that is not on the edge of the board has exactly $4$ red squares among its $8$ neighboring squares. Determine the number of red-painted squares on the board.

Show that there exists a set $\mathcal{C}$ of $2020$ distinct, positive integers that satisfies simultaneously the following properties: $\bullet$ When one computes the greatest common divisor of each pair of elements of $\mathcal{C}$, one gets a list of numbers that are all distinct. $\bullet$ When one computes the least common multiple of each pair of elements of $\mathcal{C}$, one gets a list of numbers that are all distinct.

Positive real numbers $x_1,x_2,...,x_n$ satisfy the condition $\sum_{i=1}^n x_i \le \sum_{i=1}^n x_i ^2$ . Prove the inequality $\sum_{i=1}^n x_i^t \le \sum_{i=1}^n x_i ^{t+1}$ for all real numbers $t > 1$.