We are given a convex quadrilateral and point $M$ inside it . The perimeter of the quadrilateral has length $L$ while the lengths of the diagonals are $D_1$ and $D_2$. Prove that the sum of the distances from $M$ to the vertices of the quadrilateral are not greater than $L + D_1 + D_2$ . (V. Prasolov)

Given a rectangular grid, split into $m\times n$ squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions: [list] [*]All squares touching the border of the grid are coloured black. [*]No four squares forming a $2\times 2$ square are coloured in the same colour. [*]No four squares forming a $2\times 2$ square are coloured in such a way that only diagonally touching squares have the same colour.[/list] Which grid sizes $m\times n$ (with $m,n\ge 3$) have a valid colouring?

Find all non-negative integers $x, y$ and primes $p$ such that $$3^x+p^2=7 \cdot 2^y.$$

Let $\omega$ denote the excircle tangent to side $BC$ of triangle $ABC$. A line $\ell$ parallel to $BC$ meets sides $AB$ and $AC$ at points $D$ and $E$, respectively. Let $\omega'$ denote the incircle of triangle $ADE$. The tangent from $D$ to $\omega$ (different from line $AB$) and the tangent from $E$ to $\omega$ (different from line $AC$) meet at point $P$. The tangent from $B$ to $\omega'$ (different from line $AB$) and the tangent from $C$ to $\omega'$ (different from line $AC$) meet at point $Q$. Prove that, independent of the choice of $\ell$, there is a fixed point that line $PQ$ always passes through.

A ballot box contains the votes for the election of two candidates $A$ and $B$. It is known that candidate $A$ has $6$ votes and candidate $B$ has $9$. Find the probability that, when carrying out the scrutiny, candidate $B$ always goes first.

Are there natural $a, b >1000$ , such that for any $c$ that is a perfect square, the three numbers $a, b$ and $c$ are not the lengths of the sides of a triangle?

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 $$\limsup_{n\to\infty}\frac{g(n)}{\sqrt{n}}=1$$