Found problems: 85335
There are $n$ girls $G_1,\ldots, G_n$ and $n$ boys $B_1,\ldots,B_n$. A pair $(G_i,B_j)$ is called $\textit{suitable}$ if and only if girl $G_i$ is willing to marry boy $B_j$. Given that there is exactly one way to pair each girl with a distinct boy that she is willing to marry, what is the maximal possible number of suitable pairs?
There are $n$ lamps in a row. Some of which are on. Every minute all the lamps already on go off. Those which were off and were adjacent to exactly one lamp which was on will go on. For which $n$ one can find an initial configuration of lamps which were on, such that at least one lamp will be on at any time?
Three circles $\omega_1,\omega_2,\omega_3$ are tangent to line $l$ at points $A,B,C$ ($B$ lies between $A,C$) and $\omega_2$ is externally tangent to the other two. Let $X,Y$ be the intersection points of $\omega_2$ with the other common external tangent of $\omega_1,\omega_3$. The perpendicular line through $B$ to $l$ meets $\omega_2$ again at $Z$. Prove that the circle with diameter $AC$ touches $ZX,ZY$.
[i]Proposed by Iman Maghsoudi - Siamak Ahmadpour[/i]
Find the smallest positive integer $n$ for which $0 <\sqrt[4]{n}- [\sqrt[4]{n}]< 10^{-5}$
.
Alexis and Joshua are walking along the beach when they decide to draw symbols in the sand. Alex draws only stars and only draws them in pairs while Joshua draws only squares in trios. "Let's see how many rows of $15$ adjacent symbols we can make this way," suggests Josh. Alexis is game for the task and the two get busy drawing. Some of their rows look like
\[\begin{array}{ccccccccccccccc} \vspace{10pt}*&*&*&*&*&*&*&*&*&*&*&*&\blacksquare&\blacksquare&\blacksquare\\\vspace{10pt}\blacksquare&\blacksquare&\blacksquare&*&*&*&*&*&*&*&*&*&*&*&*\\\vspace{10pt}\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&*&*&*&*&*&*&\blacksquare&\blacksquare&\blacksquare \\\vspace{10pt}\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare\\\vspace{10pt} *&*&*&*&*&*&\blacksquare&\blacksquare&\blacksquare&*&*&*&*&*&*\end{array}\]
The twins decide to count each of the first two rows above as distinct rows, even though one is the mirror image of the other. But the last of the rows above is its own mirror image, so they count it only once. Around an hour later, the twins realized that they had drawn every possible row exactly once using their rules of stars in pairs and squares in trips. How many rows did they draw in the sand?
The $8 \times 8$ board is covered with the same shape as in the picture to the right (each of the shapes can be rotated $90^o$) so that any two do not overlap or extend beyond the edge of the chessboard. Determine the largest possible number of fields of this chessboard can be covered as described above.
[img]https://cdn.artofproblemsolving.com/attachments/e/5/d7f44f37857eb115edad5ea26400cdca04e107.png[/img]
Let $n \geq 3$ be an integer. At a MEMO-like competition, there are $3n$ participants, there are n languages spoken, and each participant speaks exactly three different languages. Prove that at least $\left\lceil\frac{2n}{9}\right\rceil$ of the spoken languages can be chosen in such a way that no participant speaks more than two of the chosen languages.
[b]Note.[/b] $\lceil x\rceil$ is the smallest integer which is greater than or equal to $x$.
Let $ABC$ be an acute triangle so that $2BC = AB + AC,$ with incenter $I{}.$ Let $AI{}$ meet $BC{}$ at point $A'.{}$ The perpendicular bisector of $AA'{}$ meets $BI{}$ and $CI{}$ at points $B'{}$ and $C'{}$ respectively. Let $AB'{}$ intersect $(ABC)$ at $X{}$ and let $XI{}$ intersect $AC'{}$ at $X'{}.$ Prove that $2\angle XX'A'=\angle ABC.{}$
[i](Proposed by Kang Taeyoung, South Korea)[/i]
Let $a$ and $b$ be integers such that $a - b = a^2c - b^2d$ for some consecutive integers $c$ and $d$. Prove that $|a - b|$ is a perfect square.
Prove that for all primes \( p \) such that \( p \equiv 3 \pmod{4} \) or \( p \equiv 5 \pmod{8} \), there exist integers
\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]
such that
\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]
$ 1650$ students are arranged in $ 22$ rows and $ 75$ columns. It is known that in any two columns, the number of pairs of students in the same row and of the same sex is not greater than $ 11$. Prove that the number of boys is not greater than $ 928$.
Find all positive integers $a$ and $b$ for which there are three consecutive integers at which the polynomial \[ P(n) = \frac{n^5+a}{b} \] takes integer values.
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $ x,y$:
$ x f(x)\minus{}yf(y)\equal{}(x\minus{}y)f(x\plus{}y)$.
Show that there are no integers $x, y, z$, and $t$ such that $$\sqrt[3]{x^5 + y^5 + z^5 + t^5} = 2016.$$
$\displaystyle 2n$ students $\displaystyle (n \geq 5)$ participated at table tennis contest, which took $\displaystyle 4$ days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this:
- there is only one winner;
- there are $\displaystyle 3$ students on the second place;
- no student lost all $\displaystyle 4$ matches.
How many students won only a single match and how many won exactly $\displaystyle 2$ matches? (In the above conditions)
Find the integer $x$ for which $135^3+138^3=x^3-1.$
In all tetrahedron $ABCD$ holds
[list=1]
[*] $(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(h_a-r)^2}{(h_a^n-r^n)(h_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}$
[*] $(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(r_a-r)^2}{(r_a^n-r^n)(r_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}$
[/list]
for all $n\in \mathbb{N}^*$
Prime $p>2$ and a polynomial $Q$ with integer coefficients are such that there are no integers $1 \le i < j \le p-1$ for which $(Q(j)-Q(i))(jQ(j)-iQ(i))$ is divisible by $p$. What is the smallest possible degree of $Q$?
[i](Proposed by Anton Trygub)[/i]
For positive integers $ m$ and $ n$, let $ f\left(m,n\right)$ denote the number of $ n$-tuples $ \left(x_1,x_2,\dots,x_n\right)$ of integers such that $ \left|x_1\right| \plus{} \left|x_2\right| \plus{} \cdots \plus{} \left|x_n\right|\le m$. Show that $ f\left(m,n\right) \equal{} f\left(n,m\right)$.
Does there exist a polynomial $P(x)$ with integral coefficients such that
a) $P(\sqrt[3]{25 }+ \sqrt[3]{5}) = 220\sqrt[3]{25} + 284\sqrt[3]{5}$ ?
b) $P(\sqrt[3]{25 }+ \sqrt[3]{5}) = 1184\sqrt[3]{25} + 1210\sqrt[3]{5}$ ?
Sixteen squares of an $8\times 8$ chessboard are chosen so that there are exactly lwo in each row and two in each column. Prove that eight white pawns and eight black pawns can be placed on these sixteen squares so that there is one white pawn and one black pawn in each row and in cach colunm.
a) Let $n$ be a positive integer. Prove that $ n\sqrt {x-n^2}\leq \frac {x}{2}$ , for $x\geq n^2$.
b) Find real $x,y,z$ such that: $ 2\sqrt {x-1} +4\sqrt {y-4} + 6\sqrt {z-9} = x+y+z$
Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2=3$. Prove that
\[ xyz(x+y+z)+2021\geqslant 2024xyz\]
Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed:
$$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$.
The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this.
Prove that Eduardo has a winning strategy.
[i]Proposed by Amine Natik, Morocco[/i]
A triangle $XY Z$ and a circle $\omega$ of radius $2$ are given in a plane, such that $\omega$ intersects segment $\overline{XY}$ at the points $A$, $B$, segment $\overline{Y Z}$ at the points $C$, $D$, and segment $\overline{ZX}$ at the points $E$, $F$. Suppose that $XB > XA$, $Y D > Y C$, and $ZF > ZE$. In addition, $XA = 1$, $Y C = 2$, $ZE = 3$, and $AB = CD = EF$. Compute $AB$.