Found problems: 85335
[list=a]
[*] In the right picture there is a square with four congruent circles inside it. Each circle is tangent to two others, and to two of the edges of the square. Evaluate the ratio between the blue part and white part of the square's area.
[*] In the left picture there is a regular hexagon with six congruent circles inside it. Each circle is tangent to two others, and to one of the edges on the hexagon in its midpoint. Evaluate the ratio between the blue part and white part of the hexagon's area.
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[img]https://i.imgur.com/fAuxoc9.png[/img]
The incircle of $\triangle ABC$ touch $AB$,$BC$,$CA$ at $K$,$L$,$M$. The common external tangents to the incircles of $\triangle AMK$,$\triangle BKL$,$\triangle CLM$, distinct from the sides of $\triangle ABC$, are drawn. Show that these three lines are concurrent.
Andryusha has $100$ stones of different weight and he can distinguish the stones by appearance, but does not know their weight. Every evening, Andryusha can put exactly $10$ stones on the table and at night the brownie will order them in increasing weight. But, if the drum also lives in the house then surely he will in the morning change the places of some $2$ stones.Andryusha knows all about this but does not know if there is a drum in his house. Can he find out?
Let $Oxyz$ be a trihedron whose edges $x,y, z$ are mutually perpendicular. Let $C$ be the point on the ray $z$ with $OC = c$. Points $P$ and $Q$ vary on the rays $x$ and $y$ respectively in such a way that $OP+OQ = k$ is constant. For every $P$ and $Q$, the circumcenter of the sphere through $O,C,P,Q$ is denoted by $W$. Find the locus of the projection of $W$ on the plane O$xy$. Also find the locus of points $W$.
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that:
- $f$ is strictly increasing,
- there exists $M \in \mathbb{N}$ such that $f(x+1) - f(x) < M$ for all $x \in \mathbb{N}$,
- for every $x \in \mathbb{Z}$, there exists $y \in \mathbb{Z}$ such that
\[
f(y) = \frac{f(x) + f(x + 2024)}{2}.
\]
[i]Proposed by Pavle Martinović[/i]
Let $x$ and $y$ be rational numbers, such that $x^{5}+y^{5}=2x^{2}y^{2}$. Prove that $1-xy$ is the square of a rational number.
[color=darkred]For any $ m \in \mathbb{N}^*$ solve the ecuation
\[ \left\{\left( x \plus{} \frac {1}{m}\right) ^3\right\} \equal{} x^3
\]
[/color]
Two isosceles triangles of the same area are located as shown in the figure. Find the angle $x$.
[img]https://cdn.artofproblemsolving.com/attachments/a/6/f7dbfd267274781b67a5f3d5a9036fb2905156.png[/img]
Solve in integer:$x^6+x^2y=y^3+2y^2$
Find the area of a square inscribed in an equilateral triangle, with one edge of the square on an edge of the triangle, if the side length of the triangle is $ 2\plus{}\sqrt{3}$.
Let $S$ be a finite set of integers. Prove that there exists a number $c$ depending on $S$ such that for each non-constant polynomial $f$ with integer coefficients the number of integers $k$ satisfying $f(k)\in S$ does not exceed $\max(\deg f,c)$.
An ant can move from a vertex of a cube to the opposite side of its diagonal only on the edges and the diagonals of the faces such that it doesn’t trespass a second time through the path made. Find the distance of the maximum journey this ant can make.
Evaluate $\int^4_1\frac{x-2}{(x^2+4)\sqrt x}dx$
For a prime number $p$. Can the number of n positive integers that make the expression \[\dfrac{n^3+np+1}{n+p+1}\] an integer be $777$?
It is possible to write the numbers $111$, $112$, $121$, $122$, $211$, $212$, $221$ and $222$ at the vertices of a cube, so that the numbers written in adjacent vertices match at most in one digit?
Barney Schwinn notices that the odometer on his bicycle reads $1441$, a palindrome, because it reads the same forward and backward. After riding $4$ more hours that day and $6$ the next, he notices that the odometer shows another palindrome, $1661$. What was his average speed in miles per hour?
$\textbf{(A)}\ 15\qquad
\textbf{(B)}\ 16\qquad
\textbf{(C)}\ 18\qquad
\textbf{(D)}\ 20\qquad
\textbf{(E)}\ 22$
$100$ heaps of stones lie on a table. Two players make moves in turn. At each move, a player can remove any non-zero number of stones from the table, so that at least one heap is left untouched. The player that cannot move loses. Determine, for each initial position, which of the players, the first or the second, has a winning strategy.
[i]K. Kokhas[/i]
[b]EDIT.[/b] It is indeed confirmed by the sender that empty heaps are still heaps, so the third post contains the right guess of an interpretation.
A student is asked to measure $30.0 {\text{g}}$ of methanol $(d=0.7914 \text{g/mL at 25}^{\circ}\text{C})$ but has only a graduated cylinder with which to measure it. What volume of methanol should the student use to obtain the required ${30.0 \text{g}}$?
${ \textbf{(A)}\ 23.7 \text{mL} \qquad\textbf{(B)}\ 30.0 \text{mL} \qquad\textbf{(C)}\ 32.4 \text{mL} \qquad\textbf{(D)}\ 37.9 \text{mL} }$
Let $ABC$ be a triangle with orthocenter $H$, $\Gamma$ its circumcircle, and $A' \neq A$, $B' \neq B$, $C' \neq C$ points on $\Gamma$. Define $l_a$ as the line that passes through the projections of $A'$ over $AB$ and $AC$. Define $l_b$ and $l_c$ similarly. Let $O$ be the circumcenter of the triangle determined by $l_a$, $l_b$ and $l_c$ and $H'$ the orthocenter of $A'B'C'$. Show that $O$ is midpoint of $HH'$.
[color=blue][size=150]PERU TST IMO - 2006[/size]
Saturday, may 20.[/color]
[b]Question 02[/b]
Find all pairs $(a,b)$ real positive numbers $a$ and $b$ such that:
$[a[bn]]= n-1,$
for all $n$ positive integer.
Note: [x] denotes the integer part of $x$.
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[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88510]Spanish version[/url]
$\text{\LaTeX}{}$ed by carlosbr
On the infinite surface of the sea floats a black and bounded oil slick. After every minute the slick and the sea change according to the following law: at each point $P$ of the sea (or of the slick), a disk $D$ of radius $1$ is considered centered on $ P$. If more than half of the area inside the disk $D$ is black, the $P$ point will remain black for the next minute. If more than half of the area inside the disk $D$ is dark blue, the point $P$ will be dark blue for the next minute. In the event that both the clean and the contaminated area within the disk $D$ are the same, its center $P$ will not change color. Can that stain "live" forever or will it disappear at some point?
Let $ABC$ be a triangle with $BC = a$, $CA = b$, and $AB = c$. The $A$-excircle is tangent to $\overline{BC}$ at $A_1$; points $B_1$ and $C_1$ are similarly defined.
Determine the number of ways to select positive integers $a$, $b$, $c$ such that
[list]
[*] the numbers $-a+b+c$, $a-b+c$, and $a+b-c$ are even integers at most 100, and
[*] the circle through the midpoints of $\overline{AA_1}$, $\overline{BB_1}$, and $\overline{CC_1}$ is tangent to the incircle of $\triangle ABC$.
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Two squids are forced to participate in a game. Before it begins, they will be informed of all the rules, and can discuss their strategies freely. Then, they will be locked in separate rooms, and be given distinct positive integers no larger than $2023$ as their IDs respectively. The two squids then take turns alternatively; on one's turn, the squid chooses one of the following:
1. announce a positive integer, which will be heard by the other squid;
2. declare which squid has the larger ID. If correct, they win and are released together; otherwise, they lose and are fried together.
Find the smallest positive integer $N$ so that, no matter what IDs the squids have been given, they can always win in a finite number of turns, and the sum of the numbers announced during the game is no larger than $N$.
Describe a construction of quadrilateral $ABCD$ given:
(i) the lengths of all four sides;
(ii) that $AB$ and $CD$ are parallel;
(iii) that $BC$ and $DA$ do not intersect.
The diagram below shows an equilateral triangle and a square of side length $2$ joined along an edge. What is the area of the shaded triangle?
[asy]
fill((2,0) -- (2,2) -- (1, 2 + sqrt(3)) -- cycle, gray);
draw((0,0) -- (2,0) -- (2,2) -- (1, 2 + sqrt(3)) -- (0,2) -- (0,0));
draw((0,2) -- (2,2));
[/asy]