Found problems: 85335
Let $a$, $b$ and $c$ be positive real numbers, prove that \[\frac{2a^2 b^2}{a^5+b^5}+\frac{2b^2 c^2}{b^5+c^5}+\frac{2c^2 a^2}{c^5+a^5}\le\frac{a+b}{2ab}+\frac{b+c}{2bc}+\frac{c+a}{2ca}\]
Four points $B,E,A,F$ lie on line $AB$ in order, four points $C,G,D,H$ lie on line $CD$ in order, satisfying:
$$\frac{AE}{EB}=\frac{AF}{FB}=\frac{DG}{GC}=\frac{DH}{HC}=\frac{AD}{BC}.$$
Prove that $FH\perp EG$.
What's the smallest possible value of $$\frac{(x+y+|x-y|)^2}{xy}$$ over positive real numbers $x, y$?
For a given parameter $m$, solve the equation
\[x(x + 1)(x + 2)(x + 3) + 1 - m = 0.\]
A square-shaped pizza with side length $30$ cm is cut into pieces (not necessarily rectangular). All cuts are parallel to the sides, and the total length of the cuts is $240$ cm. Show that there is a piece whose area is at least $36$ cm$^2$
Let a unit square $\mathcal D$ be given in the plane. For any point $X$ in the plane denote $\mathcal D_X$ the image of $\mathcal D$ in rotation with respect to origin $X$ by $+90^\circ.$ Find the locus of all $X$ such that the area of union $\mathcal D\cup\mathcal D_X$ is at most 1.5.
Given a square $ABCD$, let us consider an equilateral triangle $KLM$, whose vertices $K$, $L$ and $M$ belong to the sides $AB$, $BC$ and $CD$ respectively. Find the locus of the midpoints of the sides $KL$ for all possible equilateral triangles $KLM$.
Note: The set of points that satisfy a property is called a locus.
There are $20$ balls in a bag. $a$ of them are red, $b$ of them are white, and $c$ of them are black. It is known that
$ \bullet$ if we double the white balls, the probability of drawing one red ball is $\dfrac 1{25}$ less than the probability of drawing one red ball at the beginning,
and
$ \bullet$ if we remove all red balls, the probability of drawing one white ball is $\dfrac 1{16}$ more than the probability of drawing one white ball at the beginning.
Find $a,b,c$.
Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$
[i]Proposed by Daniel Liu[/i]
A valid lottery ticket is formed by choosing $5$ distinct numbers from $1, 2,3,..., 90$. What is the probability that the winning ticket contains at least two consecutive numbers?
For a permutation $ \sigma\in S_n$ with $ (1,2,\dots,n)\mapsto(i_1,i_2,\dots,i_n)$, define
\[ D(\sigma) \equal{} \sum_{k \equal{} 1}^n |i_k \minus{} k|
\]
Let
\[ Q(n,d) \equal{} \left|\left\{\sigma\in S_n : D(\sigma) \equal{} d\right\}\right|
\]
Show that when $ d \geq 2n$, $ Q(n,d)$ is an even number.
The nonzero numbers $x{}$ and $y{}$ satisfy the inequalities $x^{2n}-y^{2n}>x$ and $y^{2n}-x^{2n}>y$ for some natural number $n{}$. Can the product $xy$ be a negative number?
[i]Proposed by N. Agakhanov[/i]
Determine all pairs $ (n,p)$ of positive integers, where $ p$ is prime, such that $ 3^p\minus{}np\equal{}n\plus{}p$.
How many distinct triangles are there with prime side lengths and perimeter $100$?
[i]Proposed by Muztaba Syed[/i]
[hide=Solution][i]Solution.[/i] $\boxed{0}$
As the perimeter is even, $1$ of the sides must be $2$. Thus, the other $2$ sides are congruent by Triangle Inequality. Thus, for the perimeter to be $100$, both of the other sides must be $49$, but as $49$ is obviously composite, the answer is thus $\boxed{0}$.[/hide]
Given triangle $ ABC$. Let the tangent lines of the circumcircle of $ AB$ at $ B$ and $ C$ meet at $ A_0$. Define $ B_0$ and $ C_0$ similarly.
a) Prove that $ AA_0,BB_0,CC_0$ are concurrent.
b) Let $ K$ be the point of concurrency. Prove that $ KG\parallel BC$ if and only if $ 2a^2\equal{}b^2\plus{}c^2$.
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)
Let $ x_1 , x_2, x_3 ...$ a succession of positive integers such that for every couple of positive integers $(m,n)$ we have $ x_{mn} \neq x_{m(n+1)}$ . Prove that there exists a positive integer $i$ such that $x_i \ge 2017 $.
Set $A$ consists of $m$ consecutive integers whose sum is $2m$, and set $B$ consists of $2m$ consecutive integers whose sum is $m$. The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is $99$. Find $m$.
For what smallest $n$ can a $n \times n$ square be cut into squares $40 \times 40$ and $49 \times 49$ so that squares of both types are present?
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
All the positive integers from 1 till 1000 are written on the blackboard in some order and there is a collection of cards each containing 10 numbers. If there is a card with numbers $1\le a_1<a_2<\ldots<a_{10}\le1000$ in collection then it is allowed to arrange in increasing order the numbers at places $a_1, a_2, \ldots, a_{10},$ counting from left to right. What is the smallest amount of cards in the collection which enables us to arrange in
increasing order all the numbers for any initial arrangement of them?
The points $A, B, C, D$ lie in that order on a circle. The segments $AC$ and $BD$ intersect at the point $P$. The point $B'$ lies on the line $AB$ such that $A$ is between $B$ and $B'$ and $|AB'| = |DP |$. The point $C'$ lies on the line $CD$ such that $D$ is between $C$ and $C'$ lies and $|DC' | = |AP|$. Prove that $\angle B'PC' = \angle ABD'$.
[img]https://cdn.artofproblemsolving.com/attachments/2/2/7ec65246ff5ecfebc25ca13f3709d1791ceb6c.png[/img]
=
[b]p1.[/b] In triangle $ABC$, the median $BM$ is drawn. The length $|BM| = |AB|/2$. The angle $\angle ABM = 50^o$. Find the angle $\angle ABC$.
[b]p2.[/b] Is there a positive integer $n$ which is divisible by each of $1, 2,3,..., 2018$ except for two numbers whose difference is$ 7$?
[b]p3.[/b] Twenty numbers are placed around the circle in such a way that any number is the average of its two neighbors. Prove that all of the numbers are equal.
[b]p4.[/b] A finite number of frogs occupy distinct integer points on the real line. At each turn, a single frog jumps by $1$ to the right so that all frogs again occupy distinct points. For some initial configuration, the frogs can make $n$ moves in $m$ ways. Prove that if they jump by $1$ to the left (instead of right) then the number of ways to make $n$ moves is also $m$.
[b]p5.[/b] A square box of chocolates is divided into $49$ equal square cells, each containing either dark or white chocolate. At each move Alex eats two chocolates of the same kind if they are in adjacent cells (sharing a side or a vertex). What is the maximal number of chocolates Alex can eat regardless of distribution of chocolates in the box?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each. Along this line, the midpoint of the base of one triangle is a vertex of the next. The area of the region of the plane that is covered by the union of the five triangular regions is
[asy]
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pair C=origin, N=dir(0), B=dir(20), A=dir(135), M=dir(180), P=(3/7)*dir(C--N);
draw((0,0)--(1,sqrt(3))--(2,0)--(3,sqrt(3))--(4,0)--(5,sqrt(3))--(6,0));
draw((1,0)--(2,sqrt(3))--(3,0)--(4,sqrt(3))--(5,0));
draw((-1.5,0)--(7.5,0));
[/asy]
$ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 10\sqrt{3}\qquad\textbf{(E)}\ 12\sqrt{3} $
A non-negative integer $p$ is a [i]3-choice[/i] if $\dfrac{k(k-1)(k-2)}{6}$ for some positive integer $k$. Let $p$ and $q$ be 3-choices with $p<q$. Show there is an integer $n$ such that $p \leq n^2 < q$.
What is the smallest positive integer $n$ such that $2016n$ is a perfect cube?