Found problems: 85335
Let $a,b,c$ be real positive numbers. Prove that the following inequality holds:
\[{
\sum_{\rm cyc}\sqrt{5a^2+5c^2+8b^2\over 4ac}\ge 3\cdot \root 9 \of{8(a+b)^2(b+c)^2(c+a)^2\over (abc)^2}
}\]
How many numbers of $3$ digits have their central digit greater than any of the other two? How many of them have also three different digits?
Right triangle $ABC$ has a right angle at $A.$ Points $D$ and $E$ respectively lie on $\overline{AC}$ and $\overline{BC}$ so that $\angle BDA \cong \angle CDE.$ If the lengths $DE,$ $DA,$ $DC,$ and $DB,$ in this order, form an arithmetic sequence of distinct positive integers, then the set of all possible areas of $\triangle ABC$ is a subset of the positive integers. Compute the smallest element in this set that is greater than $1000.$
One point of the plane is called $rational$ if both coordinates are rational and $irrational$ if both coordinates are irrational. Check whether the following statements are true or false:
[b]a)[/b] Every point of the plane is in a line that can be defined by $2$ rational points.
[b]b)[/b] Every point of the plane is in a line that can be defined by $2$ irrational points.
This maybe is not algebra so sorry if I putted it in the wrong category!
Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign "$+$" or a multiplication sign "$\times$" between each pair of adjacent digits. (For example, her expression could be $4 + 3\times 3$, with value 13.) Each positive digit is equally likely, each arithmetic sign ("$+$" or "$\times$") is equally likely, and all choices are independent. What is the expected value (average value) of her expression?
In the beginning, there is a circle with three points on it. The points are colored (clockwise): Green, blue, red. Jonathan may perform the following actions, as many times as he wants, in any order:
[list]
[*] Choose two adjacent points with [u]different[/u] colors, and add a point between them with one of the two colors only.
[*] Choose two adjacent points with [u]the same[/u] color, and add a point between them with any of the three colors.
[*] Choose three adjacent points, at least two of them having the same color, and delete the middle point.
[/list]
Can Jonathan reach a state where only three points remain on the circle, colored (clockwise): Blue, green, red?
Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ is tangent to the line $CD$, and the circle with diameter $CD$ is tangent to the line $AB$. Prove that the two intersection points of these circles and the point $AC \cap BD$ are collinear.
A number $n$ is called a Niven number, named for Ivan Niven, if it is divisible by the sum of its digits. For example, $24$ is a Niven number. Show that it is not possible to have more than $20$ consecutive Niven numbers.
Find all real solutions $(x,y)$ to the equation $y^4+2y^2+8x^2+16x^4 = 24xy-8$.
The closed interval $A = [0, 50]$ is the union of a finite number of closed intervals, each of length $1$. Prove that some of the intervals can be removed so that those remaining are mutually disjoint and have total length greater than $25$.
Note: For reals $a\le b$, the closed interval $[a, b] := \{x\in \mathbb{R}:a\le x\le b\}$ has length $b-a$; disjoint intervals have [i]empty [/i]intersection.
Let $a,b,c$ be positive reals s.t. $a^2+b^2+c^2=1$. Prove the following inequality \[ \sum \frac{a}{a^3+bc} >3 . \]
[i]Proposed by A. Khrabrov[/i]
Let $a$ and $b$ be real numbers with $0 \le a, b \le 1$.
(a) Prove that $ \frac{a}{b+1} +\frac{b}{a+1} \le 1.$
(b) Find the case of equality.
From the set of integers $ \{1,2,3,\ldots,2009\}$, choose $ k$ pairs $ \{a_i,b_i\}$ with $ a_i<b_i$ so that no two pairs have a common element. Suppose that all the sums $ a_i\plus{}b_i$ are distinct and less than or equal to $ 2009$. Find the maximum possible value of $ k$.
In trapezoid $ABCD$, the bisectors of angles $A$ and $D$ intersect at point $E$ lying on the side of $BC$. These bisectors divide the trapezoid into three triangles into which the circles are inscribed. One of these circles touches the base $AB$ at the point $K$, and two others touch the bisector $DE$ at points $M$ and $N$. Prove that $BK = MN$.
[b]Q9.[/b] Evaluate the integer part of the number
\[H= \sqrt{1+2011^2+ \frac{2011^2}{2012^2}}+ \frac{2011}{2012}.\]
Two equal non-intersecting wooden disks, one gray and one black, are glued to a plane. A triangle with one gray side and one black side can be moved along the plane so that the disks remain outside the triangle, while the colored sides of the triangle are tangent to the disks of the same color (the tangency points are not the vertices). Prove that the line that contains the bisector of the angle between the gray and black sides always passes through some fixed point of the plane.
(Egor Bakaev, Pavel Kozhevnikov, Vladimir Rastorguev) (Senior version[url=https://artofproblemsolving.com/community/c6h2102856p15209040] here[/url])
An odd positive integer $n$ can be expressed as the sum of two or more consecutive integers in exactly $2023$ ways. Find the greatest possible nonnegative integer $k$ such that $3^k$ is a factor of the least possible value of $n$.
Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$
(1) Find $f(x).$
(2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.
How many [b]distinguishable[/b] rearrangements of the letters in CONTEST have both the vowels first? (For instance, OETCNST is a one such arrangements but OTETSNC is not.)
$ \textbf{(A)}\ 60\qquad
\textbf{(B)}\ 120\qquad
\textbf{(C)}\ 240\qquad
\textbf{(D)}\ 720\qquad
\textbf{(E)}\ 2520$
Let $x_1$ and $x_2$ be such that $x_1 \neq x_2$ and $3x_i^2-hx_i=b$, $i=1, 2$. Then $x_1+x_2$ equals
$ \textbf{(A)}\ -\frac{h}{3} \qquad\textbf{(B)}\ \frac{h}{3} \qquad\textbf{(C)}\ \frac{b}{3} \qquad\textbf{(D)}\ 2b \qquad\textbf{(E)}\ -\frac{b}{3} $
A positive integer $n$ is said to possess Property ($A$) if there exists a positive integer $N$ such that $N^2$ can be written as the sum of the squares of $n$ consecutive positive integers. Is it true that there are infinitely many positive integers which possess Property ($A$)? Justify your answer.
(As an example, the number $n = 2$ possesses Property ($A$) since $5^2 = 3^2 + 4^2$).
A polygon with sides running along the sides of the squares was cut out of an endless chessboard. A segment of the perimeter of a polygon is called black if the polygon adjacent to it from the inside is which cell is black, respectively white if the cell is white. Let $A$ be the number of black segments on the perimeter, and $B$ be the number of white ones, Let the polygon consist of $a$ black and $b$ white cells. Prove that $A-B = 4(a -b)$.
Find all positive integers $n$ for which there exist positive integers $a, b,$ and $c$ that satisfy the following three conditions:
$\bullet$ $a+b+c=n$
$\bullet$ $a$ is a divisor of $b$ and $b$ is a divisor of $c$
$\bullet$ $a < b < c$
Let $ n$ be a positive integer, and $ k\leq n\minus{}1$, $ k\in \mathbb{N}$. Denote $ a_k\equal{}k!(1\plus{}\frac12\plus{}\frac13\plus{}\cdots\plus{}\frac1k)$.
Prove that the number $ k! \cdot\left[\binom{n\minus{}1}{k}\minus{}(\minus{}1)^k\right]\plus{}(\minus{}1)^k\cdot a_k \cdot n$ is divisible by $ n^2$.
Prove that $\int_0^4 \left(1-\cos \frac{x}{2}\right)e^{\sqrt{x}}dx\leq -2e^2+30.$