Found problems: 85335
Solve equation $$x^2-\sqrt{a-x}=a$$ where $x$ is real number and $a$ is real parameter
The numbers $1, 2,..., 2020$ are written on the blackboard. Venus and Serena play the following game. First, Venus connects by a line segment two numbers such that one of them divides the other. Then Serena connects by a line segment two numbers which has not been connected and such that one of them divides the other. Then Venus again and they continue until there is a triangle with one vertex in $2020$, i.e. $2020$ is connected to two numbers that are connected with each other. The girl that has drawn the last line segment (completed the triangle) is the winner. Which of the girls has a winning strategy?
(Tomáš Bárta, Czech Republic)
Two cyclists leave simultaneously a point $P$ in a circular runway with constant velocities $v_1, v_2 (v_1 > v_2)$ and in the same sense. A pedestrian leaves $P$ at the same time, moving with velocity $v_3 = \frac{v_1+v_2}{12}$ . If the pedestrian and the cyclists move in opposite directions, the pedestrian meets the second cyclist $91$ seconds after he meets the first. If the pedestrian moves in the same direction as the cyclists, the first cyclist overtakes him $187$ seconds before the second does. Find the point where the first cyclist overtakes the second cyclist the first time.
A positive integer $m$ is called a [i]beautiful [/i] integer if that there exists a positive integer $n$ such that $m = n^2+ n + 1$. Prove that there are infinitely many [i]beautiful [/i] integers with square factors, and the square factors of different beautiful integers are relative prime.
A zerg player can produce one zergling every minute and a protoss player can produce one zealot every $2.1$ minutes. Both players begin building their respective units immediately from the beginning of the game. In a ght, a zergling army overpowers a zealot army if the ratio of zerglings to zealots is more than $3$. What is the total amount of time (in minutes) during the game such that at that time the zergling army would overpower the zealot army?
The edges of a graph with $2n$ vertices ($n \ge 4$) are colored in blue and red such that there is no blue triangle and there is no red complete subgraph with $n$ vertices. Find the least possible number of blue edges.
In triangle $ABC, AB = AC$. A line is drawn through $A$ parallel to $BC$. Outside triangle $ABC$, a circle is drawn tangent to this line, to the line $BC$, to $AB$ and to the incircle of $ABC$. If the radius of this circle is $1$ , determine the inradius of $ABC$.
(RK Gordin)
Let $n$ be a positive integer. Let $a_1$, $a_2$, $\ldots\,$, $a_n$ be real numbers such that $-1 \le a_i \le 1$ (for all $1 \le i \le n$). Let $b_1$, $b_2$, $\ldots$, $b_n$ be real numbers such that $-1 \le b_i \le 1$ (for all $1 \le i \le n$). Prove that
\[
\left| \prod_{i=1}^n a_i - \prod_{i=1}^n b_i \right|
\le \sum_{i = 1}^n \left| a_i - b_i \right| \, .
\]
Inscribed in a circle is a quadrilateral having sides of lengths $25,~39,~52,$ and $60$ taken consecutively. The diameter of this circle has length
$\textbf{(A) }62\qquad\textbf{(B) }63\qquad\textbf{(C) }65\qquad\textbf{(D) }66\qquad \textbf{(E) }69$
Let $f :\mathbb{R} \to \mathbb{R}$ a derivable function such that $f(0) = 0$ and $|f'(x)| \leq |f(x)\cdot log |f(x)||$ for every $x \in \mathbb{R}$ such that $0 < |f(x)| < 1/2.$ Prove that $f(x) = 0$ for every $x \in \mathbb{R}$.
Consider partitions of an $n \times n$ square (composed of $n^2$ unit squares) into rectangles with one integer side and the other side equal to $1$. What is the largest possible number of such partitions among which no two have an identical rectangle at the same place?
Find a polynomial $Q(x)$ such that $(2x^2 - 6x + 5)Q(x)$ is a polynomial with all positive coefficients.
The line passing through the center of the equilateral triangle $ ABC $ intersects the lines $ AB $, $ BC $ and $ CA $ at the points $ {{C} _ {1}} $, $ {{A} _ {1}} $ and $ {{B} _ {1}} $, respectively. Let $ {{A} _ {2}} $ be a point that is symmetric $ {{A} _ {1}} $ with respect to the midpoint of $ BC $; the points $ {{B} _ {2}} $ and $ {{C} _ {2}} $ are defined similarly. Prove that the points $ {{A} _ {2}} $, $ {{B} _ {2}} $ and $ {{C} _ {2}} $ lie on the same line tangent to the inscribed circle of the triangle $ ABC $.
(Serdyuk Nazar)
Let $A$ and $B$ be two points inside a circle $C$. Show that there exists a circle that contains $A$ and $B$ and lies completely inside $C$.
Each face of a $7 \times 7 \times 7$ cube is divided into unit squares. What is the maximum number of squares that can be chosen so that no two chosen squares have a common point?
[i]A. Chukhnov[/i]
Find the minimum possible value of the largest of $xy$, $1-x-y+xy$, and $x+y-2xy$ if $0\leq x \leq y \leq 1$.
Let $ a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that
(a) The polynomial $ p(x) \equal{} x^{n} \minus{} a_{1}x^{n \minus{} 1} \plus{} ... \minus{} a_{n \minus{} 1}x \minus{} a_{n}$ has preceisely 1 positive real root $ R$.
(b) let $ A \equal{} \sum_{i \equal{} 1}^n a_{i}$ and $ B \equal{} \sum_{i \equal{} 1}^n ia_{i}$. Show that $ A^{A} \leq R^{B}$.
Let $m, n \ge 2$ be distinct positive integers. In an infinite grid of unit squares, each square is filled with exactly one real number so that
[list]
[*]In each $m \times m$ square, the sum of the numbers in the $m^2$ cells is equal.
[*]In each $n \times n$ square, the sum of the numbers in the $n^2$ cells is equal.
[*]There exist two cells in the grid that do not contain the same number.
[/list]
Let $S$ be the set of numbers that appear in at least one square on the grid. Find, in terms of $m$ and $n$, the least possible value of $|S|$.
[i]Kiran Reddy[/i]
For each positive integer $n$, decide whether it is possible to tile a square with exactly $n+1$ similar rectangles, each with a positive area and aspect ratio $1:n$.
Let $ABC$ arbitrary triangle ($AB \neq BC \neq AC \neq AB$) And O,I,H it's circum-center, incenter and ortocenter (point of intersection altitudes). Prove, that
1) $\angle OIH > 90^0$(2 points)
2)$\angle OIH >135^0$(7 points)
balls for 1) and 2) not additive.
Let $k\geq 2$ be a positive integer and $x_1,x_2,\dots ,x_k\in (0,1)$. Also, let $m_1,m_2,\dots ,m_k$ and $n_1,n_2,\dots ,n_k$ be integers. Define $$A=x_1^{m_1}x_2^{m_2}\dots x_k^{m_k},\quad B=x_1^{n_1}x_2^{n_2}\dots x_k^{n_k}.$$ Let $$C=x_1^{\min(m_1,n_1)}x_2^{\min(m_2,n_2)}\dots x_k^{\min(m_k,n_k)}$$ $$D=x_1^{\max(m_1,n_1)}x_2^{\max(m_2,n_2)}\dots x_k^{\max(m_k,n_k)}.$$ Prove that $A+B\leq C+D$. When does equality hold?
[i]Dorel Miheț[/i]
[i]Triangular numbers[/i] are numbers of the form $1 + 2 + . . . + n$ with positive integer $n$, that is $1, 3, 6, 10$, . . . . Find the largest non-triangular positive integer number that cannot be represented as the sum of distinct triangular numbers.
[i]Proposed by A. Golovanov[/i]
Let $p$ be a prime number and $n, k$ and $q$ natural numbers, where $q\le \frac{n -1}{p-1}$ should be. Let $M$ be the set of all integers $m$ from $0$ to $n$, for which $m-k$ is divisible by $p$. Show that $$\sum_{m \in M} (-1) ^m {n \choose m}$$ is divisible by $p^q$.
Find the area of the set $A = \{(x, y)\ |\ 1 \leq x \leq e,\ 0 \leq y \leq f (x)\}$, where \begin{tabular}{ c| c c c c |}
&1 & 1& 1 & 1\\
$f(x)$=& $\ln x$ & 2$\ln x$ & 3$\ln x$ & 4$\ln x$ \\
&${(\ln x)}^2$ & $4{(\ln x)}^2 $& $9{(\ln x)}^2 $& $16{(\ln x)}^2$\\
&${(\ln x)}^3$ & $8{(\ln x)}^3$ &$ 27{(\ln x)}^3$ &$ 64{(\ln x)}^3$
\end{tabular}
If $10\%$ of $\left(x+10\right)$ is $\left(x-10\right)$, what is $10\%$ of $x$?
$\text{(A) }\frac{11}{90}\qquad\text{(B) }\frac{9}{11}\qquad\text{(C) }1\qquad\text{(D) }\frac{11}{9}\qquad\text{(E) }\frac{110}{9}$