Found problems: 85335
Find the slope of a line passing through the point $(0,\ 1)$ with which the area of the part bounded by the line and the parabola $y=x^2$ is $\frac{5\sqrt{5}}{6}.$
Given a positive integer $n$. Consider a $3 \times n$ grid, a set $S$ of squares is called [i]connected[/i] if for any points $A \neq B$ in $S$, there exists an integer $l \ge 2$ and $l$ squares $A=C_1,C_2,\dots ,C_l=B$ in $S$ such that $C_i$ and $C_{i+1}$ shares a common side ($i=1,2,\dots,l-1$).
Find the largest integer $K$ satisfying that however the squares are colored black or white, there always exists a [i]connected[/i] set $S$ for which the absolute value of the difference between the number of black and white squares is at least $K$.
Let $[AD]$ and $[CE]$ be internal angle bisectors of $\triangle ABC$ such that $D$ is on $[BC]$ and $E$ is on $[AB]$. Let $K$ and $M$ be the feet of perpendiculars from $B$ to the lines $AD$ and $CE$, respectively. If $|BK|=|BM|$, show that $\triangle ABC$ is isosceles.
Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer.
[i]Proposed by fattypiggy123[/i]
Let $1\le k\le n$. Suppose that the sequence $a_1, a_2, \ldots, a_n$ satisfies $0\le a_1 \le a_2 \le \ldots \le a_k$ and $0 \le a_n \le a_{n-1} \le \ldots \le a_k$. The sequence $b_1, b_2, \ldots, b_n$ is the nondecreasing permutation of $a_1, a_2, \ldots, a_n$. Prove that
\[\sum_{i=1}^n \sum_{j=1}^n (j-i)^2a_ia_j \le \sum_{i=1}^n \sum_{j=1}^n (j-i)^2b_ib_j \]
Betty Lou and Peggy Sue take turns flipping switches on a $100 \times 100$ grid. Initially, all switches are "off". Betty Lou always flips a horizontal row of switches on her turn; Peggy Sue always flips a vertical column of switches. When they finish, there is an odd number of switches turned "on'' in each row and column. Find the maximum number of switches that can be on, in total, when they finish.
Prove that the set of all the points with both coordinates begin rational numbers can be written as a reunion of two disjoint sets $ A$ and $ B$ such that any line that that is parallel with $ Ox$, and respectively $ Oy$ intersects $ A$, and respectively $ B$ in a finite number of points.
For each integer $n$, determine the smallest real number $M_n$ such that
\[\frac{1}{a_1}+\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots+\frac{a_{n-1}}{a_n} \le M_n\]
for any $n$-tuple $(a_1,a_2,\dots,a_n)$ of integers such that $1<a_1<a_2<\dots<a_n$.
A trapezoid has side lengths as indicated in the figure (the sides with length $11$ and $36$ are parallel). Calculate the area of the trapezoid.[img]https://1.bp.blogspot.com/-5PKrqDG37X4/XzcJtCyUv8I/AAAAAAAAMY0/tB0FObJUJdcTlAJc4n6YNEaVIDfQ91-eQCLcBGAsYHQ/s0/1995%2BMohr%2Bp1.png[/img]
Given is a natural number $n \geq 3$. What is the smallest possible value of $k$ if the following statements are true?
For every $n$ points $ A_i = (x_i, y_i) $ on a plane, where no three points are collinear, and for any real numbers $ c_i$ ($1 \le i \le n$) there exists such polynomial $P(x, y)$, the degree of which is no more than $k$, where $ P(x_i, y_i) = c_i $ for every $i = 1, \dots, n$.
(The degree of a nonzero monomial $ a_{i,j} x^{i}y^{j} $ is $i+j$, while the degree of polynomial $P(x, y)$ is the greatest degree of the degrees of its monomials.)
i) Given line segments $A,B,C,D$ with $A$ the longest, construct a quadrilateral with these sides and with $A$ and $B$ parallel, when possible.
ii) Given any acute-angled triangle $ABC$ and one altitude $AH$, select any point $D$ on $AH$, then draw $BD$ and extend until it intersects $AC$ in $E$, and draw $CD$ and extend until it intersects $AB$ in $F$. Prove that $\angle AHE = \angle AHF$.
For any set $ A $ we say that two functions $ f,g:A\longrightarrow A $ are [i]similar,[/i] if there exists a bijection $ h:A\longrightarrow A $ such that $ f\circ h=h\circ g. $
[b]a)[/b] If $ A $ has three elements, construct a finite, arbitrary number functions, having as domain and codomain $ A, $ that are two by two similar, and every other function with the same domain and codomain as the ones determined is similar to, at least, one of them.
[b]b)[/b] For $ A=\mathbb{R} , $ show that the functions $ \sin $ and $ -\sin $ are similar.
Show that it is possible to express $1$ as a sum of $6$, and as a sum of $9$ reciprocals of odd positive integers. Generalize the problem.
On a circle, distinct points $A_1, ... , A_{16}$ are chosen. Consider all possible convex polygons all of whose vertices are among $A_1, ... , A_{16}$ . These polygons are divided into $2$ groups, the first group comprising all polygons with $A_1$ as a vertex, the second group comprising the remaining polygons. Which group is more numerous?
If $a_1 ,a_2 ,\ldots, a_n$ are complex numbers such that
$$ |a_1| =|a_2 | =\cdots = |a_n| =r \ne 0,$$
and if $T_s$ denotes the sum of all products of these $n$ numbers taken $s$ at a time, prove that
$$ \left| \frac{T_s }{T_{n-s}}\right| =r^{2s-n}$$
whenever the denominator of the left-hand side is different from $0$.
Based on the fact that only $10\%$ of the total volume of an iceberg can be seen above the surface of the water, if $500\text{ m}^3$ of a certain iceberg is seen, then what is the total volume of the iceberg?
$\textbf{(A) } 5\qquad\textbf{(B) } 50\qquad\textbf{(C) } 500\qquad\textbf{(D) } 5{,}000\qquad\textbf{(E) } \text{imposssible to determined}$
Let $P(x)=x^2+bx+c$. Suppose $P(P(1))=P(P(-2))=0$ and $P(1)\neq P(-2)$. Then $P(0)=$
[list=1]
[*] $-\frac{5}{2}$
[*] $-\frac{3}{2}$
[*] $-\frac{7}{4}$
[*] $\frac{6}{7}$
[/list]
The number $4$ has an odd number of odd positive divisors, namely $1$, and an even number of even positive divisors, namely $2$ and $4$. Is there a number with an odd number of even positive divisors and an even number of odd positive divisors?
Given two parallelograms $ABCD$ and $AECF$ with common diagonal $AC$, where $E$ and $F$ lie inside parallelogram $ABCD$. Show: The circumcircles of the triangles $AEB$, $BFC$, $CED$ and $DFA$ have one point in common.
Let $a$, $b$ and $c$ be real numbers larger than $1$. Prove the inequality $$\frac{ab}{c-1}+\frac{bc}{a - 1}+\frac{ca}{b -1} \ge 12.$$
When does equality hold?
[i](Karl Czakler)[/i]
[b]Problem:[/b]For a positive integer $ n$,let $ V(n; b)$ be the number of decompositions of $ n$ into a
product of one or more positive integers greater than $ b$. For example,$ 36 \equal{} 6.6 \equal{}4.9 \equal{} 3.12 \equal{} 3 .3. 4$, so that $ V(36; 2) \equal{} 5$.Prove that for all positive integers $ n$; b it holds that $ V(n;b)<\frac{n}{b}$. :)
Nico wants to write the $100$ integers from $1$ to $100$ around a circle in some order and without repetition, such that they have the following property: when moving around the circle clockwise, the sum of the $100$ distances between each number and its next number is equal to $198$. Determine in how many ways the $100$ numbers can be ordered so that Nico achieves his goal.
[b]Note:[/b] The distance between two numbers $a$ and $b$ is equal to the absolute value of their difference: $|a - b|$.
The width of a lane in a circular running track is $1.22$ meters. One loop in the first lane (shortest lane) is $400$ meters. Thus $12.5$ loops makes it a $5{,}000$ meter distance. Which lane should an athlete run in if they want to make $12$ loops as close to the $5{,}000$ meter distance as possible? \[\rm a. ~second\qquad \mathrm b. ~third \qquad \mathrm c. ~fourth \qquad\mathrm d. ~fifth \qquad\mathrm e. ~sixth\]
A circle radius $320$ is tangent to the inside of a circle radius $1000$. The smaller circle is tangent to a diameter of the larger circle at a point $P$. How far is the point $P$ from the outside of the larger circle?
The bisector of the angle $BAC$of the acute triangle $ABC$ ( $AC \ne AB$) intersects its circumscribed circle for the second time at the point $W$. Let $O$ be the center of the circumscribed circle $\Delta ABC$. The line $AW$ intersects for the second time the circumcribed circles of triangles $OWB$ and $OWC$ at the points $N$ and $M$, respectively. Prove that $BN + MC = AW$.
(Mitrofanov V., Hilko D.)