Found problems: 85335
The circles ${{w} _ {1}}$ and ${{w} _ {2}}$ intersect at points $P$ and $Q$. Let $AB$ and $CD$ be parallel diameters of circles ${ {w} _ {1}}$ and ${{w} _ {2}} $, respectively. In this case, none of the points $A, B, C, D$ coincides with either $P$ or $Q$, and the points lie on the circles in the following order: $A, B, P, Q$ on the circle ${{w} _ {1} }$ and $C, D, P, Q$ on the circle ${{w} _ {2}} $. The lines $AP$ and $BQ$ intersect at the point $X$, and the lines $CP$ and $DQ$ intersect at the point $Y, X \ne Y$. Prove that all lines $XY$ for different diameters $AB$ and $CD$ pass through the same point or are all parallel.
(Serdyuk Nazar)
$ \frac{(.2)^{3}}{(.02)^{2}}= $
$ \text{(A)}\ .2\qquad\text{(B)}\ 2\qquad\text{(C)}\ 10\qquad\text{(D)}\ 15\qquad\text{(E)}\ 20 $
Aisha went shopping. At the first store she spent $40$ percent of her money plus four dollars. At the second store she spent $50$ percent of her remaining money plus $5$ dollars. At the third store she spent $60$ percent of her remaining money plus six dollars. When Aisha was done shopping at the three stores, she had two dollars left. How many dollars did she have with her when she started shopping?
At one school, $85$ percent of the students are taking mathematics courses, $55$ percent of the students are taking history courses, and $7$ percent of the students are taking neither mathematics nor history courses. Find the percent of the students who are taking both mathematics and history courses.
Let $ p \geq 1$ be a real number and $ \mathbb{R}_\plus{}\equal{}(0, \infty)$. For which continuous functions $ g : \mathbb{R}_\plus{} \rightarrow \mathbb{R}_\plus{}$ are following functions all convex? \[ M_n(x)\equal{}\left[ \frac{\sum_{i\equal{}1}^n g(\frac{x_i}{x_{i\plus{}1}}) x_{i\plus{}1}^p}{\sum_{i\equal{}1}^n g(\frac{x_i}{x_{i\plus{}1}})} \right ]^\frac 1p ,\] \[ x\equal{}(x_1,\ldots, x_{n\plus{}1}) \in \mathbb{R}_\plus{} ^ {n\plus{}1} , \; n\equal{}1,2,\ldots\]
[i]L. Losonczi[/i]
Determine all real values of $x$ for which
\[\frac{1}{\sqrt{x} + \sqrt{x - 2}} + \frac{1}{\sqrt{x} + \sqrt{x + 2}} = \frac{1}{4}.\]
Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that
\[f(x)+f\left(1-\frac1x\right)=\arctan x\]
for all real $x\ne 0.$ (As usual, $y=\arctan x$ means $-\pi/2<y<\pi/2$ and $\tan y=x.$) Find
\[\int_0^1f(x)\,dx.\]
Let $f(x)=\frac{9^x}{9^x + 3}$. Evaluate $\sum_{i=1}^{1995}{f\left(\frac{i}{1996}\right)}$.
Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ that satisfy the following two conditions:
[list]$\bullet\ f(n)$ is a perfect square for all $n\in\mathbb{Z}_{>0}$
$\bullet\ f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}_{>0}$.[/list]
In trapezoid $ABCD$,$M$ is the midpoint of the base $AD$.Point $E$ lies on the segment $BM$.It is known that $\angle ADB=\angle MAE=\angle BMC$.Prove that the triangle $BCE $ is isosceles.
A cell phone plan costs $\$20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay?
$ \textbf{(A)}\ \$ 24.00 \qquad
\textbf{(B)}\ \$ 24.50\qquad
\textbf{(C)}\ \$ 25.50\qquad
\textbf{(D)}\ \$ 28.00\qquad
\textbf{(E)}\ \$ 30.00$
Consider a polynomial with coefficients of real numbers $ \phi(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ with three positive real roots. Assume that $ \phi(0)<0$, prove that \[ 2b^3\plus{}9a^2d\minus{}7abc \le 0.\]
[i]Hery Susanto, Malang[/i]
Let x; y; z be real numbers, satisfying the relations
$x \ge 20$
$y \ge 40$
$z \ge 1675$
x + y + z = 2015
Find the greatest value of the product P = $xy z$
Let $n$ be a positive integer. Find all polynomials $P$ with real coefficients such that $$P(x^2+x-n^2)=P(x)^2+P(x)$$ for all real numbers $x$.
There is a shape like this (Attachment down below)
Find the number of triangles made by choosing 3 vertex from the 8 vertex in the attachment.
The shortest distances between an interior diagonal of a rectangular parallelepiped, P, and the edges it does not meet are $2\sqrt{5}$, $\frac{30}{\sqrt{13}}$, and $\frac{15}{\sqrt{10}}$. Determine the volume of $P$.
The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function $ f(x) \equal{} ax^{2} \plus{} bx \plus{} c$ are equal. Their common value must also be which of the following?
$ \textbf{(A)}\ \text{the coefficient of }x^{2}\qquad \textbf{(B)}\ \text{the coefficient of }x$
$ \textbf{(C)}\ \text{the y \minus{} intercept of the graph of }y \equal{} f(x)$
$ \textbf{(D)}\ \text{one of the x \minus{} intercepts of the graph of }y \equal{} f(x)$
$ \textbf{(E)}\ \text{the mean of the x \minus{} intercepts of the graph of }y \equal{} f(x)$
A square board is dissected into $n^2$ rectangular cells by $n-1$ horizontal and $n-1$ vertical lines. The cells are painted alternately black and white in a chessboard pattern. One diagonal consists of $n$ black cells which are squares. Prove that the total area of all black cells is not less than the total area of all white cells.
Let $n\in\mathbb{Z}^+$. Arrange $n$ students $A_1,A_2,...,A_n$ on a circle such that the distances between them are.equal. They each receives a number of candies such that the total amount of candies is $m\geq n$. A configuration is called [i]balance[/i] if for an arbitrary student $A_i$, there will always be a regular polygon taking $A_i$ as one of its vertices, and every student standing at the vertices of this polygon has an equal number of candies.
a) Given $n$, find the least $m$ such that we can create a balance configuration.
b) In a [i]move[/i], a student can give a candy to the student standing next to him (no matter left or right) on one condition that the receiver has less candies than the giver. Prove that if $n$ is the product of at most $2$ prime numbers and $m$ satisfies the condition in a), then no matter how we distribute the candies at the beginning, one can always create a balance configuration after a finite number of moves.
Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that
\[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})\]
Five unit squares are arranged in a plus shape as shown below:
[asy]
size(3cm);
real s=0.1;
draw(s*(0,1)--s*(0,2));
draw(s*(1,0)--s*(1,3));
draw(s*(2,0)--s*(2,3));
draw(s*(3,1)--s*(3,2));
draw(s*(1,0)--s*(2,0));
draw(s*(0,1)--s*(3,1));
draw(s*(0,2)--s*(3,2));
draw(s*(1,3)--s*(2,3));
[/asy]
What is the area of the smallest circle containing the interior and boundary of the plus shape?
[i]2020 CCA Math Bonanza Team Round #3[/i]
The circle $C_1$ and $C_2$ are externally tangent to each other at $T$. A line passing through $T$ meets $C_1$ at $A$ and meets $C_2$ at $B$. The line which is tangent to $C_1$ at $A$ meets $C_2$ at $D$ and $E$. If $D \in [AE]$, $|TA|=a$, $|TB|=b$, what is $|BE|$?
$
\textbf{(A)}\ \sqrt{a(a+b)}
\qquad\textbf{(B)}\ \sqrt{a^2+b^2+ab}
\qquad\textbf{(C)}\ \sqrt{a^2+b^2-ab}
\qquad\textbf{(D)}\ \sqrt{a^2+b^2}
\qquad\textbf{(E)}\ \sqrt{(a+b)b}
$
Let $P(x)=a x^{75}+b$ be a polynomial where \(a\) and \(b\) are coprime integers in the set \(\{1,2,\dots,151\}\), and suppose it satisfies the following condition: there exists at most one prime \(p\) such that for every positive integer \(k\), \(p\mid P(k)\). Prove that for every prime \(q \neq p\) there exists a positive integer \(k\) for which $q^2 \mid P(k).$
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]
In a multiplication table, the entry in the $i$-th row and the $j$-th column is the product $ij$ From an $m\times n$ subtable with both $m$ and $n$ odd, the interior $(m-2) (n-2)$ rectangle is removed, leaving behind a frame of width $1$. The squares of the frame are painted alternately black and white. Prove that the sum of the numbers in the black squares is equal to the sum of the numbers in the white squares.