Found problems: 85335
In the figure, $AB \perp BC$, $BC \perp CD$, and $BC$ is tangent to the circle with center $O$ and diameter $AD$. In which one of the following cases is the area of $ABCD$ an integer?
[asy]
size(170);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair O=origin, A=(-1/sqrt(2),1/sqrt(2)), B=(-1/sqrt(2),-1), C=(1/sqrt(2),-1), D=(1/sqrt(2),-1/sqrt(2));
draw(unitcircle);
dot(O);
draw(A--B--C--D--A);
label("$A$",A,dir(A));
label("$B$",B,dir(B));
label("$C$",C,dir(C));
label("$D$",D,dir(D));
label("$O$",O,N);
[/asy]
$ \textbf{(A)}\ AB=3, CD=1\qquad\textbf{(B)}\ AB=5, CD=2\qquad\textbf{(C)}\ AB=7, CD=3\qquad\textbf{(D)}\ AB=9, CD=4\qquad\textbf{(E)}\ AB=11, CD=5 $
Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a, b, c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$.
Let $a,b,c$ be distinct nonzero real numbers. If the equations $ax^3+bx+c=0$, $bx^3+cx+a=0,$ and $cx^3+ax+b=0$ have a common root, prove that at least one of these equations has three real roots(not necessarily distinct).
Two circles, $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ meet $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1,O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.
Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy equation:
$$ f(x^3+y^3) =f(x^3) + 3x^2f(x)f(y) + 3f(x)f(y)^2 + y^6f(y) $$
for all reals $x,y$
Let $\alpha,\beta,\gamma$ be angles of a triangle. Determine all real triplets $x,y,z$ satisfying the system
\begin{align*}
x\cos\beta+\frac1z\cos\alpha &=1, \\
y\cos\gamma+\frac1x\cos\beta &=1, \\
z\cos\alpha+\frac1y\cos\gamma &=1.
\end{align*}
Let $ a, b, c $ be positive real numbers such that $ ab+bc+ca=1 $. Prove that
\[ \sqrt{ a^2 + b^2 + \frac{1}{c^2}} + \sqrt{ b^2 + c^2 + \frac{1}{a^2}} + \sqrt{ c^2 + a^2 + \frac{1}{b^2}} \ge \sqrt{33} \]
Determine all function $f:\mathbb{R}\to\mathbb{R}$ such that $xf(y)+yf(x)\leqslant xy$ for all $x,y\in\mathbb{R}$.
Consider an equilateral triangular grid $G$ with $20$ points on a side, where each row consists of points spaced $1$ unit apart. More specifically, there is a single point in the first row, two points in the second row, ..., and $20$ points in the last row, for a total of $210$ points. Let $S$ be a closed non-self-intersecting polygon which has $210$ vertices, using each point in $G$ exactly once. Find the sum of all possible values of the area of $S$.
Find the sum of all possible values of $f(2)$ such that
$f(x)f(y)-f(xy) = \frac{y}{x}+\frac{x}{y}$, for every positive real numbers $x,y$
$ \textbf{(A)}\ \frac{5}{2}
\qquad\textbf{(B)}\ -\frac{5}{4}
\qquad\textbf{(C)}\ \frac{5}{4}
\qquad\textbf{(D)}\ \frac{3}{2}
\qquad\textbf{(E)}\ \text{None}
$
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \]
cannot be divided by $5$.
Suppose that $ 4^{x_1} \equal{} 5, 5^{x_2} \equal{} 6, 6^{x_3} \equal{} 7,...,127^{x_{124}} \equal{} 128$. What is $ x_1x_2 \cdots x_{124}$?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ \frac {5}{2}\qquad
\textbf{(C)}\ 3\qquad
\textbf{(D)}\ \frac {7}{2}\qquad
\textbf{(E)}\ 4$
Let $M$ be a set of $n \ge 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $AB$ and $CD$ sharing a common interior point and replace them by the segments $AC$ and $BD$ if none of them is present at this moment. Prove that it is impossible to perform $n^3 /4$ or more such moves.
[i]Proposed by Vladislav Volkov, Russia[/i]
How many ordered triples of integers $(a, b, c)$ satisfy the following system?
$$
\begin{cases} ab + c &= 17 \\ a + bc &= 19 \end{cases}
$$
$$
\mathrm a. ~ 2\qquad \mathrm b.~3\qquad \mathrm c. ~4 \qquad \mathrm d. ~5 \qquad \mathrm e. ~6
$$
Let $ABC$ a scalene triangle and $AD, BE, CF$ your angle bisectors, with $D$ in the segment $BC, E$ in the segment $AC$ and $F$ in the segment $AB$. If $\angle AFE = \angle ADC$.
Determine $\angle BCA$.
The sum of all the roots of $ 4x^3\minus{}8x^2\minus{}63x\minus{}9\equal{}0$ is:
$ \textbf{(A)}\ 8 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \minus{}8 \qquad
\textbf{(D)}\ \minus{}2 \qquad
\textbf{(E)}\ 0$
The first term of an arithmetic series of consecutive integers is $ k^2 \plus{} 1$. The sum of $ 2k \plus{} 1$ terms of this series may be expressed as:
$ \textbf{(A)}\ k^3 \plus{} (k \plus{} 1)^3\qquad
\textbf{(B)}\ (k \minus{} 1)^3 \plus{} k^3\qquad
\textbf{(C)}\ (k \plus{} 1)^3\qquad \\
\textbf{(D)}\ (k \plus{} 1)^2\qquad
\textbf{(E)}\ (2k \plus{} 1)(k \plus{} 1)^2$
If $\sqrt{x+2}=2$, then $(x+2)^2$ equals
$\text{(A)}\ \sqrt{2} \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 16$
A student divides all $30$ marbles into $5$ boxes numbered $1, 2, 3, 4, 5$ (after being divided, there may be a box with no marbles).
a) How many ways are there to divide marbles into boxes (are two different ways if there is a box with a different number of marbles)?
b) After dividing, the student paints those $30$ marbles by a number of colors (each with the same color, one color can be painted for many marbles), so that there are no $2$ marbles in the same box. have the same color and from any $2$ boxes it is impossible to choose $8$ marbles painted in $4$ colors. Prove that for every division, the student must use no less than $10$ colors to paint the marbles.
c) Show a division so that with exactly $10$ colors the student can paint the marbles that satisfy the conditions in question b).
The Bell Zoo has the same number of rhinoceroses as the Carlton Zoo has lions. The Bell Zoo has three more elephants than the Carlton Zoo has lions. The Bell Zoo has the same number of elephants as the Carlton Zoo has rhinoceroses. The Carlton Zoo has two more elephants than rhinoceroses. The Carlton Zoo has twice as many monkeys as it has rhinoceroses, elephants, and lions combined, and it has two more penguins than monkeys. The Bell Zoo has two-thirds as many monkeys as the Carlton Zoo has penguins. The Bell Zoo has two more penguins than monkeys but only half as many lions as penguins. The total of the numbers of rhinoceroses, elephants, lions, monkeys, and penguins in the Bell Zoo is $48$. Find the total of the numbers of rhinoceroses, elephants, lions, monkeys, and penguins in the Carlton Zoo.
Solve the inequality
$$(x-1)(x^2-1)(x^3-1)\cdot ...\cdot (x^{100}-1)(x^{101}-1)\ge 0$$
A unit square is cut by $n$ straight lines . Prove that in at least one of these parts one can completely fit a square with side $\frac{1}{n+1}$
[hide=original wording]Одиничний квадрат розрізано $n$ прямими на частини. Доведіть, що хоча б в одній з цих частин можна повністю розмістити квадрат зі стороною $\frac{1}{n+1}$[/hide]
[hide=notes]
The selection panel jury made a mistake because the solution known to it turned out to be incorrect. As it turned out, the assertion of the problem is still correct, although it cannot be proved by simple methods, see. article:
Keith Ball. Тhe plank problem for symmetric bodies // Іпѵепііопез МаіЬешаІіеае. — 1991. — Ѵоі. 104, по. 1. — Р. 535-543. [url]https://arxiv.org/abs/math/9201218[/url][/hide]
Let $\triangle ABC$ be a triangle with circumcircle $\Gamma$, and let $I$ be the center of the incircle of $\triangle ABC$. The lines $AI$, $BI$ and $CI$ intersect $\Gamma$ in $D \ne A$, $E \ne B$ and $F \ne C$. The tangent lines to $\Gamma$ in $F$, $D$ and $E$ intersect the lines $AI$, $BI$ and $CI$ in $R$, $S$ and $T$, respectively. Prove that
\[\vert AR\vert \cdot \vert BS\vert \cdot \vert CT\vert = \vert ID\vert \cdot \vert IE\vert \cdot \vert IF\vert.\]
Marie is painting a $4 \times 4$ grid of identical square windows. Initially, they are all orange but she wants to paint $4$ of them black. How many ways can she do this up to rotation and reflection?
Calculate all real numbers $r $ with the following properties:
If real numbers $a, b, c$ satisfy the inequality$ | ax^2 + bx + c | \le 1$ for each $x \in [ - 1, 1]$, then they also satisfy the inequality $| cx^2 + bx + a | \le r$ for each $ x \in [- 1, 1]$.