Found problems: 85335
Let $ABC$ be a triangle with $\angle ABC=90^{\circ}$. Let $D$ and $E$ be the feet from $B$ and $C$ to the median from $A$, respectively. Suppose $DE=4$ and $CD=5$. Find the area of $ABC.$
Let triangle $ABC$ have $\angle A = 70^\circ, \angle B = 60^\circ$, and $\angle C = 50^\circ$. Extend altitude $BH$ past $H$ to point $D$ so that $BD = BC$. Find $\angle BDA$ in degrees.
Let $ABC$ be an acute triangle. Let $D$ be a point inside side $BC$. Let $E$ be the foot from $D$ to $AC$, and let $F$ be a point on $AB$ so that $FE\perp AB$. It is given that the lines $AD, BE, CF$ concur. $M_A, M_B, M_C$ are the midpoints of sides $BC, AC, AB$ respectively, and $O$ is the circumcenter of $ABC$. Moreover, we define $P=EF\cap M_AM_B, S=DE\cap M_AM_C$. Prove that $O, P, S$ are collinear.
A metro network with $n \ge 2$ stations, where each station is connected to each of the others by a one-way line, is said to be [i]dispersed [/i]i f there are two stations $A$ and $B$ such that it is not possible to go from $A$ to $B$ through is from the network. If a network is [i]dispersed[/i], but it is possible to choose a station $A$ and reverse the direction of all lines to and from $A$ so that the new network is no longer dispersed, the network is said to be [i]correctable[/i]. Indicates all integers $n$ for which there is a network with $n$ stations, [i]dispersed [/i]and not [i]correctable[/i].
Let $S$ be a finite set with $n{}$ $(n>1)$ elements, $M{}$ the set of all subsets of $S{}$ and a function $f:M\rightarrow\mathbb{R}$, that verifies the relation $f(A\cap B)=\min\{f(A),f(B)\}, \forall A,B\in M$. Show that $$\sum_{A\in M} (-1)^{n-|A|}\cdot f(A)=f(S)-\max\{f(A)|A\in M, A\neq S\},$$ where$|A|$ is the number of elements of subset $A{}$.
Solve the following equation in positive integers:
$x^{3} = 2y^{2} + 1 $
The $12$ numbers on a clock face are rearranged. Show that we can still find three adjacent numbers whose sum is $21$ or more.
Cells of a $2000\times2000$ board are colored according to the following rules:
1)At any moment a cell can be colored, if none of its neighbors are colored
2)At any moment a $1\times2$ rectangle can be colored, if exactly two of its neighbors are colored.
3)At any moment a $2\times2$ squared can be colored, if 8 of its neighbors are colored
(Two cells are considered to be neighboring, if they share a common side). Can the entire $2000\times2000$ board be colored?
[I]Proposed by K. Kohas[/i]
What is the largest prime factor of 8091?
The reciprocal of $ \left(\frac{1}{2}+\frac{1}{3}\right) $ is
$ \text{(A)}\ \frac{1}{6}\qquad\text{(B)}\ \frac{2}{5}\qquad\text{(C)}\ \frac{6}{5}\qquad\text{(D)}\ \frac{5}{2}\qquad\text{(E)}\ 5 $
Natural number $N$ is given. Let $p_N$ - biggest prime, that $ \leq N$. On every move we replace $N$ by $N-p_N$. We repeat this until we get $0$ or $1$. If we get $1$ then $N$ is called as good, else is bad. For example, $95$ is good because we get $95 \to 6 \to 1$.
Prove that among numbers from $1$ to $1000000$ there are between one quarter and half good numbers
For a positive constant $a$, find the minimum value of $f(x)=\int_0^{\frac{\pi}{2}} |\sin t-ax\cos t|dt.$
Let $ I$ be the incircle centre of triangle $ ABC$ and $ \omega$ be a circle within the same triangle with centre $ I.$ The perpendicular rays from $ I$ on the sides $ \overline{BC}, \overline{CA}$ and $ \overline{AB}$ meets $ \omega$ in $ A', B'$ and $ C'.$ Show that the three lines $ AA', BB'$ and $ CC'$ have a common point.
Consider for a real number $t>1$, $I(t)=\int_{-4}^{4t-4} (x-4)\sqrt{x+4}\ dx.$
Find the minimum value of $I(t)\ (t>1).$
Let $ n$ be a positive integer and $ [ \ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied:
$ (1)$ $ n$ is not a perfect square;
$ (2)$ $ a^{3}$ divides $ n^{2}$.
$16$ different natural numbers are written on the board, none of which exceeds $30$. Prove that there must be two coprime numbers among the written numbers.
For what values of the variable $x$ does the following inequality hold: \[ \dfrac{4x^2}{(1-\sqrt{2x+1})^2}<2x+9 \ ? \]
The sequences $(x_n),(y_n),(z_n),n\in\mathbb N$, are defined by the relations
$$x_{n+1}=\frac{2x_n}{x_n^2-1},\qquad y_{n+1}=\frac{2y_n}{y_n^2-1},\qquad z_{n+1}=\frac{2z_n}{z_n^2-1},$$where $x_1=2$, $y_1=4$, and $x_1y_1z_1=x_1+y_1+z_1$.
(a) Show that $x_n^2\ne1$, $y_n^2\ne1$, $z_n^2\ne1$ for all $n$;
(b) Does there exist a $k\in\mathbb N$ for which $x_k+y_k+z_k=0$?
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?
$ \textbf{(A)}\ \frac{1}{5} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{2}{5} \qquad
\textbf{(D)}\ \frac{2}{3} \qquad
\textbf{(E)}\ \frac{4}{5}$
(a) Find all positive integers $ n$ for which $ 2^n\minus{}1$ is divisible by $ 7$.
(b) Prove that there is no positive integer $ n$ for which $ 2^n\plus{}1$ is divisible by $ 7$.
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
$\textbf{(A) } 7 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11 $
$AK$ and $BL$ are altitudes of an acute triangle $ABC$. Point $P$ is chosen on the segment $AK$ so that $LK=LP$. The parallel to $BC$ through $P$ meets the parallel to $PL$ through $B$ at point $Q$. Prove that $\angle AQB = \angle ACB$.
[i](S. Berlov)[/i]
The first question was asked in Set 4. The second question was asked in Set 5.
Question) Eshaan the Elephant has a long memory. He remembers that out of the integers $0, 1, 2, \dots, 15$, one of them is special. Submit to the grader an ordered 4-tuple of subsets of $0, 1, 2, \dots, 15$ and they will tell you whether the special number is in each. You can then submit your guess for the special number on the next round for points. (You might want to write down a copy of your submission somewhere other than your answer sheet. Note that this question itself is not worth any points, though the corresponding problem in Set 5 is.)
Question) Eshaan the Elephant has a long memory. He remembers that out of the integers $0, 1, 2, \dots, 15$, one of them is special. You have submitted an ordered 4-tuple of subsets of $0, 1, 2, \dots, 15$. Here is your reply from the grader.
\begin{tabular}{|c|c|c|c|}
\hline
1 & 2 & 3 & 4 \\ \hline
Y/N & Y/N & Y/N & Y/N \\ \hline
\end{tabular}
What is the special number?
[i]2016 CCA Math Bonanza Lightning #5.1[/i]
The Hawking Space Agency operates $n-1$ space flights between the $n$ habitable planets of the Local Galaxy Cluster. Each flight has a fixed price which is the same in both directions, and we know that using these flights, we can travel from any habitable planet to any habitable planet.
In the headquarters of the Agency, there is a clearly visible board on a wall, with a portrait, containing all the pairs of different habitable planets with the total price of the cheapest possible sequence of flights connecting them. Suppose that these prices are precisely $1,2, ... , \binom{n}{2}$ monetary units in some order. prove that $n$ or $n-2$ is a square number.
The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of $1$ and $2016$ is closest to which integer?
$\textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 45 \qquad
\textbf{(C)}\ 504 \qquad
\textbf{(D)}\ 1008 \qquad
\textbf{(E)}\ 2015 $