Found problems: 85335
While out for a stroll, you encounter a vicious velociraptor. You start running away to the northeast at $ 10 \text{m/s}$, and you manage a three-second head start over the raptor. If the raptor runs at $ 15\sqrt{2} \text{m/s}$, but only runs either north or east at any given time, how many seconds do you have until it devours you?
Harry Potter is sitting $2.0$ meters from the center of a merry-go-round when Draco Malfoy casts a spell that glues Harry in place and then makes the merry-go-round start spinning on its axis. Harry has a mass of $\text{50.0 kg}$ and can withstand $\text{5.0} \ g\text{'s}$ of acceleration before passing out. What is the magnitude of Harry's angular momentum when he passes out?
(A) $\text{200 kg} \cdot \text{m}^2\text{/s}$
(A) $\text{330 kg} \cdot \text{m}^2\text{/s}$
(A) $\text{660 kg} \cdot \text{m}^2\text{/s}$
(A) $\text{1000 kg} \cdot \text{m}^2\text{/s}$
(A) $\text{2200 kg} \cdot \text{m}^2\text{/s}$
Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$. The midpoint of $BC$ is denoted by $M$. $AH$ intersects the circumcircle at $D \ne A$ and $DM$ intersects circumcircle of $\vartriangle ABC$ at $T\ne D$. Now, assume the reflection points of $M$ with respect to $AB,AC,AH$ are $F,E,S$. Show that the midpoints of $BE,CF,AM,TS$ are concyclic.
[img]https://3.bp.blogspot.com/-v7D_A66nlD0/XnYNJussW9I/AAAAAAAALeQ/q6DMQ7w6QtI5vLwBcKqp4010c3XTCj3BgCK4BGAYYCw/s1600/imoc2019g2.png[/img]
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?
Three unit circles have a common point $O.$ The other points of (pairwise) intersection are $A, B$ and $C$. Show that the points $A, B$ and $C$ are located on some unit circle.
The line passing through vertex $A$ of triangle $ABC$ and parallel to $BC$ meets the circumcircle of $ABC$ for the second time at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the perpendiculars from $A_1, B_1, C_1$ to $BC, CA, AB$ respectively concur.
Given two circles $\omega_1$ and $\omega_2$ where $\omega_2$ is inside $\omega_1$. Show that there exists a point $P$ such that for any line $\ell$ not passing through $P$, if $\ell$ intersects circle $\omega_1$ at $A,B$ and $\ell$ intersects circle $\omega_2$ at $C,D$, where $A,C,D,B$ lie on $\ell$ in this order, then $\angle APC=\angle BPD$.
What is the sum of the mean, median, and mode of the numbers, $2,3,0,3,1,4,0,3$?
$ \textbf{(A)}\ 6.5 \qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 7.5\qquad\textbf{(D)}\ 8.5\qquad\textbf{(E)}\ 9 $
Given a finite sequence of real numbers $a_1,a_2,\dots ,a_n$ ($\ast$), we call a segment $a_k,\dots ,a_{k+l-1}$ of the sequence ($\ast$) a “[i]long[/i]”(Chinese dragon) and $a_k$ “[i]head[/i]” of the “[i]long[/i]” if the arithmetic mean of $a_k,\dots ,a_{k+l-1}$ is greater than $1988$. (especially if a single item $a_m>1988$, we still regard $a_m$ as a “[i]long[/i]”). Suppose that there is at least one “[i]long[/i]” among the sequence ($\ast$), show that the arithmetic mean of all those items of sequence ($\ast$) that could be “[i]head[/i]” of a certain “[i]long[/i]” individually is greater than $1988$.
Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$.
Prove that for any positive integer one can place all it's divisor on a circle such that among any two neighbours one is a multiple of the other
The number $123454321$ is written on a blackboard. Evan walks by and erases some (but not all) of the digits, and notices that the resulting number (when spaces are removed) is divisible by $9$. What is the fewest number of digits he could have erased?
[i]Ray Li[/i]
Assume that $a, b,$ and $c$ are positive real numbers for which $(a+b)(a+c)(b+c) = 1$. Prove that $ab+bc+ca \leq\frac{3 }{4}.$
Let $f : \mathbb{R}^+\to\mathbb{R}$ satisfying $f(x)=f(x+2)+2f(x^2+2x)$. Prove that if for all $x>1400^{2021}$, $xf(x) \le 2021$, then $xf(x) \le 2021$ for all $x \in \mathbb {R}^+$
Proposed by [i]Navid Safaei[/i]
Let $a,b,c$ be a real numbers such that this equations:
$a^2x + b^2y + c^2z = 1$
$xy + yz + xz = 1$
have only one solution $(x, y, z)$ in real numbers. Prove that $a, b, c$ are sides of the triangle
If $ab+\sqrt{ab+1}+\sqrt{a^2+b}\sqrt{a+b^2}=0$, find the value of $b\sqrt{a^2+b}+a\sqrt{b^2+a}$
Let $ x,y,z $ be numbers distinct from $ -1 $ that verify the equation
$$ \frac{1}{1+a} +\frac{1}{1+b} +\frac{1}{1+c} =\frac{3}{2} . $$
Prove that if $ abc=1, $ then $ a $ or $ b $ or $ c $ is equal to $ 1. $
In a triangle $ABC$, let $a = BC, b = AC$ and let $m_a,m_b$ be the corresponding medians. Find all real numbers $k$ for which the equality $m_a+ka = m_b +kb$ implies that $a = b$.
Solve in the real numbers the equation $ \lfloor ax \rfloor -\lfloor (1+a)x \rfloor = (1+a)(1-x) . $
[i]Dumitru Acu[/i]
Find the $n$-th derivative with respect to $x$ of
$$\int_{0}^{x} \left(1+\frac{x-t}{1!}+\frac{(x-t)^{2}}{2!}+\ldots+\frac{(x-t)^{n-1}}{(n-1)!}\right)e^{nt} dt.$$
$ABCD$ is a convex quadrilateral with $AB=36$, $CD=9$, $DA=39$, and $BD=15$. Given that $\angle{C}$ is right, compute the area of $ABCD$.
[i]2018 CCA Math Bonanza Team Round #4[/i]
Adamand Topher are playing a game in which each of them starts with $2$ pickles. Each turn, they flip a fair coin: if it lands heads, Topher takes $1$ pickle from Adam; if it lands tails, Adam takes $2$ pickles from Topher. (If Topher has only $1$ pickle left, Adam will just take it.) What’s the probability that Topher will have all $4$ pickles before Adam does?
Three circles are constructed on the medians of a triangle as on diameters. It is known that they intersect in pairs. Let $C_1$ be the intersection point of the circles built on the medians $AM_1$ and $BM_2$, which is more distant from the vertex $C$. Points $A_1$ and $B_1$ are defined similarly. Prove that the lines $AA_1, BB_1$ and $CC_1$ intersect at one point.
(D. Tereshin)
A star-polygon is drawn on a clock face by drawing a chord from each number to the firth number counted clockwise from that number. That is, chords are drawn from $ 12$ to $ 5,$ from $ 5$ to $ 10,$ from $ 10$ to $ 3,$ and so on, ending back at $ 12.$ What is the degree measure of the angle at each vertex in the star-polygon?
$ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 30 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 60$
(A.Zaslavsky) Given two triangles $ ABC$, $ A'B'C'$. Denote by $ \alpha$ the angle between the altitude and the median from vertex $ A$ of triangle $ ABC$. Angles $ \beta$, $ \gamma$, $ \alpha'$, $ \beta'$, $ \gamma'$ are defined similarly. It is known that $ \alpha \equal{} \alpha'$, $ \beta \equal{} \beta'$, $ \gamma \equal{} \gamma'$. Can we conclude that the triangles are similar?