Found problems: 85335
All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$
Let $k$ be a nonzero natural number and $m$ an odd natural number . Prove that there exist a natural number $n$ such that the number $m^n+n^m$ has at least $k$ distinct prime factors.
Prove that there exists infinitely many positive integers $n$ such that $n, n+1$, and $n+2$ can be written as the sum of two perfect squares.
Let $\triangle ABC$ have $\angle ABC=67^{\circ}$. Point $X$ is chosen such that $AB = XC$, $\angle{XAC}=32^\circ$, and $\angle{XCA}=35^\circ$. Compute $\angle{BAC}$ in degrees.
[i]Proposed by Raina Yang[/i]
Circle $C$ with radius $2$ has diameter $\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C,$ externally tangent to circle $D,$ and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$ and can be written in the form $\sqrt{m} - n,$ where $m$ and $n$ are positive integers. Find $m+n$.
Solve the equation $$2(x-6)=\dfrac{x^2}{(1+\sqrt{x+1})^2}$$
There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer $ n$ in between 1 and 25. Betül draws $ n$ balls from the bag one by one such that no ball is put back to the bag after it is drawn. If exactly 2 of the $ n$ balls are blue and the second blue ball is drawn at $ n^{th}$ order, Ahmet wins, otherwise Betül wins. To increase the possibility to win, Ahmet must choose
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 23$
The pressure $ (P)$ of wind on a sail varies jointly as the area $ (A)$ of the sail and the square of the velocity $ (V)$ of the wind. The pressure on a square foot is $ 1$ pound when the velocity is $ 16$ miles per hour. The velocity of the wind when the pressure on a square yard is $ 36$ pounds is:
$ \textbf{(A)}\ 10\frac {2}{3} \text{ mph} \qquad\textbf{(B)}\ 96 \text{ mph} \qquad\textbf{(C)}\ 32\text{ mph} \qquad\textbf{(D)}\ 1\frac {2}{3} \text{ mph} \qquad\textbf{(E)}\ 16 \text{ mph}$
A set $X$ consisting of $n$ positive integers is called $\textit{good}$ if the following condition holds:
For any two different subsets of $X$, say $A$ and $B$, the number $s(A) - s(B)$ is not divisible by $2^n$.
(Here, for a set $A$, $s(A)$ denotes the sum of the elements of $A$)
Given $n$, find the number of good sets of size $n$, all of whose elements is strictly less than $2^n$.
Let $\phi =\frac{1}{2019}$. Define
$$g_n =\begin{cases} 0 & \text{ if} \,\,\,\, round (n\phi) = round \,\,\,\, ((n - 1)\phi) \\
1 & \text{ otherwise} .\end{cases}.$$
where round $(x)$ denotes the round function.
Compute the expected value of $g_n$ if $n$ is an integer chosen from interval $[1, 2019^2]$.
Suppose that $ a,b,$ and $ \sqrt[3]{a}\plus{}\sqrt[3]{b}$ are rational numbers. Prove that $ \sqrt[3]{a}$ and $ \sqrt[3]{b}$ are also rational.
Ryan got $80\%$ of the problems on a $25$-problem test, $90\%$ on a $40$-problem test, and $70\%$ on a $10$-problem test. What percent of all problems did Ryan answer correctly?
$ \textbf{(A)}\ 64 \qquad\textbf{(B)}\ 75\qquad\textbf{(C)}\ 80\qquad\textbf{(D)}\ 84\qquad\textbf{(E)}\ 86 $
In acute triangle $ABC$, $AB > AC$. Let $M$ be the midpoint of side $BC$. The exterior angle bisector of $\widehat{BAC}$ meet ray $BC$ at $P$. Point $K$ and $F$ lie on line $PA$ such that $MF \perp BC$ and $MK \perp PA$. Prove that $BC^2 = 4 PF \cdot AK$.
[asy]
defaultpen(fontsize(10)); size(7cm);
pair A = (4.6,4), B = (0,0), C = (5,0), M = midpoint(B--C), I = incenter(A,B,C), P = extension(A, A+dir(I--A)*dir(-90), B,C), K = foot(M,A,P), F = extension(M, (M.x, M.x+1), A,P);
draw(K--M--F--P--B--A--C);
pair point = I;
pair[] p={A,B,C,M,P,F,K};
string s = "A,B,C,M,P,F,K";
int size = p.length;
real[] d; real[] mult; for(int i = 0; i<size; ++i) { d[i] = 0; mult[i] = 1;}
string[] k= split(s,",");
for(int i = 0;i<p.length;++i) {
label("$"+k[i]+"$",p[i],mult[i]*dir(point--p[i])*dir(d[i]));
}[/asy]
Let $O, H$ be the orthocenter and circumcenter of of an acute-angled triangke $ABC$ with $AB<AC$.Let $K$ be the midpoint of $AH$.The line through $K$ perpendicular to $OK$ meet $AB$ and the tangent to the circumcircle at $A$ at $X$ and $Y$ respectively. Prove that $\angle XOY=\angle AOB$
Ana has $22$ coins. She can take from her friends either $6$ coins or $18$ coins, or she can give $12$ coins to her friends. She can do these operations many times she wants. Find the least number of coins Ana can have.
How many integers between $2$ and $100$ inclusive [i]cannot[/i] be written as $m \cdot n,$ where $m$ and $n$ have no common factors and neither $m$ nor $n$ is equal to $1$? Note that there are $25$ primes less than $100.$
There are 2021 light bulbs in a row, labeled 1 through 2021, each with an on/off switch. They all start in the off position when 1011 people walk by. The first person flips the switch on every bulb; the second person flips the switch on every 3rd bulb (bulbs 3, 6, etc.); the third person flips the switch on every 5th bulb; and so on. In general, the $k$th person flips the switch on every $(2k - 1)$th light bulb, starting with bulb $2k - 1$. After all 1011 people have gone by, how many light bulbs are on?
Let $S$ be a set of $49$-digit numbers $n$, with the property that each of the digits $1, 2, 3, \dots, 7$ appears in the decimal expansion of $n$ seven times (and $8, 9$ and $0$ do not appear). Show that no two distinct elements of $S$ divide each other.
A checkerboard is $91$ squares long and $28$ squares wide. A line connecting two opposite vertices of the checkerboard is drawn. How many squares does the line pass through?
Let $ABCD$ be a rectangle with $AB=10$ and $AD=5.$ Suppose points $P$ and $Q$ are on segments $CD$ and $BC,$ respectively, such that the following conditions hold:
[list]
[*] $BD \parallel PQ$
[*] $\angle APQ=90^{\circ}.$
[/list]
What is the area of $\triangle CPQ?$
[i]Proposed by Kyle Lee[/i]
Let $n > 1$ be an odd integer. On an $n \times n$ chessboard the center square and four corners are deleted.
We wish to group the remaining $n^2-5$ squares into $\frac12(n^2-5)$ pairs, such that the two squares in each pair intersect at exactly one point
(i.e.\ they are diagonally adjacent, sharing a single corner).
For which odd integers $n > 1$ is this possible?
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
Maria Ivanovna drew on the blackboard a right triangle $ABC$ with a right angle $B$. Three students looked at her and said:
$\bullet$ Yura said: "The hypotenuse of this triangle is $10$ cm."
$\bullet$ Roma said: "The altitude drawn from the vertex $B$ on the side $AC$ is $6$ cm."
$\bullet$ Seva said: "The area of the triangle $ABC$ is $25$ cm$^2$."
Determine which of the students was mistaken if it is known that there is exactly one such person.
Let $D$ be an arbitrary point on side $BC$ of triangle $ABC$. Circles $\omega_1$ and $\omega_2$ pass through $A$ and $D$ in such a way that $BA$ touches $\omega_1$ and $CA$ touches $\omega_2$. Let $BX$ be the second tangent from $B$ to $\omega_1$, and $CY$ be the second tangent from $C$ to $\omega_2$. Prove that the circumcircle of triangle $XDY$ touches $BC$.
A convex polygon $G$ is placed inside a convex polygon $ F$ so that their boundaries have no common points. A segment $s$ joining two points on the boundary of $F$ is called a support chord for $G$ if s contains a side or only a vertex of $G$. Prove that
(a) there exists a support chord for $G$ such that its midpoint lies on the boundary of $G$,
(b) there exist at least two such chords.
(P Pushkar)