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231,038 | Let \(a_1\) be a natural number not divisible by \(5\). The sequence \(a_1,a_2,a_3,\dots\) is defined by
\[
a_{n+1}=a_n+b_n,
\]
where \(b_n\) is the last digit of \(a_n\). Prove that the sequence contains infinitely many powers of two. [quote="outback"]Let $ a_1$ be a natural number not divisible by $ 5$. The sequence ... |
2,310,426 | Find the number of complex solutions to
\[
\frac{z^3 - 1}{z^2 + z - 2} = 0.
\] [quote=GameBot]Find the number of complex solutions to
\[\frac{z^3 - 1}{z^2 + z - 2} = 0.\][/quote]
Firstly, let us factor the top with difference of cubes. We rewrite as $(z-1)(z^2+z+1)$. The bottom is just $(z+2)(z-1)$, so the fraction re... |
2,310,435 | Let \(ABC\) be a triangle with circumcircle \((O)\). Let \(P\) be an arbitrary interior point of \((O)\) with \(P\neq O\). Let \(X,Y,Z\) be the symmedian points of triangles \(BPC,\; CPA,\; APB\), respectively. Let \(D,E,F\) be the intersections of the tangents to \((O)\) at \(B\) and \(C\), at \(C\) and \(A\), and at ... |
2,310,441 | Prove that
\[
\operatorname{Area}(\triangle ABC)=(s-a)r_a,
\]
where \(s=\dfrac{a+b+c}{2}\), \(a=BC\), \(b=CA\), \(c=AB\), and \(r_a\) is the exradius of the A-excircle. [quote=franzliszt]You have $[ABC]=rs$ where $r$ is the inradius. Consider the homothety sending the incircle to the excircle with scale factor $\frac{s... |
231,045 | A rectangular cow pasture is enclosed on three sides by a fence and the fourth side is part of the side of a barn that is \(400\) feet long. The fence costs \(\$5\) per foot, and \(\$1{,}200\) altogether. To the nearest foot, find the length of the side parallel to the barn that will maximize the area of the pasture. T... |
23,105 | Problem: Find all positive integer $n$ and prime number $p$ such that :
Any $a_1,a_2,...,a_n\in\{1,2,...,p-1\}$ we have $\sum_{k=1}^na_k^2\not\equiv0\pmod{p}$
Repaired ....! I think that $n<3$ and $p=4k+3$.
Let's first show that for $n\ge 3$, all residues $\pmod p$ can be written as $\sum_{i=1}^k a_i^2$. Let $A... |
231,050 | A rectangular piece of paper \(ADEF\) is folded so that corner \(D\) meets the opposite edge \(EF\) at \(D'\), forming a crease \(BC\) where \(B\) lies on edge \(AD\) and \(C\) lies on edge \(DE\). If \(AD=25\) cm, \(DE=16\) cm, and \(AB=5\) cm, find \(BC^2\). mathwizarddude, i believe your wrong.
[u]Here is my solu... |
2,310,536 | For each positive integer \(n\), the mean of the first \(n\) terms of a sequence is \(n\). What is the 2008th term of the sequence? [hide=Better Solution]We notice that: $$a_1+a_2+\cdots+a_n=n^2.$$ Therefore: $$a_1+a_2+\cdots+a_{2008}=2008^2.$$ Similarly, $$a_1+a_2+\cdots+a_{2007}=2007^2.$$ Hence, $$a_{2008}=2008^2-200... |
231,054 | Two regular polygons with the same number of sides have side lengths 48 m and 55 m, respectively. A third regular polygon with the same number of sides has area equal to the sum of the areas of the first two. What is the side length of the third polygon? [hide="most likely incorrect"]The area of the regular $ n$-gon is... |
2,310,555 | Let \(f:\mathbb{R}\to\mathbb{R}\) be a differentiable function such that
\[
\lim_{x\to 2}\frac{x^2+x-6}{\sqrt{1+f(x)}-3}\ge 0
\]
and \(f(x)\ge -1\) for all \(x\in\mathbb{R}\). If the line \(6x-y=4\) intersects \(y=f(x)\) at \(x=2\), find \(f'(2)\). If $y=f(x)$ intersects the line $6x-y=4$, then $f(2)=8$ (i). Upon insp... |
2,310,567 | Solve the equation
\[
- x^2 = \frac{3x+1}{x+3}.
\]
Enter all solutions, separated by commas. Cross multiplying gives $-x^2(x+3)=3x+1$. Expanding the LHS results in $-x^3-3x^2=3x+1$. Now, we add $x^3+3x^2$ to both sides. This gives $x^3+3x^2+3x+1=0$. We can factor $x^3+3x^2+3x+1$ as $(x+1)^3$, so the answer is just $\bo... |
2,310,595 | If \(a\) and \(b\) are complex numbers such that \(|a| = 6\) and \(|b| = 4\), find \(\left| \frac{a}{b} \right|\). [quote=OlympusHero][quote=GameBot]If $a$ and $b$ are complex numbers such that $|a| = 6$ and $|b| = 4,$ then find $\left| \frac{a}{b} \right|.$[/quote]
@above, nice cheese.
Notice that $\frac{|a|}{|b|}=\... |
231,062 | Determine whether \(100,\!895,\!598,\!169\) is prime. Provide a solution. [quote="stevenmeow"]uhh testing either 1) all odd #s
or 2) all primes in a database
yeah thats probably how
[/quote]
I would bet that's not how it was done. There are several (advanced) techniques for determining [url=http://en.wikipedia.org... |
2,310,658 | Find all real values of \(r\) that satisfy
\[
\frac{1}{r}>\frac{1}{r-1}+\frac{1}{r-4}.
\]
Give your answer in interval notation. ans is [hide]$(-\infty,-2)\cup(0,1)\cup(2,4)$[/hide], you multiply both sides of the inequality by [hide]$r(r-1)(r-4)$[/hide], being careful of the sign and doing casework on whether you flip... |
2,310,660 | Let \(a,b,c\) be nonnegative real numbers such that \(a+b+c=1\). Find the maximum value of
\[
\frac{ab}{a+b}+\frac{ac}{a+c}+\frac{bc}{b+c}.
\] [quote=GameBot]Let $a,$ $b,$ $c$ be nonnegative real numbers such that $a + b + c = 1.$ Find the maximum value of
\[\frac{ab}{a + b} + \frac{ac}{a + c} + \frac{bc}{b + c}.\][/q... |
2,310,680 | For constants \(x\) and \(a\), the third, fourth, and fifth terms in the expansion of \((x+a)^n\) are \(84\), \(280\), and \(560\), respectively. Find \(n\). We can see that the third, fourth, and fifth terms are $\binom n2x^{n-2}a^2=84,\binom n3x^{n-3}a^3=280,$ and $\binom n4x^{n-4}a^4=560,$ which is a system of equat... |
2,310,688 | Solve
\[
\frac{1}{x-5}>0.
\]
Enter your answer using interval notation. [quote=bestzack66][quote name="HIA2020" url="/community/p18362019"]
Not too good at this, but $(5,infinity]$?
Am I right?
[/quote]
when we write interval notation, we write ) or ( before/after infinity.[/quote]
thanks :) |
2,310,694 | Given triangle \(ABC\) inscribed in \((O)\). Let \(M\) be the midpoint of \(BC\), and let \(H\) be the foot of the perpendicular from \(A\) to \(BC\). Let \(OH\) meet \(AM\) at \(P\). Prove that \(P\) lies on the radical axis of \((BOC)\) and the nine-point circle of triangle \(ABC\). Let $\Omega$ be the nine-point cir... |
2,310,715 | If \(\log_2 x + \log_2 x^2 = 6\), find the value of \(x\). [quote=Dragonslayer118]The answer is 4[/quote]
Dragonslayer118, the community appreciates that you want to participate, but remember, in the FAQ, it says that if you don't have anything to add to the discussion, you don't need to post anything. |
2,310,753 | Find constants \(A,B,C,\) and \(D\) so that
\[
\frac{x^3 + 3x^2 - 12x + 36}{x^4 - 16} = \frac{A}{x - 2} + \frac{B}{x + 2} + \frac{Cx + D}{x^2 + 4}.
\]
Enter the ordered quadruple \((A,B,C,D)\). [quote=dajeff][hide=solution]
As in the AoPS solution, we multiply both sides by $(x-2)(x+2)(x^2+4)$ and substitute $x=2,-2$ t... |
2,310,754 | Two real numbers \(x\) and \(y\) satisfy \(x-y=4\) and \(x^3-y^3=28\). Compute \(xy\). Cubing gives $x^3-3x^2y+3xy^2-y^3=64$. So $-3x^2y+3xy^2=36$. Dividing each side by $-3$ gives $x^2y-xy^2=-12$, so $xy(x-y)=-12$. Therefore, $xy=-3$.
lol i forgot a negative sign :P |
2,310,758 | The fourth-degree polynomial equation
\[
x^4 - 7x^3 + 4x^2 + 7x - 4 = 0
\]
has four real roots \(a,b,c,d\). What is the value of
\[
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}?
\]
Express your answer as a common fraction. [quote=eduD_looC]We have $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{abc+acd+abd+bc... |
2,310,764 | Let \(E\) and \(F\) be points on side \(BC\) of triangle \(\triangle ABC\). Points \(K\) and \(L\) are chosen on segments \(AB\) and \(AC\), respectively, so that \(EK\parallel AC\) and \(FL\parallel AB\). The incircles of \(\triangle BEK\) and \(\triangle CFL\) touch segments \(AB\) and \(AC\) at \(X\) and \(Y\), resp... |
2,310,792 | Given vectors \(\mathbf{a}\) and \(\mathbf{b}\) such that \(\|\mathbf{a}\| = 6\), \(\|\mathbf{b}\| = 8\), and \(\|\mathbf{a} + \mathbf{b}\| = 11\). Find \(\cos\theta\), where \(\theta\) is the angle between \(\mathbf{a}\) and \(\mathbf{b}\). [quote=OlympusHero]Given vectors $\mathbf{a}$ and $\mathbf{b}$ such that $\|\m... |
2,310,805 | The roots of the polynomial
\[
x^3 - 52x^2 + 581x - k
\]
are distinct prime numbers. Find \(k\). [quote=jasperE3]How did you find what the roots were?[/quote]
[hide=spoiler]
one prime must be 2 by common sense
then the other two primes add to 50
lets call thm $p$ and $q$
we have that $p + q = 50$
and that $pq + 2p + 2... |
231,084 | Show that if \(f:\mathbb{R}\to\mathbb{R}\) is a monotone increasing function, then the set of points where \(f\) is discontinuous has Lebesgue measure zero. [quote="Kalle"]I was just loosely saying that an "uncountable sum" of positive numbers is infinite. Index all the discontinuities in $ (a,b)$ as $ x_j$ where $ j$ ... |
231,086 | How many combinations of pennies, nickels, and dimes are there with a total value of 25 cents? [quote="ernie"]The formula which mewto talks about is this:
$ 3 \plus{} \frac {(\lfloor\frac {n}{5}\rfloor)(\lfloor\frac {n}{5}\rfloor \minus{} 1)}{2}$
This works for $ 10$ through $ 50$ cents, I believe.
$ n \equal{} \tex... |
2,310,868 | Find the sum of the real solutions for \(x\) to the equation
\[
\frac{1}{x-1} + \frac{1}{x+1} = 17.
\] [hide=Sol]$\frac{1}{x-1} + \frac{1}{x+1} = 17$
$(x+1)+(x-1)=17(x-1)(x+1)$
$2x=17(x^2-1)$
$2x=17x^2-17$
$17x^2-2x-17=0$
$x=\frac{2\pm\sqrt{2^2+4\cdot17\cdot17}}{34}=\frac{2\pm\sqrt{1160}}{34}=\frac{1\pm\sqrt{290}}{17}$... |
23,109 | Find the minimal constant \(k\) such that for all positive real numbers \(a,b,c\) the following inequality holds:
\[
k\frac{a^2+b^2+c^2}{ab+bc+ca}+\frac{8abc}{(a+b)(b+c)(c+a)} \ge 1+k.
\] The minimal k such that the inequality
$k\frac{a^2+b^2+c^2}{bc+ca+ab}+\frac{8abc}{\left(b+c\right)\left(c+a\right)\left(a+b\right... |
2,310,918 | The four positive integers \(a,b,c,d\) satisfy
\[
a\times b\times c\times d = 10!.
\]
Find the smallest possible value of \(a+b+c+d\). Wrong.
[hide=Solution.]Note that $\sqrt[4]{10!}\approx43.6,$ so we try to find factors of $10!$ around that number, and $a=40,b=42,c=45,d=48$ works, giving $a+b+c+d=\boxed{175}.$
Then... |
2,310,935 | Solve the nonlinear system
\[
\begin{cases}
\dfrac{x^2}{16}+\dfrac{y^2}{9}=1,\\[6pt]
3x+4y=12.
\end{cases}
\]
Solution:
From \(3x+4y=12\) we have \(y=\dfrac{12-3x}{4}=3-\dfrac{3x}{4}\). Substitute into the ellipse equation:
\[
\frac{x^2}{16}+\frac{\bigl(3-\tfrac{3x}{4}\bigr)^2}{9}=1.
\]
Compute the second term:
\[
\b... |
2,310,955 | Find the smallest solution to the equation
\[
\frac{2x}{x-2}+\frac{2x^2-24}{x}=11.
\] Why hasn't anyone posted a solution yet? Are we all scared of the cubic? [img]https://artofproblemsolving.com/assets/images/smilies/tongue.gif[/img]
[hide=Solution.]The Integer Root Theorem says that for an integer $r$ to be a root... |
231,097 | Let \(a,b,c\ge 0\). Prove that
\[
\sum_{\text{cyc}}\frac{1}{a^2+ab+b^2}\ge\frac{9}{(a+b+c)^2}.
\] [quote="conan_naruto236"]
Suppose $ a \plus{} b \plus{} c \equal{} 1$. We have :
$ \sum\frac {1}{1 \minus{} (ab \plus{} bc \plus{} ca) \minus{} a} \geq 9$
$ \leftrightarrow 1 \minus{} 4(ab \plus{} bc \plus{} ca) \plus{} 9a... |
2,310,975 | In a geometric sequence \(a_1,a_2,a_3,\dots\), where all the terms are positive, \(a_5-a_4=576\) and \(a_2-a_1=9\). Find \(a_1+a_2+a_3+a_4+a_5\). Because $a_1, a_2, a_3, \cdots, a_n$ form a geometric series, we can write the sequence as $a_1, a_1 \times r, a_1 \times r^2, \cdots.$ Thus, $a_1 \times r^4 - a_1 \times r^3... |
2,310,982 | Let \(a_1,a_2,a_3\) be the first three terms of a geometric sequence. If \(a_1=1\), find the smallest possible value of \(4a_2+5a_3\). Because $a_1,a_2, a_3$ form a geometry sequence, and $a_1 =1,$ we can write the sequence as $1,r,r^2.$ Thus, we seek to find the minimum value of $4r+5r^2.$ The minimum value of this oc... |
2,310,983 | Find all values of \(z\) such that
\[
z^4 - 4z^2 + 3 = 0.
\]
Enter all the solutions, separated by commas. [hide = Solution]
Two numbers that multiply to $3$ and add to $-4$ are $-3$ and $-1$ so this factors to $(z^2 - 3)(z^2 - 1).$ The solutions are then $\pm \sqrt{3}$ and $\pm 1.$ [/hide] |
23,110 | Find the best constant \(k\) such that for all positive real numbers \(a,b,c\) the following holds:
\[
\frac{a^3}{ka^2+b^2}+\frac{b^3}{kb^2+c^2}+\frac{c^3}{kc^2+a^2}\ge\frac{a+b+c}{1+k}.
\] [quote="manlio"]Find the best costant $ k$ such that for all positive reals $ x,y,z$ we have
$ \frac {x^3}{kx^2 \plus{} y^2} \plu... |
2,311,001 | The function \(f(x)\) satisfies
\[
f(x) + 2f(1 - x) = 3x^2
\]
for all real numbers \(x\). Find \(f(3)\). Very good, @above! :thumbup: This is called a cyclic function. When you see questions like this, you want to first get $f(3)$ in the first $f$ and then $f(3)$ in the second $f$, it will give you a system of equation... |
231,102 | 1. Construct a circle tangent to two given lines that passes through a given point.
2. Construct a circle tangent to a given line and passing through two given points. [quote]Get the perpendiclar lines on B,C repectively [/quote]
the two lines through B,C perpendicular to which other two lines??? To WY (see my pict... |
2,311,025 | Find the ordered pair \((a,b)\) of positive integers, with \(a<b\), for which
\[
\sqrt{1+\sqrt{21+12\sqrt{3}}}=\sqrt{a}+\sqrt{b}.
\] [quote=pog][quote=OlympusHero]$\sqrt{21+12\sqrt{3}}=3+2\sqrt{3}$. Now we need $\sqrt{4+2\sqrt{3}}$ which is $\sqrt{3}+1$. So it is $\boxed{(1,3)}$.[/quote]
Could you elaborate on how you... |
2,311,075 | Two real numbers \(x\) and \(y\) satisfy
\[
x - y = 4
\]
and
\[
x^3 - y^3 = 28.
\]
Find \(xy\). I confused.
1) [url=https://artofproblemsolving.com/community/c63h2310754]Same Alcumus question?[/url]
2) My solution wrong somehow :what?:
Cubing gives $x^3-3x^2y+3xy^2-y^3=64$. So $-3x^2y+3xy^2=36$. Dividing each side ... |
2,311,081 | The equation \(x^3 - 4x^2 + 5x - \frac{19}{10} = 0\) has real roots \(r,\ s,\) and \(t.\) Find the length of the long diagonal of a box with side lengths \(r,\ s,\) and \(t.\) [hide=Solution]Lets first write what we know using Vieta's. We have $$r+s+t = 4$$, $$rs+rt+st = 5$$, and $rst = \dfrac{19}{10}$. We want to find... |
231,109 | Let \(ABC\) be a triangle. Construct outside it two similar isosceles triangles \(ABD\) and \(ACE\), with \(AB=AD\) and \(AC=AE\). Let \(I=BE\cap CD\), and let \(O\) be the circumcenter of triangle \(IDE\). Prove that \(AO\perp BC\). [hide="Solution"]
Let $ T$ be the midpoint of arc $ DE$ on the circumcircle of $ \tria... |
2,311,102 | Find the sum of all solutions to \(2^{|x|} + 3|x| = 18.\) See. Let's say $a$ is a solution. We know
$$2^{|a|} + 3|a| = 18$$.
Now since $|a|=|-a|$, we can substitute to get $2^{|-a|} + 3|-a| = 18$, which implies that $-a$ is also a solution.
EDIT: @wamofan :P Get it? |
231,117 | The line \(y - x\sqrt{3} + 3 = 0\) cuts the parabola \(y^{2} = x + 2\) at \(A\) and \(B\). If \(P=(\sqrt{3},0)\), find the value of \(PA\cdot PB\). hello,inserting $ y\equal{}x\sqrt{3}\minus{}3$ in $ y^2\equal{}x\plus{}2$ we get the quadratic equation
$ 3x^2\minus{}6x\sqrt{3}\minus{}x\plus{}7\equal{}0$, with the solut... |
2,311,181 | The ellipse
\[
\frac{(x-6)^2}{25} + \frac{(y-3)^2}{9} = 1
\]
has two foci. Find the one with the larger \(x\)-coordinate. Enter your answer as an ordered pair, like \((2,1)\). [hide=Sol]Standard form:
$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$
$h=6$, $k=3$, $a=5$, $b=3$
Distance from center to foci is $\sqrt{a^2-b^2}=... |
2,311,213 | Compute the exact value of the expression
\[
\left|\pi - \left|\pi - 7\right|\right|.
\]
Write your answer using only integers and \(\pi\), without any absolute value signs. Starting from the inner absolute value signs, we know $|\pi-7|$ can we written as $7-\pi$. Now we have $|\pi-(7-\pi)|.$ Simplifying this expressio... |
2,311,254 | Let \(t\) be a real number such that
\[
\begin{aligned}
\lfloor t \rfloor &= 4,\\
\lfloor t+\{t\}\rfloor &= 4,\\
\lfloor t+2\{t\}\rfloor &= 5,
\end{aligned}
\]
where \(\{t\}=t-\lfloor t\rfloor\). Find all possible values of \(t\). (Enter your answer in interval notation.) [hide=alternate way, hint]you could start by br... |
2,311,260 | Find the smallest solution to the equation
\[
\frac{1}{x-2} + \frac{1}{x-4} = \frac{3}{x-3}.
\] [hide=Solution]After some manipulation and factoring, you'll reach a quadratic solution $$x = \frac{6 \pm \sqrt{6^2 - 4 \cdot 6}}{2} = 3 \pm \sqrt{3},$$ whose least is $\boxed{3-\sqrt{3}}.$ [I don't why the topic is FE and i... |
23,113 | For \(a,b,c\) positive real numbers, prove that
\[
\frac{a^4}{a^3+b^3}+\frac{b^4}{b^3+c^3}+\frac{c^4}{c^3+a^3}\ge\frac{a+b+c}{2}.
\] it's the same as $\sum_{cyc} a\frac{a^3-b^3}{a^3+b^3}\geq 0$.
1. case: $\frac{a}{b},\frac{b}{c},\frac{c}{a}\leq \frac 12 (\sqrt{13}+1)$. in that case we can use $\frac{1-x^3}{1+x^3}\ge... |
2,311,321 | The function \(f\) takes nonnegative integers to real numbers, such that \(f(1)=1\), and
\[
f(m+n)+f(m-n)=\frac{f(2m)+f(2n)}{2}
\]
for all nonnegative integers \(m\ge n\). Find the sum of all possible values of \(f(10)\). [hide=different sol]We notice that $n=0$ gives $f(2m)=4f(m).$ The fakesolve way would be to notice... |
2,311,349 | Let \(a\) and \(b\) be real numbers. One of the roots of \(x^3 + a x + b = 0\) is \(1 + i\sqrt{3}\). Find \(a + b\). The [hide=Alcumus solution]Since the coefficients are real, another root is $1 - i \sqrt{3}.$ By Vieta's formulas, the sum of the roots is 0, so the third root is $-2.$ Hence, the cubic polynomial is
\be... |
231,136 | Let \(G\) be a non-commutative group. Consider all bijections \(a\mapsto a'\) of \(G\) onto itself such that \((ab)'=b'a'\) for all \(a,b\in G\) (i.e. the anti-automorphisms of \(G\)). Prove that these mappings together with the automorphisms of \(G\) constitute a group which contains the group of automorphisms of \(G\... |
2,311,366 | In the coordinate plane, the graph of
\[
\lvert x+y-1\rvert + \bigl\lvert\,\lvert x\rvert - x\bigr\rvert + \bigl\lvert\,\lvert x-1\rvert + x-1\bigr\rvert = 0
\]
is a certain curve. Find the length of this curve. [hide=Solution]
Each of the absolute values must equal $0$, so we have $x+y-1=0$, $|x|-x=0$, and $|x-1|+x-1=... |
231,137 | Solve the equation
\[
\sin x+\sin 2x+\sin 3x=1.
\] I've tried but i couldn't get the solution...however here it is
$ \sin x \plus{} \sin 2x \plus{} \sin 3x \equal{} 1$
$ \sin x \plus{} 2\sin x \cos x \plus{} \sin x(3\minus{}4\sin^2 x) \equal{} 1$
$ \sin x (1 \plus{} 2\cos x \plus{} 3 \minus{} 4 \sin^2 x) \equal{} ... |
2,311,372 | Let real numbers \(a,b,c\) satisfy
\[
a+b+c=ab+bc+ca>0.
\]
Prove that
\[
\sqrt{a^{2}-a+1}+\sqrt{b^{2}-b+1}+\sqrt{c^{2}-c+1}\ge a+b+c.
\] [quote=anhduy98]Let three real numbers $ a, b, c $ satisfy : $ a + b + c = ab + bc + ca> 0 $ . Prove that:$$\sqrt{a^2-a+1}+\sqrt{b^2-b+1}+\sqrt{c^2-c+1}\ge a+b+c$$[... |
2,311,374 | Evaluate the infinite geometric series:
\[
\frac{1}{3}+\frac{1}{6}+\frac{1}{12}+\frac{1}{24}+\dots
\] [quote=Stormbreaker7984][quote=selinapan][hide]Do you need a solution or hint.... since it's an alcumus problem I'm assuming hint
Use the geometric series sum formula, $\frac{a}{1-r}$, where a is your first term and ... |
231,138 | Let $2n$ distinct points on a circle be given. Arrange them into disjoint pairs in an arbitrary way and join each pair by a chord. Determine the probability that no two of these $n$ chords intersect. (All possible arrangements into pairs are equally likely.) The total number $ a_n$ of "good" pairings of $ 2n$ vertices ... |
231,142 | Let \(n\) be a positive integer. Prove that, for \(0<x<\dfrac{\pi}{n+1}\),
\[
\sin x - \frac{\sin 2x}{2} + \frac{\sin 3x}{3} - \cdots + (-1)^{n+1}\frac{\sin n x}{n} - \frac{x}{2}
\]
is positive if \(n\) is odd and negative if \(n\) is even. $ f_n(x) \equal{} \sin{x} \minus{} \frac {\sin{2x}}{2} \plus{} \cdots \plus{} (... |
2,311,429 | If \(x\) is a real number and \(\lfloor x \rfloor = -9\), how many possible values are there for \(\lfloor 5x \rfloor\)? Let's say our test values were $-9.1,-9.2,-9.3,...,-9.9$ (note that these numbers satisfy $\lfloor x \rfloor = -9$).
If we plug in $-9.1$ and $-9.2$ into $\lfloor 5x \rfloor,$ we get $-46.$
If we pl... |
2,311,431 | Solve
\[
\frac{1}{x+9}+\frac{1}{x+7}=\frac{1}{x+10}+\frac{1}{x+6}.
\] Why should this be in Intermediate Algebra? This is so simple!
[hide=My Steps]
1. Multiply all terms by all 4 denominators
2. Simplify
3. Subtract $2x^3$ from both sides
4. Subtract $48x^2$ from both sides
5. Subtract $382x$ from both sides
6. Subt... |
2,311,445 | Let \(x\diamondsuit y=xy+x+y+1\) for positive real \(x,y\). Given \(x+y=4\), compute the maximum possible value of \(x\diamondsuit y\). [hide=Sol]$x\diamondsuit y=xy+x+y+1=xy+5$, so we want to maximize $xy$, and this is done when $x=y=2$. $x\diamondsuit y=\boxed9$[/hide] |
2,311,447 | The function \(f(x)=x+1\) generates the sequence
\[
1,2,3,4,\dots
\]
in the sense that plugging any number in the sequence into \(f(x)\) gives the next number in the sequence.
What rational function \(g(x)\) generates the sequence
\[
\frac{1}{2},\ \frac{2}{3},\ \frac{3}{4},\ \frac{4}{5},\ \dots
\]
in this manner? Let ... |
231,147 | Let \(a,b,c\in[1,2]\). Find the maximum value of
\[
(a-b)^4+(b-c)^4+(c-a)^4+(a-b)(b-c)(c-a).
\] [quote="Dr Sonnhard Graubner"]hello, i have found with calculus $ P_{Max} \equal{} 2$ for $ a \equal{} 1,b \equal{} 1,c \equal{} 2$.
Sonnhard.[/quote]
Yes, :lol: ,and the proof ?(note the equality holds also when $ a\equal{... |
231,148 | Find a closed-form expression for the sum
\[
1^k + 2^k + \cdots + n^k,
\]
where \(k\) and \(n\) are positive integers. Thank you very much hsiljak.
I have another question about this.
Is it true that $ 1^k\plus{}2^k\plus{}...\plus{}n^k$ is a polynomial of degree $ k\plus{}1$, there is no constant term, the least... |
2,311,489 | Let \(a_n\) be the first (leading) digit of \(2^n\).
Prove that the sequence \((a_n)_{n\ge1}\) is not periodic. Let $a_n=k$ be the first digit of $2^n$, then $k\cdot 10^s \le 2^n <(k+1)10^s \implies s+\log k\le \log(2^n)\implies (k+1)10^s
\implies \alpha_n:={\rm fr}(n\log 2)\in [\log k,\log(k+1))$.
Calculating modulo... |
231,149 | For any sets \(C\) and \(D\), prove that
\[
\text{i)}\quad C = (C\cup D)\cap D,
\qquad
\text{ii)}\quad C = (C\cap D)\cup C.
\] Well, what can I use in the proof? If you can use the axioms of Boolean algebra, then simple absorption (or the distribution) rule should do the trick-they're both applicable here (BTW, check y... |
2,311,493 | Solve \(\tan x = \sin x\) for \(0 \le x \le 2\pi.\) Enter all the solutions, separated by commas. [hide=solution]Expressing $\tan x$ as $\dfrac{\sin x}{\cos x}$ gives $$\dfrac{\sin x}{\cos x} = \sin x.$$
If $\sin x = 0$, then the equality holds, giving $x = 0, \pi, 2\pi$.
If $\sin x \neq 0$, then simplifying the equa... |
2,311,494 | Let \(\mathbf{a}=\begin{pmatrix}2\\1\\5\end{pmatrix}.\) Find the vector \(\mathbf{b}\) such that
\[
\mathbf{a}\cdot\mathbf{b}=11
\]
and
\[
\mathbf{a}\times\mathbf{b}=\begin{pmatrix}-13\\-9\\7\end{pmatrix}.
\] [quote=ilovepizza2020][quote=mop][quote=cryptographer][hide=AoPS Solution #1]Let $\mathbf{b} = \begin{pmatrix} ... |
2,311,498 | The points \((0,0)\), \((a,11)\), and \((b,37)\) are the vertices of an equilateral triangle. Find the value of \(ab\). Compute the matrix for rotation by $60$ degrees about the origin (which is a linear transformation). Then, solve for $a$ (the $y$ coordinate of the rotation of $(a, 11)$ should be $37$). Then solve fo... |
2,311,499 | Find the curve defined by the equation
\[
r = 2.
\]
(A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola
Enter the letter of the correct option. [quote=GameBot]Find the curve defined by the equation
\[r = 2.\]
(A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola
Enter the letter of the correct optio... |
2,311,504 | Find the vector \(\mathbf{v}\) such that
\[
\begin{pmatrix}
2 & 3 & -1\\[4pt]
0 & 4 & 5\\[4pt]
4 & 0 & -2
\end{pmatrix}
\mathbf{v}
=
\begin{pmatrix}
2\\[4pt]
27\\[4pt]
-14
\end{pmatrix}.
\] Oh, no! An empty GameBot thread?
Let $\mathbf{\text v} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}.$
By the matrix dot product r... |
2,311,506 | For real numbers \(t\) where \(\tan t\) and \(\sec t\) are defined, the point
\[
(x,y)=(\tan t,\sec t)
\]
is plotted. All the plotted points lie on what kind of curve?
(A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola
Enter the letter of the correct option. [hide=Solution]Note that $$\sec^2{t}-\tan^2{t}=... |
2,311,507 | When \(\begin{pmatrix} a \\ b \end{pmatrix}\) is projected onto \(\begin{pmatrix} \sqrt{3} \\ 1 \end{pmatrix}\), the resulting vector has magnitude \(\sqrt{3}\). Also, \(a = 2 + b\sqrt{3}\). Enter all possible values of \(a\), separated by commas. [hide=Why does Alcumus solution always overkill?]
You just draw the line... |
231,151 | Να βρεθεί ο φυσικός αριθμός \(n\) αν ο αριθμός
\[
A=2^{17}+17\cdot 2^{12}+2^{n}
\]
είναι τέλειο τετράγωνο ακεραίου αριθμού. [b][color=red]Καλησπέρα μετά από καιρό! Καταρχήν εύχομαι σε όλους καλή σχολική και καλή ΟΛΥΜΠΙΑΚΗ χρονιά! Καλή επιτυχία στον ερχόμενο διαγωνισμό της ΕΜΕ και καλή σταδιοδρομία στους (πλέον) φοιτητέ... |
2,311,522 | For real numbers \(t\), the point
\[
(x,y) = (5\cos 2t,\, 3\sin 2t)
\]
is plotted. All the plotted points lie on what kind of curve?
(A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola
Enter the letter of the correct option. [hide=Solution]We have $x=5\cos2t$ and $y=3\sin2t.$ Then, $x^2=25\cos^22t$ and... |
2,311,547 | If \(a\ge b\ge c\ge 0\) and \(a^2+b^2+c^2=3\), prove that
\[
abc-1+\sqrt{\tfrac{2}{3}}\,(a-c)\ge 0.
\] [quote=csav10]If $a\ge b\ge c\ge 0$ and $a^2+b^2+c^2=3$, then
$abc-1+\sqrt\frac 2{3}\ (a-c)\ge 0$.[/quote]
$$LHS \ge abc-1+\sqrt{\frac{2}{3}}\sqrt{(a-c)^2+(a-b)(c-b)}=r-1+\sqrt{\frac{2(p^2-3q)}{3}}$$
$\bullet Let : ... |
2,311,557 | The matrix
\[
\mathbf{M} = \begin{pmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{pmatrix}
\]
satisfies \(\mathbf{M}^T \mathbf{M} = \mathbf{I}\). Find \(x^2 + y^2 + z^2\).
For a matrix \(\mathbf{A}\), \(\mathbf{A}^T\) is the transpose of \(\mathbf{A}\). Here,
\[
\mathbf{M}^T = \begin{pmatrix} 0 & x & x \\ 2y & y ... |
231,156 | \[
\int \log_{3} x \, dx
\]
\[
\int_{0}^{\pi} \sin^{2} x \, dx
\]
\[
\int \frac{x^{2}}{x^{2}-4x+5} \, dx
\] While it's true that these don't belong here (for future reference, try http://www.artofproblemsolving.com/Forum/index.php?f=296 ) I'll provide hints for the other two that JRav hasn't done.
1) I'll assume... |
2,311,563 | Simplify \(\sin(x-y)\cos y + \cos(x-y)\sin y.\) if you write it out it becomes ${cos(y)*sin(x)*cos(y) - cos(x)*sin(y)*cos(y)}$ +
${sin(y)*cos(x)*cos(y)+sin(x)*sin(y)*sin(y)}$
which simplified to $sin(x)cos^2(y)+sin(x)sin^2(y).$
factor out the $sin(x)$ to get
$sin(x)*{cos^2(y)+sin^2(y)} = sin(x)*1$ :D |
2,311,567 | Does there exist any function \(f:\mathbb{C}\to\mathbb{C}\) such that
\[
f(f(z)) = z^{2}\quad\text{for all }z\in\mathbb{C}?
\] How would you define $i^{\sqrt{2}}$ ? ;)
In fact, the answer is no. Note first that
$$f(z^2) = f(f(f(z))) = f(z)^2 \ \forall z \in \mathbb{C}.$$
This implies that $f(0)^2 = f(0)$ and $f(1)^2 = ... |
2,311,570 | In triangle \(ABC\), \(\sin A = \frac{3}{5}\) and \(\cos B = \frac{5}{13}\). Find \(\cos C\). We recognise the sines and cosines as 3-4-5 triangle and 5-12-13 right triangle. Now, WLOG, we choose AC to be 5. Then we use the cosine law to find that the answer is [hide]$\boxed{\frac{16}{65}}$[/hide]. idk how to hide atta... |
2,311,577 | Compute \(\arcsin\!\left(\frac{1}{\sqrt{2}}\right)\). Express your answer in radians. [hide=Solution]
$\sin = \dfrac{\text{opposite}}{\text{hypotenuse}}$, so the opposite is $1$ and the hypotenuse is $\sqrt{2}$. This should remind you of a $45-45-90$ triangle, and $45$ degrees is $\boxed{\dfrac{\pi}{4}}$ radians.
[/hid... |
2,311,578 | Find the area of the parallelogram generated by
\[
\begin{pmatrix}3\\1\\-2\end{pmatrix}
\quad\text{and}\quad
\begin{pmatrix}1\\-3\\4\end{pmatrix}.
\] [hide]
The area of a parallelogram is equal to twice the area of one of the triangles formed by drawing its diagonal. Thus, the area is $\frac{ab\sin\theta}{2}\cdot2=ab\s... |
2,311,581 | Find the matrix \(\mathbf{R}\) such that for any vector \(\mathbf{v}\), \(\mathbf{R}\mathbf{v}\) is the reflection of \(\mathbf{v}\) through the \(xy\)-plane. [hide=solution sketch] Find the images for $i, j, k$, then write it out as a matrix. (You could make a general point but this is easier since no variables). It g... |
2,311,587 | Solve over \(\mathbb{C}\) the system of equations
\[
\begin{cases}
x\bigl(y^{2}+1\bigr)=2y,\\[4pt]
y\bigl(z^{2}+1\bigr)=2z,\\[4pt]
z\bigl(x^{2}+1\bigr)=2x.
\end{cases}
\] [quote=vralex]Solve over $\mathbb{C}$ the system of equations $x(y^2+1)=2y, y(z^2+1)=2z, z(x^2+1)=2x$.[/quote]
Let $f(x)=\frac {2x}{x^2+1}$ so that s... |
2,311,616 | Let \(S\) be the set of complex numbers \(z\) such that \(\operatorname{Re}\!\left(\frac{1}{z}\right)=\frac{1}{6}\). This set forms a curve. Find the area of the region inside the curve. [hide]Let z = a + bi, then Re(1/z) = Re(1/(a+bi)) = a / (a^2+b^2). Given this is 1/6, we arrange to get a^2 - 6a + b^2 = 0. This is a... |
231,162 | Let p be a prime and let A = Z / p^r Z, where r ∈ N. Let G be the group of units of A (the residue classes modulo p^r represented by integers prime to p). Show that G is cyclic except when p = 2 and r ≥ 3, in which case G ≅ C2 × C_{2^{r−2}}. Show $ x\in (1 \plus{} p^k\mathbb{Z})\setminus (1 \plus{} p^{k \plus{} 1}\math... |
2,311,623 | In polar coordinates, the point \(\left(-2,\frac{3\pi}{8}\right)\) is equivalent to what other point, in the standard polar coordinate representation? Enter your answer in the form \((r,\theta)\), where \(r>0\) and \(0\le\theta<2\pi.\) Or you could simply add $\pi$ to the angle. |
2,311,633 | Solve the functional equation: find all functions \(f:\mathbb{R}\to\mathbb{R}\) satisfying
\[
f(xy)=f(x)+f(y)\quad\text{for all }x,y\in\mathbb{R}.
\] I will let $P(x,\, y)$ be the assertion of this functional equation.
Eh, this question is ill-defined and up to interpretation, because the domain and range are not stat... |
2,311,634 | Evaluate
\[
\begin{vmatrix}
\cos\alpha\cos\beta & \cos\alpha\sin\beta & -\sin\alpha\\[4pt]
-\sin\beta & \cos\beta & 0\\[4pt]
\sin\alpha\cos\beta & \sin\alpha\sin\beta & \cos\alpha
\end{vmatrix}.
\] [hide=Solution.]Expanding the determinant with focus towards the third column, we have
\begin{align*}
\begin{vmatrix} \cos... |
2,311,636 | The matrices
\[
\mathbf{A}=\begin{pmatrix}1 & x\\[4pt] y & -\frac{9}{5}\end{pmatrix}
\qquad\text{and}\qquad
\mathbf{B}=\begin{pmatrix}\frac{12}{5} & \frac{1}{10}\\[4pt] 5 & z\end{pmatrix}
\]
satisfy \(\mathbf{A}+\mathbf{B}=\mathbf{A}\mathbf{B}.\) Find \(x+y+z.\) [hide=who decided to invent matrices, i will find them]
u... |
2,311,641 | For a constant \(c\), in spherical coordinates \((\rho,\theta,\phi)\), find the shape described by the equation
\[
\theta = c.
\]
(A) Line
(B) Circle
(C) Plane
(D) Sphere
(E) Cylinder
(F) Cone
Enter the letter of the correct option. [quote=ThriftyPiano][hide=Answer]Since the angle must stay constant and everythin... |
2,311,651 | The point \((4 + 7\sqrt{3},\, 7 - 4\sqrt{3})\) is rotated \(60^\circ\) counterclockwise about the origin. Find the resulting point. [hide=sol]
Let $\mathbf{M}$ be the matrix that corresponds to the transformation. Since $\mathbf{M}$ is the rotation, we can use the formula for a rotation matrix. Therefore $\mathbf{M}=\b... |
2,311,652 | Express \(\sin(a+b)-\sin(a-b)\) as a product of trigonometric functions. [hide]
I can't believe I put $2\sin a\cos b$ as my first answer
Anyways, $\sin(a+b)-\sin(a-b)=2\sin b\cos a$ by sum to product (or trig addition formulas then cancellation of a bunch of stuff)
[/hide] |
2,311,654 | Find the values of the following expressions:
\[
{}^{\pi}C_{r},\qquad {}^{i}C_{r},\qquad {}^{n}C_{\pi},\qquad {}^{n}C_{i},
\]
\[
{}^{\pi}P_{r},\qquad {}^{i}P_{r},\qquad {}^{n}P_{\pi},\qquad {}^{n}P_{i},
\]
where \(i=\sqrt{-1}\). [quote name="HumanCalculator9" url="/community/p18375315"]
I think this involves the gamma ... |
2,311,661 | The volume of the parallelepiped generated by \(\begin{pmatrix}2\\3\\4\end{pmatrix}, \begin{pmatrix}1\\k\\2\end{pmatrix},\) and \(\begin{pmatrix}1\\2\\k\end{pmatrix}\) is \(15\). Find \(k\), where \(k>0\). [hide=Solution]Note that the volume of the parallelpiped is $$\left|\begin{matrix}2&3&4\\1&k&2\\1&2&k\end{matrix}\... |
2,311,672 | When every vector on the line \(y=\tfrac{5}{2}x+4\) is projected onto a certain vector \(\mathbf{w}\), the result is always the vector \(\mathbf{p}\). Find the vector \(\mathbf{p}\). [hide=sol]The same vector p will be the projection of w onto the line.
The head of p will be the closest point on $y=\frac{5}{2}x+4$ to t... |
2,311,678 | Let \(a,b,c,d\) be nonzero integers such that
\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix}^2
=
\begin{pmatrix} 7 & 0 \\ 0 & 7 \end{pmatrix}.
\]
Find the smallest possible value of \(|a|+|b|+|c|+|d|\). first post
so you just expand out the left hand side by multiplying the matrices, and you get that $a+d=0$. The only... |
2,311,685 | Find the point of intersection of the line
\[
\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}
\]
and the plane
\[
x-y+z=5.
\] [hide=Solution]Multiplying by $12$ gives $4(x-2)=3(y+1)=z-2$, or $4x-8=3y+3=z-2$. From $4x-8=z-2$, we have $4x=z+6$, so $x=\frac{z+6}{4}$. From $3y+3=z-2$, we have $3y=z-5$, so $y=\frac{z-5}{3}$. The... |
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