21
votes
Accepted
Symmetry in Probability (AMC 12A 2023)
The idea in Solution 1, although not explicitly stated, is that if the frog has not reached or passed $10$, irrespective of how much further Flora needs to go, the probability that the next jump ...
- 129k
8
votes
Accepted
Minimizing $\sum_{cyc}\frac{\sqrt{5a+8bc}}{8a+5bc}$ with $\sum_{cyc}ab=1$
Some thoughts.
Let
$$x := 5a + 8bc, \quad y := 5b + 8ca, \quad z := 5c + 8ab,$$
$$u := 8a + 5bc, \quad v := 8b + 5ca, \quad w := 8c + 5ab.$$
Let
$$A := \frac{2\sqrt{2} - \sqrt{5}}{3}, \quad B := \frac{...
- 36.1k
8
votes
Accepted
How to find $X$ with these given values?
Applying the law of sines:
$ \dfrac{a}{\sin(30^\circ) } = \dfrac{b}{\sin(120^\circ - x)} $
and
$ \dfrac{a}{\sin(x)} = \dfrac{b}{\sin(30^\circ)} $
Dividing these two out,
$ \dfrac{\sin(x)}{\sin(30^\...
- 19k
6
votes
Accepted
If $z_1+z_2+z_3=2, z_1^2+z_2^2+z_3^2=3$ and $z_1z_2z_3=4$, find $\frac1{z_1z_2+z_3-1}+\frac1{z_2z_3+z_1-1}+\frac1{z_3z_1+z_2-1}$
Note that: $$\sum_{cyc} \frac{1}{ab+c-1} = \sum_{cyc} \frac{1}{ab+c-(a+b+c-1)} = \sum_{cyc} \frac{1}{(a-1)(b-1)}$$ So we have to evaluate a decently easy value, it becomes:
$$\frac{a+b+c-3}{(a-1)(b-1)(...
- 1,031
6
votes
Accepted
Number of words with 8 letters using an alphabet of 3 consonants and 2 vowels with constraints
Recurrence approach. Let $c(n)$ be the number of legal words such that the $n$-th letter is a consonant and let $v(n)$ be the number of legal words such that the $n$-th letter is a vowel. Then the ...
- 145k
6
votes
Accepted
Closed form of $\prod_{n=0}^{\infty}\frac1{1+x^{2^n}}$
This is a telescoping product.
$$(1-x^{2^n})(1+x^{2^n}) = 1 - x^{2^{n+1}}.$$
Therefore,
$$(1-x) \prod_{n=0}^N (1+x^{2^n}) = 1 - x^{2^{N+1}}.$$
The rest of the details I leave to you as an exercise.
- 129k
5
votes
Accepted
Finding $\small{\max\limits_{ab+bc+ca=1}\dfrac{\sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}}{\sqrt{3a+3b+3c+10}} }$
Proof. It suffices to prove that
$$\sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}\le \frac{3}{4}
\sqrt{3a+3b+3c+10}$$
or
$$\left(\sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}\right)^2 \le \frac{9}{16}
(3a+3b+3c+10).\tag{1}...
- 36.1k
5
votes
Symmetry in Probability (AMC 12A 2023)
Based on comments:
There is nothing special about $10$ or even $\frac12$.
In the process which is Flora's progress rightwards, each position has a probability $p$ of being landed on and a probability $...
- 154k
5
votes
How long does it take to get to "Olympiad-Bronze-level" of math problem solving ability from no competition experience?
Maybe, but it took me 3 full years to make it from the moderate problem solving ability (I had practiced occasional problem solving for 7 years as a schoolboy before) to the IMO level with rather ...
- 17.8k
4
votes
Accepted
Find all natural numbers $x$ and $y$ such that $\frac{\sqrt{x} + \sqrt{y}}{\sqrt[3]{x^2 + y^2}}$ is a natural number
We have $x^2 + y^2 \ge \frac12(x + y)^2$ and $\sqrt x + \sqrt y \le \sqrt{2(x + y)}$.
Therefore $\frac{\sqrt x + \sqrt y}{\sqrt[3]{x^2 + y^2}} \le \frac{\sqrt{2(x + y)}}{\sqrt[3]{\frac12(x + y)^2}} = \...
- 22k
4
votes
If $z_1+z_2+z_3=2, z_1^2+z_2^2+z_3^2=3$ and $z_1z_2z_3=4$, find $\frac1{z_1z_2+z_3-1}+\frac1{z_2z_3+z_1-1}+\frac1{z_3z_1+z_2-1}$
OP found that
$z_i$ are the roots of $f(x) = 2x^3 -4x^2 + x - 8 $
The sum can be written as $ \sum \frac{z_i}{z_i^2 - z_i + 4 } $.
From there, divide $f(x)$ throughout by $x^2 - x + 4$ and apply ...
- 66.7k
4
votes
Partition a stable (Middle School Math)
"My daughter had this question .... she is a tad confused on this."
It is not her fault : I too am confused , because that Question is almost meaningless.
(1) When the Stable has concrete ...
- 7,313
4
votes
Accepted
BMO1 number theory question on fibonacci sequence and divisibility
Here's how I solved it.
The given problem can be formulated as: Prove that $f_n = anb^n$ mod $m$. I didn't want to think about whether various theorems hold over rings, so I'm going to initially hope ...
- 156
4
votes
If $a+b+c+abc=4,$ prove $\frac{1}{\sqrt{a^2+4bc}}+\frac{1}{\sqrt{b^2+4ac}}+\frac{1}{\sqrt{c^2+4ba}}\ge \frac{5}{4}.$
Some thoughts.
By Holder inequality, we have
\begin{align*}
&\left(\sum_{\mathrm{cyc}} \frac{1}{\sqrt{a^2 + 4bc}} \right)^2\cdot \sum_{\mathrm{cyc}} (a^2 + 4bc)(4b + 4c - bc + 4ab + 4ac)^3 \\
\...
- 36.1k
4
votes
Accepted
Proving the existence of integers $a, b, c$ such that $\left\lvert a + b\sqrt{2} + c\sqrt{3}\right\rvert < 10^{-5}$
DISCUSSION :
There are a couple of Issues with your nice approach. When those Issues are rectified , the Solution will work out.
(1) The range you have taken is $10^3$ , though the Correct range is $...
- 7,313
3
votes
Minimizing $\sum_{cyc}\frac{\sqrt{5a+8bc}}{8a+5bc}$ with $\sum_{cyc}ab=1$
Not an solution, just some thoughts
Let $f(a,b,c)$ be the expression we want to minimize. Suppose one of the variables is $0$, and the other two are inverses of each other. WLOG, assume $a=0, b=\frac{...
- 198
3
votes
Accepted
O-minimality and Putnam Competition
I think it is likely that, as @A. Rex suggested in his comment, the problem was A5 from the 2014 Putnam Competition.
The problem
The problem is stated as follows:
Let $$P_n(x) = 1 + 2x + 3x^2 + \dots ...
- 933
3
votes
Accepted
Evaluating $\int_{0}^{\pi/2}\frac{1}{1+\tan^{101}x}dx$
Let $u=\frac\pi2-x\Rightarrow\mathrm{d}u=-\mathrm{d}x$.
Then, $$\begin{align}\int_0^\frac\pi2\frac1{1+\tan^{101}x}\,\mathrm{d}x&=-\int_\frac\pi2^0\frac1{1+\tan^{101}\left(\frac\pi2-u\right)}\,\...
- 1,116
3
votes
Proving $(a^n + b^n) \sum_{k=0}^{n} \binom{n}{k} a^k b^{n-k} \leq (a^{2n} + b^{2n}) \sum_{k=0}^{n} \binom{n}{k}$
Your guess is almost correct. By noting that
\begin{align*}
(a^n + b^n) \sum_{k=0}^{n} \binom{n}{k} a^k b^{n-k}
&= a^n \sum_{k=0}^{n} \binom{n}{k} a^k b^{n-k} + b^n \sum_{k=0}^{n} \binom{n}{k} a^{...
- 160k
3
votes
A train travelling from town $A$ to town $B$ meets with an accident after 1 hour. The train is stopped for 30 minutes, after which....(AHSME 1955)
Actually the problem can be very simply solved, (actually mentally), and if you are ever planning to appear for tests for Management, etc, you must be alert for methods to solve such problems as ...
- 40k
3
votes
Accepted
Given that $a,b,c>0$ and $abc=1$, prove that $a+b+c+\frac{3}{ab+bc+ca} \geq 4$
Let $a+b+c=3u$, $ab+ac+bc=3v^2$ and $abc=w^3$.
Thus, we need to prove that:
$$3u+\frac{w^3}{v^2}\geq 4w,$$ which is true by AM-GM:
$$3u+\frac{w^3}{v^2}\geq4\sqrt[4]{\frac{u^3w^3}{v^2}}\geq4\sqrt[4]{\...
- 197k
3
votes
How long does it take to get to "Olympiad-Bronze-level" of math problem solving ability from no competition experience?
It's more about the time and effort that you put in. If you can set aside 2-4 hours a day to focus on (pretty much anything), you can make a lot of progress in a year.
After that it's having the ...
- 66.7k
3
votes
Accepted
Thailand MO $f(x)f(y)f(x-y) = x^2f(y) - y^2f(x)$
$$(1) \ \ \ \ \ f(0)=0$$
You can trivially show $f(x) = 0$ is a solution.
Otherwise,
$$(2) \ \ \ \ \ f(-t) = -f(t)$$
$$(3) \ \ \ \ f(2x)\cdot f(x) = 2x^2$$
$$(4) \ \ \ \ \ 2\cdot f(x)=f(2x)$$
Hence, $...
- 1,031
3
votes
Accepted
IMO 1987/P1 - Combinatoric approach
Your claim that $p_n(k)=\binom nk(n-k-1)!$ is not correct. In the case where $k=0$, your formula would imply that there are $(n-1)!$ permutations with zero fixed points, so $(n-1)!$ derangements of $n$...
- 72.1k
2
votes
Using complex numbers to show that a triangle whose circumcenter and centroid coincide must be equilateral
Taking the circumradius as the length unit, you can assume WLOG that the triangle is inscribed into the unit circle.
You can furthermore assume that the vertices are
$$1,e^{ib},e^{ic}\tag{1}$$
The ...
- 78.8k
2
votes
How much geometry should I know for the AMC 12
The AMC 12 includes topics through Pre-Calculus. I haven’t read that book, but I would recommend a book called Geometry by Moises E. Downs. I would recommend studying geometry, and trying to emphasize ...
- 21
2
votes
Proving $(a^n + b^n) \sum_{k=0}^{n} \binom{n}{k} a^k b^{n-k} \leq (a^{2n} + b^{2n}) \sum_{k=0}^{n} \binom{n}{k}$
We have
$$a^{2n} + b^{2n}
- a^{n + k}b^{n-k} - a^{n-k}b^{n+k}
= (a^{n + k} - b^{n + k})(a^{n - k} - b^{n - k})\ge 0.$$
(Note: The last inequality is easy.
WLOG, assume that $a \ge b$.
Then $a^{n+k} \...
- 36.1k
2
votes
Number of words with 8 letters using an alphabet of 3 consonants and 2 vowels with constraints
I think your approach is fine. First just strings of c's v's, and separating those with v's at the end of the word and those with no v at the end.
0 v's -- $1$
1 v not ending in v -- $6$
1 v ending ...
- 9,994
2
votes
Evaluating $\int_{0}^{\pi/2}\frac{1}{1+\tan^{101}x}dx$
$$I=\int_{0}^{\pi/2}\frac{1}{1+\tan^{101}x}dx$$
$$I=\int_0^{\frac{\pi}{2}}\frac{\cos^{101}(x)}{\cos^{101}(x)+\sin^{101}(x)}\,dx$$
Use the King's Property;
$$I=\int_0^{\frac{\pi}{2}}\frac{\sin^{101}(x)}...
- 2,387
2
votes
To find all functions which satisfy$f(x^3)+f^3(y)+f^3(z)=3xyz \\ \forall x+y+z=0$ and $x,y,z\in\mathbb R$
I will show how we transform this into a Cauchy's functional equation (since this is what the poster attempts to do).
Take $x=y=z=0$, then $f(0)+2f(0)^3=0$. The only real solution is $f(0)=0$.
Take $x=...
- 84
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