262
votes
Accepted
140
votes
Accepted
Is being good at mathematic contests necessary to pursue a career in mathematics or physics?
Mathematics takes place at different time-scales. If you can solve a problem in $5$ minutes that others need an hour to solve, you can probably get a good job. If you can solve a problem in a month ...
- 440k
113
votes
Accepted
7 fishermen caught exactly 100 fish and no two had caught the same number of fish. Then there are three who have together captured at least 50 fish.
Let's work with the lowest four numbers instead of the other suggestions.
Supposing there is a counterexample, then the lowest four must add to at least $51$ (else the highest three add to $50$ or ...
- 99.4k
109
votes
Prove that at a party of $25$ people there is one person knows at least twelve people.
If everyone knows everyone, then you are done.
Otherwise choose two people, A and B say, who don't know each other. These two people are part of $23$ triples. In each of these triples, either A ...
- 39.9k
81
votes
Accepted
Witty functional equation
We have $\displaystyle f(1-x)=\frac{2}{4^{1-x}+2}=\frac{2\cdot 4^x}{4+2\cdot 4^x}=\frac{4^x}{2+ 4^x}$
$\displaystyle f(x)+f(1-x)=\frac{2}{4^x+2}+\frac{4^x}{2+ 4^x}=1$
The required sum is
$$\frac{...
- 23.3k
74
votes
Is being good at mathematic contests necessary to pursue a career in mathematics or physics?
It's nice to have awards from contests on your grad school, etc. applications (and it's something to be legitimately proud of), but they have little resemblance to what mathematicians or physicists do ...
- 24.9k
74
votes
Find $xy+yz+zx$ given systems of three homogenous quadratic equations for $x, y, z$
We can obtain $yz+zx+xy=2$ simply by finding the values of $x$, $y$, and $z$. We are given $$y^2+yz+z^2=1,\qquad(1)$$$$z^2+zx+x^2=4,\qquad(2)$$$$x^2+xy+y^2=3,\,\qquad(3)$$with $x,y,z>0$. ...
- 17.9k
72
votes
Accepted
How can I answer this Putnam question more rigorously?
Why not write it the other way round?
The polynomial function
$$F(x)=\sum_{k=0}^n\frac{a_k}{k+1}x^{k+1} $$
is a differentiable function $\Bbb R\to\Bbb R$ with derivative $$F'(x)=\sum_{k=0}^na_kx^k.$$...
- 372k
67
votes
Show that there are infinitely many powers of two starting with the digit 7
I was intrigued. Not being smart enough to get a theoretic proof I used brute force. Definitely ugly. Only redeeming aspect is it is constructive.
Here is how I proceeded.
Remark: If two numbers are ...
- 19.3k
65
votes
Accepted
Drunk man with a set of keys.
The key thing here is this: let $T$ be the number of tries it takes him to open the door. Let $D$ be the event that the man is drunk. Then
$$
P(D\mid T=3)=\frac{P(T=3, D)}{P(T=3)}.
$$
Now, the event ...
- 32.1k
57
votes
Accepted
How to prove that a very large number is not prime
We have the number $10^{20}+1$. Whenever we have something in this kind of form, we need to find an odd factor of the exponent. In this case $5 \mid 20$, so we can use $5$ as the factor.
Now, we can ...
- 17.5k
57
votes
What is the algebraic intuition behind Vieta jumping in IMO1988 Problem 6?
At the heart of these so-called "Vieta-jumping" techniques are certain symmetries (reflections) on conics. These symmetries govern descent in the group of integer points of the conic. If you wish to ...
- 269k
56
votes
Accepted
How many ways can I go from 1 to 10 in the following diagram?
You can think the problem as start with the $10$ and follow the numbers in order until you reach $1$.
Then there is two ways for each step: up or left. Then there are $2^{10-1}=2^9=512$ ways. Done!
- 2,775
55
votes
Accepted
$\int_{0}^{\frac{\pi}{4}}\frac{\tan^2 x}{1+x^2}\text{d}x$ on 2015 MIT Integration Bee
There must be some problem here. Note that on $[0, \pi/4]$, $$\frac{\tan^2 x}{1+x^2} \le \frac{\tan^2 x}{1 + 0^2} = \tan^2 x,$$ therefore the definite integral is bounded above as follows: $$0 \le \...
- 129k
55
votes
Accepted
Fewest steps to reach $200$ from $1$ using only $+1$ and $×2$
Look at what the operations $[+1]$ and $[\times 2]$ do to the binary expansion of a number:
$[\times 2]$ appends a $0$, and increases the length by one, leaving the total number of $1$'s unchanged;
...
- 63.3k
53
votes
Accepted
Which is larger, $\sqrt[2015]{2015!}$ or $\sqrt[2016]{2016!}$?
Starting with:
$$\sqrt[2015]{2015!}\mid\sqrt[2016]{2016!}$$
Raise each side to the power of $2015\cdot2016$:
$$2015!^{2016}\mid2016!^{2015}$$
Divide each side by $2015!^{2015}$:
$$2015!^{1}\...
- 43k
53
votes
Find $xy+yz+zx$ given systems of three homogenous quadratic equations for $x, y, z$
To my surprise, this problem can be solved using geometry.
Identify the Euclidean plane with complex plane $\mathbb{C}$.
and let $\omega = e^{\frac{2\pi}{3}i}$ be the cubic root of unity.
Consider ...
- 121k
51
votes
Find $S = \frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b}$ if values of $a+b+c$ and $\frac1{a+b}+\frac1{b+c}+\frac1{a+c}$ are given
Multiplying the given expressions together:
\begin{align}
\frac{47}{10} &= (a+b+c)\bigg(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\bigg) \\ \\
&= \frac{a+b+c}{a+b}+\frac{a+b+c}{b+c}+\frac{a+...
- 1,536
48
votes
Accepted
Prove that there exists a triangle which can be cut into 2005 congruent triangles.
The decomposition is possible because $2005 = 5\cdot 401$ and both $5$ and $401$ are primes of the form $4k+1$. This allow $2005$ to be written as a sum of squares.
$$2005 = 22^2 + 39^2 = 18^2+41^2$$
...
- 121k
48
votes
Accepted
An interesting trigonometric integral
\begin{align*}
I&= \int_0^\pi \cos^2\cos x+\sin^2\sin x \, dx\\
&= 2\int_0^{\pi/2} \cos^2\cos x+\sin^2\sin x \, dx\\
&= 2\int_0^{\pi/2} \cos^2\sin x+\sin^2\cos x \, dx \quad(x\mapsto\pi/2-...
- 2,719
46
votes
Drunk man with a set of keys.
Let's first compute the probability that he wins on the third try in each of the two cases:
Sober: The key has to be one of the (ordered) five, with equal probability for each, so $p_{sober}=p_s=\...
- 68.3k
42
votes
Accepted
Problem from the 2020 Latvian "Sophomore's Dream" competition
Let $I$ be our integral. Substitute $t=-x \Rightarrow dt = -dx$. Then:
$$I=\int_{-a\pi}^{a\pi} \frac{\cos^5 t+1}{e^{-t}+1}\,dt=\int_{-a\pi}^{a\pi} \frac{e^t(\cos^5 t+1)}{e^{t}+1}\,dt=\int_{-a\pi}^{a\...
- 8,481
41
votes
Can a pre-calculus student prove this?
As randomgirl and Michael point out, there are counterexamples, such as $a=b=0$. However, it is true when $a \ge 1$. In principle it can be proved with just pre-calculus mathematics, but only an ...
- 536
41
votes
Accepted
What is the remainder when $6^{273} + 8^{273}$ is divided by $49$?
The binomial formula gives
$$(7\pm1)^{273}={273\choose1}7^1(\pm1)^{272}+{273\choose0}7^0(\pm1)^{273}=273\cdot 7\pm1=\pm1\qquad({\rm mod}\ 49)\ ,$$
since all other terms are divisible by $7^2$. It ...
- 225k
39
votes
Accepted
Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.
Cauchy- Schwarz works:
$$x^2+y^2+z^2=\frac{1}{3}(1^2+1^2+1^2)(x^2+y^2+z^2)\geq\frac{1}{3}(x+y+z)^2=\frac{1}{3}$$
- 197k
38
votes
Painting the plane, and finding points one unit apart
This is the subject well-known open problem called the Hadwiger-Nelson Problem. The problem asks for the exact minimum number of colors that we can color the plane with, so that no two points of ...
- 45.5k
38
votes
Accepted
Integer solutions to $x^3=y^3+2y+1$?
Hint: if $y>0$, then $y^3< y^3+2y+1< (y+1)^3$, so the RHS expression cannot be a perfect cube. A similar idea works if $y$ is a small enough negative number, but some negative numbers close ...
- 7,368
38
votes
Accepted
Putting socks and shoes on a spider
You can imagine doing this as writing a sequence, say $$3453228156467781$$
What does it mean?
It means first put sock on leg $\color{red}{3}$ and on 4-th move put shoe on leg $\color{red}{3}$
...
- 89.5k
37
votes
Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.
$x^2+y^2+z^2$ only depends on the squared distance of $(x,y,z)$ from the origin and the constraint $x+y+z=1$ tells us that $(x,y,z)$ lies in a affine plane. The problem is solved by finding the ...
- 352k
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