# Tag Info

Accepted

### A goat tied to a corner of a rectangle

Parts of three different circles.
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 ...
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 ...

### 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 ...
Accepted

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; ...
Accepted

### 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 ...
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 ...
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: $$x^2+y^2+z^2=\frac{1}{3}(1^2+1^2+1^2)(x^2+y^2+z^2)\geq\frac{1}{3}(x+y+z)^2=\frac{1}{3}$$

### 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 ...
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 ...
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 $\color{red}{3}$ ...
### 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 ...