Articles of contest math

2013 Putnam A1 Proof understanding (geometry)

Problem A1: Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a non-negative integer such that the sum of all $20$ integers is $39.$ Show that there are two faces that share a vertex […]

Ways of coloring the $7\times1$ grid (with three colors)

Hints only please! A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7 \times 1$ […]

Maths brain teaser. Fifty minutes ago it was four times as many minutes past three o'clock

Fifty minutes ago it was four times as many minutes past three o’clock. How many minutes is it to six o’clock..? I have got the solution online but have doubts in it : There are 180 minutes between 3 o’clock and 6 o’clock. Call x the number of minutes to 6 o’clock. Then it is […]

A quick question on general mathematics

I have the following question that I am currently unable to satisfactorily answer myself. My question is: Does the inequality $$\frac{a}{b} + \frac{b}{a} < \frac{f(a)}{f(b)} + \frac{f(b)}{f(a)}\tag{*}$$ imply that $a < b$, in general (i.e., for ALL functions $f$)? If the answer to my question is NO, under what conditions on the function $f$ is […]

Find the $least$ number $N$ such that $N=a^{a+2b} = b^{b+2a}, a \neq b$.

When I graphed the relation $a^{a+2b}=b^{b+2a}$ , it gives a graph similar to $y=x$. However, the question explicitly states that $a \neq b$. So does that mean that no such $N$ exists ? What happens when the problem is generalized as $N=a^{ma+nb}=b^{mb+na} $ ? Can anybody help as to what should be done ? Thanks […]

Prove by elementary methods: the plane cannot be covered by countably many copies of the letter “Y”

As a consideration from the post “Prove by "elementary methods": The plane cannot be covered by finitely-many copies of the letter "Y"”, on the basis of the remark made in previous post by the user Moishe Cohen, is it still possible to apply elementary methods to prove weaker results, namely: The plane cannot be covered […]

How to prove that $\frac x{\sqrt y}+\frac y{\sqrt x}\ge\sqrt x+\sqrt y$

I am trying to prove that $$\frac x{\sqrt y}+\frac y{\sqrt x}\ge\sqrt x+\sqrt y$$ I tried some manipulations, like multiplications in $\sqrt x$ or $\sqrt y$ or using $x=\sqrt x\sqrt x$, but I’m still stuck with that. What am I missing?

Integral $\int_0^{\pi/2} x\cot(x)dx$, Differntiation wrt parameter only.

Integrate using differentiation wrt parameter only. $$\int_0^{\pi/2} x\cot(x)dx$$ We can express this as $$\int_0^{\pi/2} x\cdot\frac{\cos(x)}{\sin(x)}dx$$ Notice we can write $u=\sin(x)$ to start but I am not sure if that will do us any good. If we use $\xi$ as a parameter, the answer is of the form.$$ \lim_{\xi \to 1}I(\xi)=\lim_{\xi \to 1}\frac{\pi}{2}\ln(\xi+1)$$ NOTE:Only use differentiation […]

Conic Sections and Complex numbers

If $\omega$ is a complex number such that |$\omega$| does not equal 1, then the complex number $$z = \omega + \frac{1}{\omega}$$ describes a conic. The distance between the foci of the conic described by $z$ is? I attempted to rewrite this in cartesian form but to no avail. How do i proceed?

Existence of Gergonne point, without Ceva theorem

The intersection at one point (called Gergonne point) of the lines from vertices of a triangle to contact points of the inscribed circle can be proved immediately using Ceva’s theorem. Is there a direct proof that does not pass through Ceva’s formula? Edit: I am hoping for a metric Euclidean proof using lengths and angles […]