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Mathematics Stack Exchange

Questions tagged [contest-math]

For questions about mathematics competitions or the questions that typically appear in math competitions. Provide enough information about the source to confirm the question doesn't come from a live contest.

This tag is intended for

  1. Questions from mathematics competitions.
  2. Inquiries about alternative proofs for problems that are from math contests.
  3. Questions that have been inspired by a contest problem, including practice problems.
  4. Questions requesting advice on competing in contests.

See this list of mathematics competitions to get an idea of the types of questions this tag is for.

Mathematics StackExchange has a policy on questions from current competitions. Questions from ongoing competitions will be locked and temporarily deleted until the end of the contest. It is a good idea to include information about a contest, such as a link to the contest webpage.

9758 questions
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Prove $4$ is the smallest value for $k$ so this equality holds

Let $a$, $b$ and $k$ be positive reals, such that $$\frac{1}{3} = k\cdot 2 \frac{ab^2}{(a + \sqrt{a^2 + b^2})^3}.$$ Prove $4$ is the smallest possible value for $k$. This problem is related to this one I asked in another question, but they are…
I'm Ingrid
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Understanding installment concept

I was solving a question which has one part of it as ...He lends the other part to his friend who pays it in two equal instalments at the rate $20$% compounded annually to be paid at the end of each of the two years. If each instalment paid by his…
Ganit
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Second meeting point in a race.

Two racers are running to and fro from points $A$ and $B$ at $3m/s$ and $7m/s$ respectively. If the distance between $A$ and $B$ is $2000m$, what is the total distance covered by both the racers, till they meet for the $2^{nd}$ time? Let $P$ and…
Ganit
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Understanding a solution to counting hexagons on a soccer ball

Each face of a soccer ball is either a pentagon or a hexagon. Each pentagonal face is adjacent to five hexagonal faces and each hexagonal face is adjacent to three pentagonal and three hexagonal faces. If the ball has 12 pentagonal faces, how many…
user200918
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The 6-digit number 2X574Y is divisible by 36. How many possible solutions are for X?

The 6-digit number 2X574Y is divisible by 36. How many possible solutions are for X? My naive approach is to write multiples of 36 and see the patterns, but many multiplies must be written to see which one has 574. I have studied Silverman's Number…
user200918
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1 answer

Olympiad Chessboard Problem

I am stumped by this contest math training problem from South Africa: Consider an 8 × 8 chessboard with the bottom left 3 × 3 squares occupied by cute little frogs. Each cute frog can jump over any frog adjacent to it, vertically or horizontally,…
MathsJourney
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The number of primes whose multiple can not be in $\{ f(m)| m\in \mathbb{Z}\}$ where $f(x)=x^4+2x^3 -2x^2 -4x +4$

Consider a polynomial $f(x)=x^4+2x^3 -2x^2 -4x +4$. Define $$A = \{ f(m) \mid m\in \mathbb{Z}\}$$ And define $$ B =\bigg\{ p \mid p \ {\rm is\ a\ prime\ and}\ \mathbb{Z}\cdot p \bigcap A =\emptyset\bigg\} $$ Hence prove that $|B|=\infty$ i.e. there…
HK Lee
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How many squares contains paint

I read a book containing competition level problems. I was unable to solve the following: Consider an $n\times m$ grid. A paintbrush whose width is the same as the width of a grid square is used to paint a line from one corner of the grid to the…
student
  • 33
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Find all sets $A$ that satisfy condition $a+b \in A \implies ab\in A$.

Find all subsets from $ \mathbb N$ such that: We say that a subset from $\mathbb N$ that it is complete (let’s call it $A$) if : $a+b \in A \implies ab\in A$. With $a,b \in \mathbb N$ Find all complete subsets from $\mathbb N$. So far I’ve found…
PNT
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At least how many different numbers are there from $a_1,a_2,\cdots,a_{1394}$?

Consider positive integers $a_1,a_2,\ldots,a_{1394}$ so that neither of two numbers of $\dfrac{a_1}{a_2},\dfrac{a_2}{a_3},\cdots,\dfrac{a_{1393}}{a_{1394}}$ be equal to each other. at least how many different numbers are there from…
Amirali
  • 1,149
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$x^3 + y^3 +3x^2 y^2 =x^3y^3$ Find all possible values to $\frac{x+y}{xy}$

$x,y \in \mathbb{R}\setminus\{0\}$ Indeed the original question said :Find all possible values to: $$\frac{1}{x} + \frac{1}{y}$$ But it’s the same thing. My Attempt: $$x^3 + y^3 +3x^2 y^2 =x^3y^3 \iff (x+y)(x^2-xy+y^2)=x^3y^3…
PNT
  • 4,164
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I need help with solving a math problem that involves clocks

Here's the question. "It is now between 10:00 and 11:00. Six minutes from now, the minute hand of a watch will be exactly opposite the place where the hour hand was three minutes ago. What is the exact time now?" It's from Art of Problem Solving…
user723556
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2 answers

How to solve $x^{x^{x^x}} = 1/3^{\sqrt{48}}$

How to solve $$x^{x^{x^x}} = \frac{1}{3^{\sqrt{48}}}$$ Attempt : Let $x^{x^{\cdots}} = y$ $$\begin{align} x^y &=y\\ y\ln(x) &= \ln(y)\\ -\ln(x) &= -\ln(y)e^{-\ln(y)}\\ -\ln(y) &= W(-\ln(x))\\ y &= e^{-W(-\ln(x))} \end{align}$$ I'll stop this. Am i…
user516076
  • 2,200
3
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There are $n$ students with a different student number. Each student number is a positive factor of $60^{60}$

In a school there are $n$ students, each with a different student number. Each student number is a positive factor of $60^{60},$ and the GCD of two student numbers is not a student number in the school. Find the greatest possible value of $n$. I am…
Ishraq Farhan
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Find the value of $E(\frac{1}{2018})+E(\frac{2}{2018})+\dots+E(\frac{2017}{2018})$

Let $E(n) = \dfrac{4^n}{4^n+2}$. If the value of $E(\frac{1}{2018})+E(\frac{2}{2018})+\dots+E(\frac{2017}{2018})=\frac{a}{b}$ (in lowest terms), find $b$. I tried solving this question using telescopic sum but was unable to solve
user740036
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