Mathematics Stack Exchange
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
4
votes
1 answer

strictly decreasing function. common points with y=x

$f:\mathbb{R}\to \mathbb{R}, ~ f'(x)<0 ~\forall x\in \mathbb{R}$. I have to prove that $f(x)=x$ has unique solution. $K(x)=f(x)-x$ is strictly decreasing $\implies K$ is 1-1 so $f(x)=x$ has at most one solution. But how do I prove that it has…
Tom Avram
  • 41
4
votes
1 answer

Junior olympiad question: Minimum value of 3 digit number divided by sum of its digits

I recently had a maths competition where we were given this problem. I solved the question, but I narrowed down the possibilities then did more of a guess and check method. I was hoping someone else could help me get the answer to this question…
Math4life
  • 341
4
votes
0 answers

How to solve this IMO competition problem?

Let $n>2$ be an integer. Let $f$ be a real-value function on a plane such that for every regular n-gon with vertices $A_1,A_2,...,A_n$, $f(A_1)+f(A_2)+...+f(A_n)=0$. Prove that $f$ is zero function. Putnam 2009 asked to prove this for the case of a…
Rikka
  • 890
4
votes
1 answer

Let $x_1,x_2,\dots ,x_{50}$ be $50$ integers such that the sum of any $6$ of them is 24, then:

Let $x_1,x_2,\dots,x_{50}$ be $50$ integers such that the sum of any $6$ of them is $24$, then which option is true the largest of $x_i$ equals $6$. the smallest of $x_i$ equals $3$. $x_{16}=x_{34}$. I am totally clueless how to proceed :( Help…
ViX28
  • 647
4
votes
1 answer

Problem PUTNAM of the day - Harvard Mathematics department

Let $f$ be a twice-differentiable real-valued function satisfying $f(x)+f''(x)= -xg(x)f'(x)$, where $g(x) \geq 0$ for all real $x$. Prove that $|f(x)|$ is bound. Honnestly I worked on this problem for a good while and I just don't know how to solve…
user1050421
  • 707
4
votes
1 answer

Is it possible to express $x^4-x^3+3x^2-4x+6$ as a product of polynomials of smaller degree with integer coefficients?

Is it possible to express $x^4-x^3+3x^2-4x+6$ as a product of polynomials of smaller degree with integer coefficients? My attempt: By equating the polynomial to $0$, one obtains $x=1\pm i, \pm\frac{1}{2}i(\sqrt{11}\mp i)$. From this, one can write…
Idonknow
  • 15,643
4
votes
1 answer

BM01 2008/09 Question 5 Sequences Problem

Determine the sequences $a_0 , a_1 , a_2 ,\dots$ which satisfy all of the following conditions: a) $a_{n+1} = 2a_n^2 − 1$ for every integer $n ≥ 0,$ b) $a_0$ is a rational number and c) $a_i =a_j$ for some $i,j$ with $i \neq j$. You can see it…
MadChickenMan
  • 793
3
votes
1 answer

finding the Prime numbers easily

I was doing some of the previous math contests and faced a question that asked me "the number of two digit primes that are still primes when the digits are reversed". I actually wrote down every two digit primes and then checked with the…
Free Spirit
  • 53
3
votes
4 answers

Show that there exists a convex hexagon in the plane such that (a) all its interior angles are equal,

Show that there exists a convex hexagon in the plane such that (a) all its interior angles are equal, (b) all its sides are 1, 2, 3, 4, 5, 6 in some order. it is the 9th question inmo 1993. i cant even start this question..
maths lover
  • 3,344
3
votes
1 answer

Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \ge \sqrt{(bc)^a(ac)^b(ab)^c}$

Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \ge \sqrt{(bc)^a(ac)^b(ab)^c}$ I tried it to do using $AM \ge GM$ but don't know how to proceed. Please help.
user2369284
  • 2,231
3
votes
1 answer

Newton's problem of cows and fields

I encountered this problem about Newton's problem of cows and fields: In a field, 17 cows can finish the whole grass in the field for 30 days. 19 cows can finish in 24 days. If a group of cows eat the grass for 6 days, then 4 cows are sold, the…
user71346
  • 4,171
3
votes
3 answers

Question from Spring 2012 AMATYC Student Mathematics League

How would you go about solving a problem like this: Let a, b, and c be positive integers which satisfy $a^3+b^3+c^2=2012$. Find $a+b+c.$ It doesn't appear that there's enough information to solve if, but test is multiple choice and the answer…
user99665
3
votes
2 answers

Contest math integer doublet equation

Can anyone help me with this? Find all ordered pairs $(x, y)$ of positive integers $x$, $y$ such that $$x^2 + 4y^2 = (2xy − 7)^2$$
user87611
3
votes
1 answer

Math contest proof equation problem

Could someone help me with this? If $m$ and $n$ are positive integers, then show that $$\frac{m}{ \sqrt n}+ \frac{m}{\sqrt[4]{n}} \neq 1$$.
user87611
3
votes
1 answer

What are the last five digits of $1+11+111+\cdots+\underbrace{11111...1}_{\text{$2002$ "1"s}}$?

Question 25 of the Australian Mathematics Competition, Junior Level, Year 2002: What are the last 5 digits of this sum? $$1+11+111+\cdots+\underbrace{11111...1}_{\text{$2002$ "1"s}}$$ Note, the last number a.k.a $11111...$ contains 2002 digits of…
Jonathan Xu
  • 373
1 2 3
…
31 32