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
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Minimum value of $f(x) = \sqrt{x^2 + (1-x)^2} + \sqrt{(1-x)^2 +(1+x)^2}$

If $f(x) = \sqrt{x^2 + (1-x)^2} + \sqrt{(1-x)^2 +(1+x)^2}$ Find the minimum value of the function I tried using the AMGM inequality and differentiation but didn't know how to solve it any ideas? This is from a math competition. ( I would like to…
bigfocalchord
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2015 AMC 10B Problem 21

The problem and solutions I've attempted to solve another AMC 10 problem, and the problem is basically like this: Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at…
space
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Olympiad problem algebra inequality

I'm having trouble solving the following inequality problem: If $n$ is positive integer greater than $1$, and $x>y>1$, then show that: $\frac{x^{n+1}-1}{x(x^{n-1}-1)} > \frac{y^{n+1}-1}{y(y^{n-1}-1)}$ Any hints? Thanks.
Michael
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Partial Order Question

Can someone explain the concept of partial order in relation to this problem? (Source: Putnam and Beyond) Problem: In some country all roads between cities are one-way and such that once you leave a city you cannot return to it again. Prove that…
Mat.S
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Partition a square

Compute the smallest positive integer $n$ such that， for any given integer $p\geq n$, we can partition a given square into $p$ number of squares (the small squares are not necessarily congruent) I think the answer is 4, clearly square can be…
nerd
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Problem from olympiad book by Arthur Engel (invariant problem)

There are $a$ white, $b$ black, and $c$ red chips on a table. In one step, you may choose two chips of different colors and replace them by a chip of the third color. If just one chip will remains at the end, its color will not depend on the…
Bluey
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Suppose entries of $A$ are either $-1$ or $1$, and $\det(A)=n!$. Find all positive numbers $n$ such that such matrix exists

A matrix $A$ is interesting if entries of $A$ are either $-1$ or $1$, and $\det(A)=n!$. Find all positive numbers $n$ such that there exists an interesting matrix of size $n \times n$. Claim: If matrix $A$ has determinant $n!$, then in its row…
Idonknow
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Prove that for any $f_1,f_2,...f_k \in I$, there exists a point $x_0 \in [a,b]$ such that $f_1(x_0)=...=f_k(x_0)=0$.

Let $C[a,b]$ be the ring of real-valued functions continuous on $[a,b]$ and let $I \subset C[a,b]$ be its proper ideal. Prove that for any $f_1,f_2,...f_k \in I$, there exists a point $x_0 \in [a,b]$ such that $f_1(x_0)=...=f_k(x_0)=0$. My attempt:…
Idonknow
  • 15,643
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Denote $y_n=\int_0^1{\frac{f^{n+1}(x)}{g^n(x)}}dx$ for all integer $n \geq 0$. Prove that $(y_n)_{n \geq 1}$ is an increasing and divergent sequence.

Let $f,g:[0,1] \rightarrow (0,\infty)$ be two distinct, continuous functions such that $$\int_0^1 f(x)dx=\int_0^1 g(x)dx$$ Denote $$y_n=\int_0^1{\frac{f^{n+1}(x)}{g^n(x)}}dx$$ for all integer $n \geq 0$. Prove that $(y_n)_{n \geq 1}$ is an…
Idonknow
  • 15,643
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2 answers

Find all functions $f(x)$ satisfying $f(x)+f^{\prime}(\pi-x)=1$ for all $x \in \mathbb{R}$.

Find all functions $f(x)$ satisfying $f(x)+f^{\prime}(\pi-x)=1$ for all $x \in \mathbb{R}$. This is a question from Moscow. I have tried $f(x)=x^m$ and it clearly does not work. Clearly $f(x)=1$ works. But I have no idea how to obtain remaining…
Idonknow
  • 15,643
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1 answer

End digit of numbers raised to a certain power

In a math competition I came across the following question: What digit does the result of 2^2006 end with? This competition tested how fast you are at solving math problems. So, I was wondering whether there is some sort of shortcut to solve…
anonymous
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Shuffling cards and laying them out in order

The numbers from 1 to 50 are printed on cards. The cards are shuffled and then laid out face up in 5 rows of 10 cards each. The cards in each row are rearranged to make them increase from left to right. The cards in each column are then…
1110101001
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2 answers

Forming Random Team and Choosing Pair of Friends

n participants of the competition were split into m teams in some manner so that each team has at least one participant. After the competition each pair of participants from the same team became friends. How will I find the minimum and the maximum…
Shahed Al Mamun
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Prove that if $abc\ne0$ and $ab+bc+ac=0$ then $a+b+c\ne0$

I tried to do proof by contradiction, but problem is how to get from $ab+bc+ac$ to $a+b+c$ Assuming $a+b+c=0$ my approachs: Adding $ab+ac+bc=0$ and $a+b+c=0$ and try to factor Deriving $$a^2+ab+ac=0\\ac+bc+c^2=0\\ab+b^2+bc=0$$ and trying to derive…
user182452
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How to write Putnam Examination proofs?

I am studying for the Putnam exam and I have learned that the graders are quite strict and will cut off points for a variety of reasons. I want to know exactly how to write a Putnam proof. How specific do I have to be? Can I skip obvious steps, or…
user1299784
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