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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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Number of People Required (Arena Survival Question)

Slaves are fighting in an arena. Each fight involves two slaves and results in one winner and one loser. Two slaves may fight one another only if they have an identical number of wins. A slave with $3$ losses will be kicked out of the arena. A…
jian ma
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"Stairstep Numbers"

I've been preparing for Mathcounts competition, but this one question confused me a bit. If a stairstep number is defined as a number whose digits are strictly increasing in value from left to right, how many positive integers containing two or…
space
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Math competition for school

I am trying to find a math competition where a 10 year old kid can participate. Can someone suggest a competition in USA?
user39359
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Left handed or Right handed?

Happy New Year! The following question is abstracted from Singapore Mathematical Olympiad 2015 Junior Round 1. Question 2: Adrian, Billy, Christopher, David and Eric are the five starters of a school’s basketball team. Two among the five shoot with…
ministic2001
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Pigeonhole Technical Problem

Problem (from Putnam and Beyond pg. 11) is: Given a set $M$ of 1985 distinct positive integers, none of which has a prime divisor greater than $26$, prove that $M$ contains at least one subset of four distinct elements whose product is the…
Mat.S
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An Olympiad problem about the solution of equations

There are positive real numbers $a_1, \dots ,a_{2013}$, that satisfy: equations $x_{k-1}-2x_{k}+x_{k+1}+a_{k}x_{k}=0(1 \le k \le 2013)$ have a solution $(x_0,x_1,\dots ,x_{2014})(x_0=x_{2014}=0)$ which is not all zeros. Prove that $a_1+\dots…
user263834
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2 answers

Can the expression be made true after replacing the blanks in this sum with 2 numbers from 0-9 and 2 symbols?

The expression is $$ 2\_\,\_\,\_\,\_5=2015 $$ you have to replace 2 of the blanks with digits (0-9), and the other 2 with one of the operations $+- \times \div$. Is it possible to make the expression true?
Erikah Wallace
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Do there exist continuous functions $f,g: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(g(x))=x^2$ and $g(f(x))=x^3$ for all $x \in \mathbb{R}$?

Do there exist continuous functions $f,g: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(g(x))=x^2$ and $g(f(x))=x^3$ for all $x \in \mathbb{R}$? My attempt: Since $x^3$ is a bijection, we have $f$ is injective and $g$ is surjective. Then I don't…
Idonknow
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Suppose a continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ is nowhere monotone. Show that there exists a local minimum for each interval.

Suppose a continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ is nowhere monotone. Show that there exists a local minimum for each interval. This question is from Moscow institute. First of all, I can't even construct a nowhere monotone…
Idonknow
  • 15,643
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BMO1 2009/10 Problem 6

Long John Silverman has captured a treasure map from Adam McBones. Adam has buried the treasure at the point $(x,y)$ with integer co-ordinates (not necessarily positive). He has indicated on the map the values of $x^2 + y$ and $x + y^2$, and these…
MadChickenMan
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Find coefficients of a quadratic by tow points and mimimum value [Monbukagakusho exam 2010]

So I was trying to solve the monbukagakusho maths exam from the year of 2010 that you can find in this link, but I can't understand the questions, I don't know what do they actually want. Here's one of the questions: The quadratic function which…
Yuran Pereira
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Question about infinite descent method in this proof

How do we conclude that:"We have therefore constructed another pair $(a_1, b_1)$ in $S(c, k)$ with $a_1 + b_1 < a + b$. However, $S(c, k)$ is contain in $Z^+ × Z^+$, so using the argument of infinite descent, we obtain our desired contradiction." in…
Victor
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If $x^2+bx+a=0$ and $x^2+ax+b=0$ have a common root $c$, Then what values of $(a,b)$ would work?

Let $a$ and $b$ be distinct integers. If $x^2+bx+a=0$ and $x^2+ax+b=0$ have a common root $c$, Then which of the following statements are true? 1) $c*(a+c)=-b$ 2) $a+b=-1$ 3) $a+b+c=0$ 4) $c=0$ Update I just tried to sub $c$ into both of the…
user2574069
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i do not how to prove this degenerate polygon

A polygon is called degenerate if one of its vertices falls on a line that joins its neighboring two vertices. In a pentagon ABCDE, AB = AE, BC = DE. P and Q are midpoints of AE and AB. PQ||CD, BD is perpendicular on both AB and DE. Prove that ABCDE…
Ahmed Rafayat
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How Find $3x^3+4y^3=7,4x^4+3y^4=16$

if postive real number $x,y$ such $$\begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases}$$ Find $x+y=?$ My try: $$4x^4-3x^3+3y^4-4y^3=9$$ $$x^3(4x-3)+y^3(3y-4)=9$$
user94270
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