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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How am I supposed to find the area of the shaded quadrilateral?

In the diagram (which is not drawn to scale) the small triangles each have the area shown. Find the area of the shaded quadrilateral.
user705240
3
votes
2 answers

A number theory problem

Let $S$ be a set of real numbers satisfying the following conditions: i. $0$ is in $S$. ii. Whenever $x$ is in $S$ then $2^x+3^x$ is in S. iii. Whenever $x^2+x^3$ is in $S$ then $x$ is in $S$. How can I prove that $S$ contains at least two distinct…
Somebody
  • 313
3
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1 answer

Don't know where to use the hypothesis

Let a be a real number such that $|a| > 2 $. Prove that if $a^{4}-4a^{2}+2$ and $a^5-5a^3+5a$ are rational numbers, then $a$ is a rational number as well. My attempt is the following. $a^5-5a^3+5a = a(a^{4}-4a^{2}+2 -a^2 +3)$. Which can be written…
user404735
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  • 10
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votes
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Spiders and beetle on a dodecahedron.

This was a question from this year's Simon Marais Mathematics Competition. I myself had no idea how to solve it nor did I find anyone who solved it.Any ideas please? Three spiders try to catch a beetle in a game. They are all initially positioned…
Reader Manifold
  • 1,297
3
votes
1 answer

MATHCOUNTS States Team Problem #9 2018 (Sum of Terms are Squares)

I've made it to states this year, but I want to know how to solve this problem. I already know since the difference between consecutive squares are changing by $2,$ I should be doing something similar in my sequence. If a certain sequence $a_1,…
Jason Kim
  • 902
3
votes
2 answers

Olympiad Mathematical of Kosovo 2011 (Problem grade 9)

A little boy wrote the numbers $1,2,3,...,2011$ on a blackboard. He picks any two numbers $x,y$ , erases them with a sponge and writes the number $ |x-y |$. This process continues until only one number is left. Prove that the number left is even
user123451241
  • 633
3
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1 answer

stars and bars application

The sum of the digits in 2017 is 10. How many numbers from 2000 to 3000 also have this property, including 2017? I used the stars and bars. The first digit (thousands digit) must be 2 (it's between 2000 and 3000), and so the rest of the 3 digits…
space
  • 4,561
3
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2 answers

On solving Double-Powers

I was solving this question, and I'm hitting a wall. $x^{x^4}=4$, then what is ${x^{x^8}}+{x^{x^2}}$? Taking $\log$, $x^4\log\{x\}=\log\{4\}$, so $1
DynamoBlaze
  • 2,781
3
votes
3 answers

Find all the solutions of $(a+1/b)(b+1/a)=4$

I tried to find out some solutions to this equation but I found that it doesn't have any solution, but I couldn't find an elegant solution for this.
Luis Martínez
  • 59
3
votes
3 answers

Maximum chocolate in a square grid

There is a $625 \times 625$ square grid. We have to place chocolates in such a way so that each row and each column contains maximum of $3$ chocolates. What is the maximum number of chocolates we can put in that grid in total?
Rangan Aryan
  • 399
3
votes
2 answers

Putnam 2016 B1 Short (and incorrect) "answer"

I took the Putnam exam for the first time on December, and I would like to discuss my (incorrect) answer for B1, which states: Let $x_0, x_1, x_2,...$ be the sequence such that $x_0=1$ and for $n \geq 0$, $$x_{n+1}=\ln (e^{x_n}-x_n)$$ Show…
Squirrel-Power
  • 3,638
3
votes
4 answers

What is the minimum value of $\dfrac {9x^2\sin^2 x+4}{x\sin x}$

Question: What is the minimum value of $$\dfrac {9x^2\sin^2x+4}{x\sin x}\tag1$$For $0
Frank
  • 5,984
3
votes
1 answer

Transforming $(2,2,2)$ into $(17,1967,1983)$

I have not dealt with problems of this type: Three integers $a$, $b$, and $c$ are written on a blackboard. Then one of them is erased and replaced by the sum of the other two diminished by 1. This operation is repeated a finite number of times until…
user37450
3
votes
2 answers

The sum of the series $1 + 2x + 3x^2 ...$

My approach : I tried using integral calculus and using infinite geometric series..however it didn't match..any trick?
Tanishka Marrott
  • 189
3
votes
3 answers

Choose an infinite GP from given terms for a particular sum

Is it possible to choose an infinite GP from amongst the terms 1, 1/2, 1/4, 1/8, 1/16 ... with a sum a) 1/5? b) 1/7 ? My approach was simply choosing terms and probable ratios to match the sum...I would be grateful if you could reveal any trick for…
Tanishka Marrott
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