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
41
votes
2 answers

Prove that there is a real number $a$ such that $\frac{1}{3} \leq \{ a^n \} \leq \frac{2}{3}$ for all $n=1,2,3,...$

Prove that there is a real number $a$ such that $\frac{1}{3} \leq \{ a^n \} \leq \frac{2}{3}$ for all $n=1,2,3,...$ Here, $\{ x \}$ denotes the fractional part of $x$. My attempt: Clearly $a$ cannot be an integer because $\{ a^n \}=0$ for all $n \in…
Idonknow
  • 15,643
28
votes
3 answers

Find the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 2012.$

Find the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 2012.$ I just know $x^{x^x}$ is increasing in $x$ and hence the equation has a unique solution, nut then I dont know how to move on, I also know…
amy tsang
  • 281
24
votes
4 answers

The number $2^{29}$ has exactly $9$ distinct digits. Which digit is missing?

The number $2^{29}$ has exactly $9$ distinct digits. Which digit is missing? I came across this question in a math competition and I am looking for how to solve this question without working it out manually. Thanks.
snivysteel
  • 1,104
11
votes
4 answers

Find coefficient of $x^8$ in $(1-2x+3x^2-4x^3+5x^4-6x^5+7x^6)^6$

Find coefficient of $x^8$ in $(1-2x+3x^2-4x^3+5x^4-6x^5+7x^6)^6$ how to do it? I think it should be $3^6$ since $(3x^2)^6=3^6x^8$. (this is false) Is this true?
user117437
  • 111
11
votes
3 answers

What value will take $f(100)$?

Let $f$ be a function from the positive integers into the positive integers and which satisfies $f(n + 1) > f(n)$ and $f(f(n)) = (f \circ f) (n) = 3n$ for all $n$. Find $f(100)$. This is one of the questions we presented in one session to contest…
user230283
10
votes
4 answers

$a^5+b^5+c^5+d^5=32$ if and only if one of $a,b,c,d$ is $2$ and others are zero.

Let $a,b,c$ and $d$ be real numbers such that $a^4+b^4+c^4+d^4=16$. Then $a^5+b^5+c^5+d^5=32$ if and only if one of $a,b,c,d$ is $2$ and others are zero. Why does this hold?
Silent
  • 6,520
10
votes
1 answer

For which $n$ does $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ imply $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}$

I'm having trouble finishing a problem on an old national competition. As the title states, the question says asks: Given $a,b,c \neq 0,a+b=c$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$, Find all integers $n$ such that…
user242594
9
votes
8 answers

Favorite Math Competition Problems

I'm running a weekly math contest for a summer camp and would like to compile a list of interesting problems. The problems may presuppose mathematical knowledge up to but not including Calculus. For my purposes, a good problem is one that emphasizes…
J. D. Null
  • 39
9
votes
2 answers

Digit-Sum of a Number.

I got a question recently, and I am unable to solve it. Find all natural numbers $N$, With sum of digits $S(N)$, where $N=2\{S(N)\}^2$ I know that $9|N-S(N)$, and since N is twice a square, it must end in $0,2,8$. But I do not know where to go…
DynamoBlaze
  • 2,781
8
votes
2 answers

Online math contests

I know from my friends who major in CS that there are many reputable online CS constests. Can you give me examples of reputable online math contests ? It would be better if they are for undergraduates.
user74194
  • 11
8
votes
2 answers

USAMTS $4/2/17$ Duck Goose Goose Problem

This is the problem on USAMTS: A teacher plays the game “Duck-Goose-Goose” with his class. The game is played as follows: All the students stand in a circle and the teacher walks around the circle. As he passes each student, he taps the student…
Jason Kim
  • 902
8
votes
2 answers

Olympiad Mathematical Kosovo 2012 (Problem grade 9)

Let $a_{1},a_{2},a_{3},\ldots,a_{2011},a_{2012}$ be integers. Exactly 29 of them are divisible by 3. Show that $a_{1}^2+a_{2}^2+a_{3}^2+\ldots+a_{2011}^2+a_{2012}^2$ is also divisible by 3.
user123451241
  • 633
8
votes
1 answer

Contest math problem proof

This is problem 23.2.8.4 from 66 MOSCOW MATHEMATICAL OLYMPIADS (author: N. Konstantinov). "A snail crawls along a straight line, always forward, at a variable speed. Several observers in succession follow its movements during $6$ minutes. Each…
Sal.Cognato
  • 1,527
7
votes
4 answers

Math Olympiad Algebraic Question Comprising Square Roots

If $m$ and $n$ are positive real numbers satisfying the equation $$m+4\sqrt{mn}-2\sqrt{m}-4\sqrt{n}+4n=3$$ find the value of $$\frac{\sqrt{m}+2\sqrt{n}+2014}{4-\sqrt{m}-2\sqrt{n}}$$ I came across this question in a Math Olympiad Competition and…
snivysteel
  • 1,104
7
votes
4 answers

Math Olympiad Algebraic Question

If both $n$ and $ \sqrt{n^2+204n} $ are positive integers, find the maximum value of $n$. I came across this question during a Math Olympiad Competition. I need help with solving the question. Thanks.
snivysteel
  • 1,104
1
2 3
…
31 32