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.

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Proof verification for $1989$ AIME $#13$(?)

Problem: Let $S\subset[1989]={1,2,...,1989}$. If two elements of $S$ have that $a-b\neq4$ or $7$, find max$|S|$. My solution is as follows: We first consider the first four integers $1,2,3,4$. Then if we add $4$ and $7$ to each of these numbers…
combinatorialist46Carey2
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Competition question - Proportion

I have this contest question that I don't quite get how to solve. At 10 am, the school flagpole cast a shadow 6m long. Next to the flagpole, the 0.5m high water tap cast a shadow 0.3m long. How tall is the flagpole in metres. I have tried using…
The Water Tap
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Mathcounts National 2014 Sprint #23

I have been attempting this problem for some time now and can't seem to figure it out. If someone could help I would be grateful. My attempt: $4b^2 + 4b + 1 = n^2$ $3b^2 + 5b + 1 = (n-2)^2$ $b^2 - b = n^2 - (n^2+4-4n) = 4n-4$ $b(b-1) =…
Smartsav10
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MATHCOUNTS NATIONAL 2014 #12

I don't know how to approach this. The correct answer is $2013$. What is the value of $$\frac{2013^3-2\cdot 2013^2\cdot 2014+3\cdot 2013\cdot 2014^2-2014^3+1}{2013\cdot 2014}\,?$$ I thought about using Binomial Theorem in some way as the format…
Smartsav10
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$m+n+p = 2017$ Solve for $(x,y,z,t)$ $x+my+nz+pt =2018$

$m, n,p \in \mathbb{N}^*$ such that $m+n+p = 2017$. Find the values of $x,y,z,t$ Such that $x+my+nz+pt =2018$ With $x,y,z,t \in \mathbb N^*$ My Attempt: Im not sure if this solution is correct because it looks easy to me. If we add a $1$ to the…
PNT
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Algebraic division of very large number and finding the remainder

Basically, I got this from a olympiad practice test and I still don't understand the logic of it, so the question asks us to find the remainder of the following: $$\frac{x^{74} + 23}{x + 1}$$ At first I thought I could just use the big brain…
AlexJaynMF
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If $a+b+2 \sqrt{ab}= 2(b-a)$, Prove that $\frac{b}{a}=9$

$a$ and $b$ are both Real numbers (different than $0$) I saw this problem in the math Olympiads of my home country What I’ve tried so far: Factoring $a+b+2 \sqrt{ab}$ to $(\sqrt{a} +\sqrt{b})^{2}$ Expanding the whole equation And thank you for your…
PNT
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$x^{2}+y^{2}=6xy$ find the value of: $\frac{x+y}{x-y}$

$x$ and $y$ are both positive real numbers : My attempt: $$x^{2}+y^{2}=6xy \Leftrightarrow x^{2}+y^{2}-2xy=4xy \Leftrightarrow (x-y)^{2}=4xy \Leftrightarrow (x-y)=\frac{4xy}{x-y} $$ A little note here: $x$ doesn’t equal to $y$. And now let’s plug…
PNT
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In each cell of a $7\times 7$ grid is a number. The sum of all the numbers in each $2\times 2$ square and each $3\times 3$ square is zero.

In each cell of a $7\times 7$ grid is a number. The sum of all the numbers in each $2\times 2$ square and each $3\times 3$ square is zero. Prove that the sum of the numbers in the 24 bordering cells is $0$ too.
user852377
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Find all $x,y,p \in \Bbb Z$ with $p$ being a prime and $x^2-3xy+p^2y^2=12p.$

Find all $x,y,p \in \Bbb Z$ with $p$ being a prime and $$x^2-3xy+p^2y^2=12p.$$ Looking at this mod $3$ one has that $$x^2+p^2y^2 \equiv0 \pmod{3}.$$ Since any square mod $3$ is either $0,1 $ we have that $x^2$ and $p^2y^2$ are both congruent to…
user713999
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Revisit 456 handshakes among 91 people

I am re-visiting the following problem which has been driving me crazy. Note: I checked the first few suggested similar mathSE queries, re handshakes, and nothing seemed on…
user2661923
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How many points inside a cube meet by lines? -- 2008 AMC Senior

All possible straight lines joining the vertices of a cube with mid-points of its edges are drawn. At how many points inside the cube do two or more of these lines meet? By symmetry, I can find 6 of these points inside a cube. The answer is 14.…
Oziter
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Find sum of all distinct numbers of the form 0.xyzxyzxyz

Given a number of the form $0.xyzxyzxyz$, where $x,y,z$ are distinct integers taking on values $\in \{0,1,2, \ldots, 9\}$, what is the sum, $S$, of all such numbers of the form $0.xyzxyzxyz$? Here is how I solved this problem. For each $x$, there…
24n8
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Math problem I’m struggling with writing down my reasoning for

The question is as such: At least how many stars should be drawn in a $4\times4$ table such that after eliminating two arbitrary columns and two arbitrary rows at least one star will remain in the table I know the answer is $7$ stars but i don’t…
user464840
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$a,b$ on the board, writing $a-b$ until someone can't write new numbers

On the occasion of the 47th Mathematical Olympiad 2016 the numbers 47 and 2016 are written on the blackboard. Alice and Bob play the following game: Alice begins and in turns they choose two numbers $a$ and $b$ with $a > b$ written on the…
Csaba Daniel Farkaš
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