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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Expected Value of Binary Numbers

A 10-digit binary number with four $1$s are chosen at random. What is its expected value? Here is what I thought to do: There are $_{10}C_4$ possible digits that could be filled with $1$. Since there are $10$ digits, that means that each digit…
Dude156
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Using algebraic identities to prove a number is not prime

$$3^{3^n}(3^{3^n}+1)+3^{3^n +1}-1$$ I want the prove that the number is not prime. I used the identity $a^3+b^3+c^3-3abc $. I couldn't simplify to the state where the factors could be observed.
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2008 AIME I Problem 6: Pascal's triangle?

A triangular array of numbers has a first row consisting of the odd integers $1,3,5,\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row…
Dude156
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2012 AIME II #8: Complex Problem

When attempting to solve this question, I multiplied the equations to extrapolate the zw. Once I did this though, I ran into some confusion. Am I allowed to use the quadratic formula here? If I am, can someone demonstrate how to solve the part…
Dude156
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2014 AIME II: Cubic Question

Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$. I assumed that the other root of p(x) would be t. From here, I used Vieta's formulas to get…
Dude156
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AIME Parabola Question

The graphs $y=3(x−h)^2+j$ and $y=2(x−h)^2+k$ have y-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer x-intercepts. Find $h$. The answer is an integer between 1 and 999. I substituted $0$ for $x$ in both…
Dude156
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Knights and liars in island

There are liars and knights in some island (liars always lie and knights always say true). $500$ people were built in the form of a rectangle $20\times 25$ ($20$ people in column and $25$ in row). During the poll, everyone stated: 1) If you do not…
RFZ
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Problem on Invariances (Arthur Engel)

In a regular pentagon, all diagonals are drawn. Initially, each vertex and each point of intersection of the diagonals is labeled by the number $1$. In one step it is permitted to change the signs of all numbers of a side or diagonal. Is it possible…
Student
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Starting with $35$ integers you may select $23$ of them and add $1$ to each. By repeating this step, one can make all $35$ integers equal. Prove this.

Problem: Starting with $35$ integers you may select $23$ of them and add $1$ to each. By repeating this step, one can make all $35$ integers equal. Prove this. My Attempt: I know that we have to find an invariant. And I think that the sum of the…
Student
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handshakes with everyone except 1

Jim and his wife Jeri attend a party with 4 other married couples. As they enter, Jim and Jeri shake hands with some of the guests, but not with each other. During the evening, each person except 1 shakes hands with some of the guests, but not with…
space
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Miscellaneous Olympiad Problem

Two brothers sold a herd of sheep which they owned. for each sheep, they sold they received as many rubles as the number of sheep originally in the herd. The money was then divided in the following manner. First, the older brother took ten rubles…
Prakhar Nagpal
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Finding the kth number when groups of numbers are listed such that they add up to a specific sum

A toy set contains blocks showing the numbers from 1 to 9. There are plenty of blocks showing each number and blocks showing the same number are indistinguishable. We want to examine the number of different ways of arranging the blocks in a sequence…
Ben Ritmico
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Venn diagram Question involving maximum and minimum values

A town has 2017 houses. Of these 2017 houses, 1820 have a dog, 1651 have a cat, and 1182 have a turtle. If x is the largest possible number of houses that have a dog, a cat, and a turtle, and y is the smallest possible number of houses that have a…
Derive Foiler
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Filling large and small jugs of water from 50L

This is Question 21 from the AMC Junior Division 2016: Angelo has a $50$ litre barrel of water and two sizes of jug to fill, large and small. Each jug, when full, holds a certain amount of litres. Angelo fills three large jugs, but does not have…
bio
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Rigorousness of USA(J)MO level proofs

I was just wondering how rigorous a USA(J)MO proof has to be to receive full credit. If a problem can be explained by logically extending an argument, will it earn as many points as a proof by induction explaining essentially the same thing? For…
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