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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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Help with distance question points A and B

Ok. I had no idea how to do the question but I tried fiddling with the triangles to see if I can get any value but only managed to get $MN$. I read the solution to this question, and it said that I should make $M$ coincide with $N$ so that we get…
Aspiring Mathlete
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What is the minimal number of weighings required to find an odd (lighter) coin out of 80?

I have $80$ coins. Among them, exactly one coin is lighter compared to all the others. I was given a physical balance, suddenly. What is the minimal number of weighings required to find the lighter coin? Can somebody tell me what is the meaning of…
Android developer
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What does the question mean by suppressing some terms?

I don’t really understand what it means by suppressing some terms. Can you provide me with an example?
Chanhyuk Park
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Given a sequence of integers whose sum is 1, prove that exactly one of the cyclic shifts has all of its partial sums positive.

I tried to understand the solution from the book for this question, but I am having hard time to understand it. How do we know that there exist two terms $x_{j}$ and $x_{j+1}$ such that $x_{j}\gt0$,$x_{j+1}\le0$ by taking indices modulo $k+1$?
Chanhyuk Park
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Help understanding two conflicting solutions

Here's the question from the 2014 CEMC Fermat contest, Q23 I added some annotations to the diagram I labelled the two points T and S and $\overline{TS}$ = $x$ Since $$\triangle PQR \sim \triangle PTS \text{ and } QR = PR = 125$$ $$\therefore TS = PS…
John Qu
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Simplifying the square root

The question is "Find the exact value of $\sqrt{97+56\sqrt{3}}$ ". It's from some regional contest back in 2013 and the answer is $7+4\sqrt{3}$. Can someone explain how they can reach the answer other than bashing in perfect squares for 97? I know…
Algi King
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$a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n}), a_1=1$, find $a_{1995}$ (craft)

I understand the solution of $a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n}), a_1=1$, find $ a_{1995}$ and I was able to derive it myself, however, in my first attempts I conjectured something very different, and I don't see the problem in my…
Itimethy
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Seeking advice on attempted Putnam problem

Way back I watched a video by 3b1b on the tetrahedron in a sphere Putnam problem and tried to solve it on my own (having forgotten the entire video). Having no experience with the Putnam, and only an easy discrete math class in terms of…
Kefir
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Suppose a and b are real numbers such that $ab^{2}=1$ and $a^{3}+3b^{3}=4$. What is the product of all possible values of $a^{3}+b^{3}$?

2019 MathCon Finals Grade 11 Part C I've been struggling with this one for quite a while, and I couldn't find any answers on the internet. If anyone can help, it would be greatly appreciated. Thank you
RandyMC
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The maximum value of the expression $x^4y^3$

If $x$ and $y$ are positive numbers, then what is the maximum value of the expression $x^4y^3$ where $4x + 6y = 28$? One way to solve this question is to find the expression of $x$ or $y$ in terms of each other from the equation given and then put…
Ganit
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Find the remainder when $11^{2013}$ is divided by $ 61$

How do I find the remainder when $11^{2013}$ is divided by $61$? Brute force? Without a calculator? How did people do that?
Commander Shepard
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Find the missing number in the circle

Some days ago I noticed one simple task from the first point of view. It's from mathematical tournament for 11-12 years old. I don't remember exact description, but I'm sure the answer has to be an integer. I found only the following dependency…
user2700840
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For $\frac{16x-3}{x^3+x} = \frac{bx+c}{x^2+1}+\frac{a}{x}$, what is a+b+c?

I only got to getting rid of the denominator and turning the equation into 16x-3 = ax^2+a+bx^2+cx, but from that on I don't know what to do.
Asha R
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Drain pipes fitted at different heights : Time and work problem

There are three drain pipes $Q1$, $Q2$ and $Q3$, all of equal capacity, fitted to a tank of height $6$ meters. The tank is in the shape of a cuboid. $Q1$ is fitted at the bottom of the tank, while $Q3$ is fitted at a height of $4 m$ above $Q1$, and…
Ganit
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How many positive integers $k$ between $1$ and $100$ are such that $\frac{7}{k+7}$ have the form $\sum_{i=0}^n\frac{a_i}{15^i}$ with $a_i\in\Bbb{I}$?

Let a fraction that can be expressed by the sum $\;\sum_{i=0}^n\frac{a_i}{15^i}$, where $a_i \in \Bbb I$ is a $15$-based fraction. Then find how many positive integer $k$ between $1$ and $100$ are there such that $\frac{7}{k+7}$ is a $15$-based…
Monai
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