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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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Split integer numbers from 1 to 31 into sets such that a maximum value in each set is the sum of the others.

This is a problem from mathematical contest for kids. Split integer numbers $1, 2,\ldots, 31$ into sets such that a maximum value in each set is the sum of the others, or show that it's impossible. My analysis so far has been that if we suppose that…
Andrey Surovtsev
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A circle with $n$ lattice points on it's boundary

Given $n$, is it possible to find a circle in the plane with exactly $n$ lattice points on it? I misread a problem that asked to show that the above has an affirmative answer with "$n$ lattice points on it" replaced with "$n$ lattice points…
mtheorylord
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Mock AMC 10 from AoPS

So, I've been having trouble with a problem from a Mock AMC 10. The question is as follows: $58$ candies are distributed among five boxes. Let $a_n$ be the number of candies in the $n$th box, starting from $1$. It is known that $3 \leq a_1 \leq 8$,…
asdf334
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An another Monovariant for IMO 1986(Problem–3)

IMO 1986 P3: To each vertex of a pentagon, we assign an integer $x_i$ with sum $s=\sum x_i>0$. If $x,y,z$ are numbers assigned to three successive vertices and if $y<0$, then we replace $(x,y,z)$ by $(x+y,-y,y+z)$. This step is repeated as long as…
user75659
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$4^x + 5^x = 6^x$

In a recent MathCounts Contest, the following question was asked If $x$ satisfies $4^x + 5^x= 6^x$ then find the greatest integer not greater than $x$ I tried to take logarithms of both sides, but then couldn't figure what to do with the $\log…
Anurag Saha
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Solving an equation with radicals in the exponent

Given the following equation:$$2^{x^{\frac{1}{12}}}+2^{x^{\frac{1}{4}}}=2\cdot2^{x^{\frac{1}{6}}}$$ I have to solve for $x$, I tried working a bit with logarithms but the plus sign is quite annoying so I didn't think that was the correct way, I…
Spasoje Durovic
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A question about Q1 from the International Mathematics Competition, 1994.

Let $A$ be an $n\times n$, symmetric, invertible matrix with real positive elements. Show that $z_n\leq n^2-2n$, where $z_n$ is the number of zero elements in $A^{-1}$. Let us prove this by contradiction. Let us suppose that $z_n\geq n^2-2n+1$.…
user67803
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A question about INMO 2017, Question 3

Find all triples $(x,a,b)$ where $x$ is a real number, and $a,b$ are integers belonging to $\{1,2,\dots,9\}$ such that $$x^2-a\{x\}+b=0$$ Here $\{x\}$ denotes the fractional part of $x$. My contention, which I know is wrong, is that the equation…
Anju George
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Find the numerical value of this expression

If $x$ is a complex number such that $x^2+x+1=0$, then the numerical value of $(x+\frac{1}{x})^2+(x^2+\frac{1}{x^2})^2+(x^3+\frac{1}{x^3})^2+\ldots+(x^{27}+\frac{1}{x^{27}})^{2}$ is equal to? A) 52 . B) 56 . C) 54. D)58 . E)None of…
nar
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Find all positive integers a,b,c,d such that $2019^a = b^3+c^3+d^3-5$

Working on a problem... Find all positive integers a,b,c,d such that $2019^a = b^3+c^3+d^3-5$ Using $(a+b+c)^3$ yields far too many terms, and I cannot come up with any solutions other than guessing and checking.
Hubert Lioni
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Difficulty understanding the following problem

Consider the following problem: Emma has three time switches. Every switch that is turned on allows an electric current to flow for three hours, then it blocks the current for three hours, allows it again for three hours, blocks it again for three…
MacropusRufus
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Let $p(x)$ be a real polynomial that is bounded below. Prove that there is a real number $x_0$ such that $p(x) ≥ p(x_0)$ for all $x$.

Let $p(x)$ be a real polynomial that is bounded below. Prove that there is a real number $x_0$ such that $p(x) ≥ p(x_0)$ for all $x$. This is listed on some practice problems for a contest math group I'm in. The question seems really easy but I'm…
IntegrateThis
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Undergraduate Math Competitions Advice

I am a freshman in college. Is there any advice for approaching undergraduate level math competitions despite a lack of extensive high school math competition experience? My participation in high school math competitions started to wane over the…
faelesterio
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Proving that $ f(1)=\frac{1-\sqrt{5}}{2}$ for this function

Let $f:(0,+\infty)\mapsto R$ be a strictly increasing function such that $\forall x\ge0,$ $$f(x)+\frac{1}{x}\ge0, \qquad f(x)f\left(f(x)+\frac{1}{x}\right)=1.$$ Show that $$f(1)=\frac{1-\sqrt{5}}{2}.$$ Please give an example that satisfies…
M.H
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Show that a given number has two identical digits(Kosovo TST 2011)

Starting with the number $7^{1996}$ we remove its first digit, and then add that digit to the rest of the number. This process continues until the result has ten digits. Show that the resulting number has two of its digits the same.
user123451241
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