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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A contest math problem involving work schedules and work rate

The problem is: It takes Danny 18 hours to complete a job alone, Ian 24 hours alone, and Roy 30 hours alone. If the three people work in the following seqence: Danny => Ian => Roy => Ian => Roy => Danny => Roy => Danny => Ian. Each person works for…
Cyh1368
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Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $yf(x)+xf(y)=(x+y)f(x^2+y^2)$

Question: Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $yf(x)+xf(y)=(x+y)f(x^2+y^2)$ for positive integers $x, y$. My attempt at a proof is the following: (1) As well if $x=y$ we have $$2xf(x)=2xf(2x^2)\Rightarrow…
TG173
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Is my proof for a 2018 Putnam problem correct?

Consider the following Putnam question: Consider a smooth function $f:\Bbb{R}\to\Bbb{R}$ such that $f\geq 0$, and $f(0)=0$ and $f(1)=1$. Prove that there exists a point $x$ and a positive integer $n$ such that $f^{(n)}(x)<0$. This is a problem…
user67803
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Certain values are not possible after a tax of 13% because of rounding find them

After applying a tax of 13%, certain values are impossible. Find those below $5.00$ For example$0.946$ would round to $0.95$ I know how to use a brute force approach but there has to be a better way. I know applying the tax is just multiplying our…
Dhdh
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Find the closest integer to $\frac{1}{\sqrt[6]{5^6+1}-\sqrt[6]{5^6-1}}$

This appeared in the Area Level of the 19th Philippine Mathematical Olympiad. Electronic calculators were not allowed during the competition of course. The closest I got to was to express it…
hpesoj626
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Prove that the area of the triangle formed by the medians is equal to $3/4$ the area of the original triangle

This is Question $2$ from this document on Olympiad Geometry. Let $ABC$ be a triangle and $M_A,M_B,M_C$ the midpoints of the sides $BC, CA, AB$, respectively. Show that the triangle with side lengths $AM_A, BM_B, CM_C$ has area $3/4$ that of the…
Anju George
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Find the sum of $x_1+x_2+x_3$ of intercept points

Suppose that the straight line $L$ meets the curve $y=3x^3-15x^2+7x-8$ in three points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$. Then $x_1+x_2+x_3=?$ A) 3 $\quad$ B) 4 $\quad$ C) 5 $\quad$ D) 6 $\quad$ E) 7 At the beginning, my…
nar
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How to find $|a|\cdot|b|\cdot|c|$ if $a+\frac 1b=b+\frac 1c=c+\frac 1a$

The question is from the Bangladesh Math Olympiad- Barisal division from $2017$. $a$, $b$ and $c$ are three separate real numbers, where $a+\frac 1b=b+\frac 1c=c+\frac 1a$. What is the product of the absolute values of $a$, $b$ and $c$? If they…
For the love of maths
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Help in understanding a statement in the solution

I have the following problem from USAMO 2006: "A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n+2^{m_n+1}$…
saisanjeev
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Reference to a theorem of T. Nagell

I have the following problem from USAMO 2006: "For an integer $m$, let $p(m)$ be the greatest prime divisor of $m$. By convention, we set $p(\pm 1)=1$ and $p(0)=\infty$. Find all polynomials $f$ with integer coefficients such that the sequence $\{…
saisanjeev
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If every point of the three dimensional space is coloured red, green or blue prove that one colour attains all distances

The problem as stated in the title is taken from "Putnam and beyond". Below is the problem as stated in the book: Every point of the three-dimensional space is coloured red, green, or blue. Prove that one of the colours attains all distances,…
user489116
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2013 AIME I #6: Empty Boxes and Books

Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her…
Dude156
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How many solutions of $a+b\sin t= t^n$ on $(0,+\infty)$

When I was in secondary school, I saw a question how many intersection of $\sin t$ and $t^n$ on $(0,+\infty)$, where $n$ is positive interger. It is not hard to answer. But general this question, how many solutions of $$ a+b\sin t= t^n $$ where…
Enhao Lan
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The 4 digit base $6$ number $abcd$ with $a>0$ and $d$ odd is a perfect square.

The 4 digit base $6$ number $abcd$ with $a>0$ and $d$ odd is a perfect square. List all possible values of $c$. (The letters are the digits of the base 6 number.) I've rewritten $abcd_6$ into $216a + 36b + 6c + d$, and I know that $d$ can be…
space
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Value of $AB^{-1}$ Given $A,B$ are in square root expression

If $\displaystyle A = \sum^{n^2-1}_{k=1}\sqrt{\sqrt{2n}+\sqrt{n+\sqrt{k}}}$ and $\displaystyle B = \sum^{n^2-1}_{k=1}\sqrt{\sqrt{2n}-\sqrt{n+\sqrt{k}}}$, Then $\displaystyle A\cdot B^{-1} = $ Try: Iam trying to convert $A+B$ into $A$ or in $B$…
DXT
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