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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.

9758 questions
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How can a junior in high school start preparing for the Putnam?

I am a junior in high school and I have a keen interest in math. I've already completed AP Calculus BC (which is equivalent to Calculus II in college) and am self-studying topics like multivariable calculus and linear algebra. I'm probably going to…
Vincent Han
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Are 4 chapters enough to score good in IMO.

I recently started reading Arthur Engel's "Problem Solving Strategies" and it's first three chapters blew my mind. Most of the time in standard maths olymiads like IMO have a majority of questions based on Invariance, Coloring proofs and Extremal…
user676047
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Ratio of 9th grade students to 10th grade students

I've been pondering on this question and I realize my understanding of percentages are probably weak so please help guide me through. Let's say 9th grade is 70% girls and tenth grade is 30% girls. If 40% of all ninth and tenth grade students are…
Star S
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Why is that particular kind of sequence relevant to the question not repeat

The following is a question from the IMO 2012 shortlist: Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x > y$ and $x$ is to the left of $y$, and replaces the pair $(x, y)$…
saisanjeev
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Find the smallest real number m for which there exist real numbers $a;b$ such that $|x^2+ax+b|\le m(x^2+1)$ for $x\in [-1;1]$

we can easily see that $m\ge 0$ then it is obvious that $m\neq 0$, because if $m=0$ the equation $x^2+ax+b=0$ will have infinitely many solutions then I have proved that $m=\frac{1}{3}$ always works but I don't if it's the smallest real number
user600785
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value of $f(2008)$ in $4$ th degree polynomial

If $f(x)$ is a $4$ th degree polynomual such that $f(2003)=24, f(2004)=-6, f(2005)=4,f(2006)=-6,f(2007)=24$ Then value of $f(2008)$ is what i try assuming $f(x)=ax^4+bx^3+cx^2+dx+e\cdots \cdots (1)$. putting $x=2003,2004,2005,2006,2007$ in…
jacky
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If $abc\neq 0$, then at least one of $ax^2+2bx+c$, $bx^2+2cx+a$, $cx^2+2ax+b$ has root

Prove that if $abc\neq 0$ then at least one of the equations $ax^2+2bx+c$, $bx^2+2cx+a$, $cx^2+2ax+b$ has root. Source: All-Russian Math Olympiad 1994. My sketch of proof: The condition $abc\neq 0$ is equivalent to $a,b,c\neq 0$. Suppose that none…
RFZ
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Find the largest real number r such that r[r] = 2019{r}

[r] denotes the greatest integer less than or equal to r {r} denotes the fractional part of r This problem is tripping me up. Obviously r has to be some sort of decimal, or else the fractional part of it would be 0. The fact that [r] is the floor of…
Matt Mcstoffel
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AMC 12 2018 A Question 2

While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $14$ each, $4$-pound rocks worth $11$ each, and $1$-pound rocks worth $2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the…
space
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$2017-(a+b+c+d)=n$ where $n=abcd_{(10)}$ (Local Math Olympiad Qualification Exam Question)

The problem text states: "'Person A' was born in 1962, in 1980 he turned 18 which is the sum of the numbers in his birth year (1+9+6+2). 'Person B' has a younger brother 'Person C', in 2017 the age of each one of them was equal to the sum of the…
Elhamer Yacine
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Find the least positive real number $k$ such that $7\sqrt{a} + 17\sqrt{b} + k\sqrt{c} \ge \sqrt{2019}$ over all positive real numbers

Working on a problem... Find the least positive real number $k$ such that $7\sqrt{a} + 17\sqrt{b} + k\sqrt{c} \ge \sqrt{2019}$ over all positive real numbers $a,b,c$ with $a+b+c=1$. Maximizing the "$a$" term doesn't seem to work, and expansion…
Hubert Lioni
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2018 AMC 12A-Cyclic Quadrilaterals

Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\overline{BC}$. Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic…
user501887
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frequencies comparison

I have a rather quiz question (sorry if this a wrong stack to ask such questions). A propeller with 3 blades makes exactly 24 spins in 1 second. Camera, that is filming it, takes 54 frames in 1 second. How many photos that differ from one another…
Bogdan
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What would be the another possible answer?

It is easy to guess that 51 is the missing number. The entries in the last row are obtained by taking multiplication of entries in third and second row and subtracting entries in first row in the same column. Is another idea possible to get another…
user61681
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