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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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Divisiblity of Algebraic Expressions

I was solving this question, but I'm hitting a wall. Find all pairs of $(x,y)$ such that $(2x+7y)|(7x+2y)$ Here is what I have done. Since we have two variables, this question has either a small number of solutions, or all solutions are of the…
DynamoBlaze
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Question on Functions and Squares.

I was solving this question, and I'm hitting a wall. Let $S_n=n^2+20n+12$, ${{n}\in{\mathbb{N}}}$. What is the sum of all possible values of $n$ for which $S_n$ is a perfect square? Here is how I have tried to…
DynamoBlaze
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There are $2^{3463}-1$ non-empty subsets to...

I'm clueless as to what this problem even wants me to do. Set theory is not my strength in math. Problem: There are $2^{3463}-1$ non-empty subsets to $\{\frac{1}{1},\frac{1}{2},\frac{1}{3},...,\frac{1}{3463}\}$. For each such subset, form the…
Parseval
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Induction. Circular track and fuel stations

The sides of a circular track contain a sequence of cans of gasoline. The total amount in the cans is sufficient to enable a certain car to make one complete circuit of the track, and it could all fit into the car's gas tank at one time. Use…
Parseval
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Direct formula for a variation of Josephus problem:

What is the direct formula for the following variation of Josephus problem? There are $n$ persons, numbered $1$ to $n$, around a circle.. Starting from $k$-th person, every second person is eliminated. Given the $n$ and $k$, determine the index of…
user7262
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Need Help Logical Question

There are three small tanks of capacity 35 L, 56 L, 84 L. Lets Find what will be the biggest capacity of a container which will measure the oil in 3 tanks in exact whole numbers. Ans=71. Please Provide Solution and Explain.
Pallab
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How many kittens at least to take

You have $7$ boxes in front of you and $140$ kittens are sitting side-by-side inside the boxes, $20$ in each box. You want to take some kittens as your pets. However the kittens are very cowardly. Each time you chose a kitten from a box, the kittens…
Rangan Aryan
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Find the range of values in a geometric progression for which $g_1< 0$ and $g_3> 4g_2 -3g_1$

Here $g_1,g_2$ and $g_3$ are three successive terms of a Geometric Progression. I assumed them to be $a/r, a$ and $ar$ and used $B^2 = A.C$ but proved futile...Any trick?
Tanishka Marrott
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Problems involving invariants

I was procrastinating here (:O) and found this question I liked - If you start with $\{3,4,12\}$, and at each step replace any two numbers $a,b$ in the set by $\frac{3a}{5}+\frac{4b}{5}$ and $\frac{4a}{5}-\frac{3b}{5}$, can you reach $\{4,6,12\}$ in…
ShakesBeer
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Which version of this question is right?

Find digits $x,y,z$ such that the equality $$\sqrt{\smash[b]{\underbrace{\overline{xx\cdots x}}_\text{$2n$}}-\smash[b]{\underbrace{\overline{yy\cdots y}}_\text{$n$}}} = \overline{\underbrace{zz\cdots z}_{n}}$$ holds for at least two values of $n…
user19405892
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Find closed form of $f(a,b,c)$

Let $$f(a,b,c)=\left|\dfrac{|b-a|}{|ab|}+\dfrac{b+a}{ab}-2c\right|+\dfrac{|b-a|}{|ab|}+\dfrac{b+a}{ab}+\dfrac{2}{c}.$$ Find closed form to $f$.
Raheem Najib
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How to measure the length of train $B$ given that trains $A,B$ are moving and that $A$ is $x$ feet long.

Train $A$ is $x$ feet long and is going east at $r_1$ mph. On a parallel track going west is train $B$ going at $r_2$ mph. f the trains take $y$ seconds to pass each other completely, how many feet long is train $B$ ? To approach this…
Idontknow
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Confused by a step in a solution to the problem

I'm pretty confused by the step $$ \prod_{n=1}^{45}\sin(2n^\circ)=\sum_{n=1}^{45}\frac{\omega^n-1}{2i\omega^{n/2}} $$ in the official solution of this problem from 2010 PUMaC Algebra A7: The expression…
wwwrqnojcm
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Probability - A random point dividing a square into $4$ parts

A point P is chosen randomly in a square. Join P with the four vertices of the square so as to divide the square into four triangles. Find, correct to 2 decimal places, the probability that all interior angles of the four triangles do not…
Nighty
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Inequality about unit vectors

$2n$ unit vectors with non-negative $y$-coordinates are given. Prove that if the sum of $x$-coordinates is an odd integer, then the sum of $y$-coordinates is at least $1$. $n=1$ case is easy; I also succeeded to prove $n=2$ case, but I cannot…
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