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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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Math contest question - prove unsolvability of equations.

Prove that the system of equations has no real solutions: $$\begin{cases} y=\sqrt{x+\sqrt{1-x}} \\ x=\sqrt{y-\sqrt{1+y}}\end{cases}. $$ This is a former problem from a national math contest which I´ve solved already. However, since my solution was…
alot
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BMO2 1996 Questio 4 - Algebra Problem

Let $a,b,c$ and $d$ be positive real numbers such that $a + b + c + d = 12$ and $abcd=27+ab+ac+ad+bc+bd+cd$. Find all possible values of $a,b,c,d$ satisfying these equations. I found this problem on someone's blog, where they had also given a proof…
MadChickenMan
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time and work question how to solve this

A,B,C are employed to do a piece of work for Rs 5290. A and B together are supposed to do 19/23 th of the work and B and C together 8/23 th of the work then A should be paid A.Rs 4250 B.Rs 3450 C.Rs 1950 D.Rs 2290 sir can you tell me how to solve…
kuberkashyap
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Finding the maximum of an expression in three variables

I am trying to find the max of $\frac{(a)(a-1)(9-a)}{2} + \frac{(b)(b-1)(9-b)}{2} + \frac{(c)(c-1)(9-c)}{2}$ subject to $a+b+c=9$ over nonnegative integers. I originally tried to see if something could work with Jensen's or calculus, and I have a…
user242594
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$Q=\{ 1,2,...n \}$. $S \subset Q$, let $p(S)$ be the product of elements of $S$, Find the sum of reciprocals $\frac{1}{p(S)}$ for all $S \subset Q$.

Consider the set $Q=\{ 1,2,...n \}$. For each $S \subset Q$, let $p(S)$ be the product of elements of $S$, Find the sum of reciprocals $\frac{1}{p(S)}$ for all $S \subset Q$. I have no idea how to start this. Any hint will be much appreciated.
Idonknow
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Assume that the sum of absolute values of all entries of $A$ equals to $1$. What is the maximal possible value of $\det(A)$?

Let $A$ be an $n \times n$ matrix and assume that the sum of absolute values of all its entries equals to $1$. What is the maximal possible value of $\det(A)$? My attempt: We know that $|a_{i,j}| \leq 1 \Rightarrow -1 \leq a_{i,j} \leq 1$ for all $1…
Idonknow
  • 15,643
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Problem using the AM-GM inequality

I am getting crazy with this one. Suppose $a_n=(1^2+2^2+3^2+\ldots+n^2)^n$ and $b_n=n^n(n!)^2$. Show that $a_n>b_n$ for all $n$. They suggest to use the AM-GM inequality.
Gian Luca
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BMO1 2008/09 Question 6 Trigonometry Problem

The obtuse-angled triangle $ABC$ has sides of length $a,b$ and $c$ opposite the angles $\angle A, \angle B$ and $\angle C$ respectively. Prove that $$a^3 \cos A + b^3 \cos B + c^3 \cos C \lt abc.$$ No real idea what to do, so any contributions are…
MadChickenMan
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Possible combinations

There are 1 to 24 numbers (1,2,3...24). How many possible combinations of 12 combine numbers will result with a sum of 146 when you add those 12 numbers? So meaning you have to combine 12 numbers from 1 to 24 and the total must be 146. You cannot…
abraham
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how can I solve this equation

I know it is an easy equation but I do not remember how we solve it I have the $Ev$ and the $N$ and I want to find the t the thing I did is that I take the exp for the both sides of the equation then I got $$ t=\dfrac{N^2}{\exp(Ev)} $$ but I have…
Dania
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Finding the invariant

There are A white, B black, and C red chips on a table. In one step, you may choose two chips of different colors and replace them by a chip of the third color. If just one chip will remain at the end, its color will not depend on the evolution of…
user197999
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No 37 question in knight of pi math tournament Dec 15, 2012

The five digit integer ABCDE, where each letter represents a digit, not necessary distinct, is divided by the numbers $2$,$3$,$4$,$5$, and $6$. The remainders are A, B, C, D, and E respectively. What is the integer ABCDE? Thanks.
Incognito Huang
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Any thoughts on how to solve this problem?

How many numbers do there exist having 2013 digits, in which every two-digit number composed of two consecutive digits is a multiple of either 17 or 23? (Taken from Singapore and Asian Schools Math Olympiad)
pirsquare
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Find the minimum value of an algebraic expression using simple algebra

Find the minimum value $2a^8+2b^6+a^4-b^3-2a^2-2$, where $a$ and $b$ are real numbers. I was told to use Lagrange multipliers but I found out this belongs to Calculus department. I tried factoring the expression but it's irreducible. Is there really…
keith_here
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Algebra contest problem

Suppose $x$ and $y$ are integers. Given $2xy+14y=-53-13x$, what does $xy$ equal? The answer is $-15$, but how do I get that? I feel like I should be able to find this.
user87611
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