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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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A Junior Olympiad like system of linear equations

Given a system of linear equations $$\begin{align}\frac{x}{3}+\frac{y}{5}+\frac{z}{9}+\frac{w}{17} &=1 \\ \frac{x}{4}+\frac{y}{6}+\frac{z}{10}+\frac{w}{18} &=\frac{1}{2} \\ \frac{x}{5}+\frac{y}{7}+\frac{z}{11}+\frac{w}{19} &=\frac{1}{3}…
kazuki
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Relationship between groups that have the same group of homomorphisms to another group

Say, there are two groups $A$ and $B$. We are given that $\mathrm{Hom}(A,G)$ and $\mathrm{Hom}(B,G)$ are isomorphic, where $G$ is another group that may or may not be trivial. What can we say about the relationship between $A$ and $B$? Not really a…
Walmart
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Find all real numbers $c$ satisfying the following condition: For all $n \in \mathbb{N}$, we have $n^c \in \mathbb{N}$.

Find all real numbers $c$ satisfying the following condition: For all $n \in \mathbb{N}$, we have $n^c \in \mathbb{N}$. My attempt: Clearly all $c \in \mathbb{N}$ works while negative integer $c$ does not work. Suppose $c=\frac{p}{q} \in…
Idonknow
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An identity satisfying the divisors of a positive integer

I saw a hard competition problem with long and ugly proof in http://solmu.math.helsinki.fi/olympia/valmennus/2013/vt2013_12var.pdf ? The question is from Australian mathematical olympiad 1985. Is there a nice way to solve the following: A positive…
Finnmath
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"Will we ever get a palindrome" 1996 Problem.

Possible Duplicate: Is it possible for the number created by ordering $1$ to $n$ where $n \geq 1$ be a palindrome? I've been thinking about this problem for a while now, and have come up with nothing useful. If anybody knows the solution, or the…
Weltschmerz
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A hammer and a nail cost $1.10, and the hammer costs one dollar more than the nail. How much does the nail cost?

The answer is 0.05. I used algebra. But my friends say, why not 0.10, and they also say, it can be that the hammer is 1.04 and the nail 0.06. How do I tell them that 0.05 is the definite answer, nothing else; I need mathematical evidence and proof…
Ali Gajani
  • 211
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how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\le 1000$

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$,For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\le 1000$ $A:38 , B:40 C:42 D:44 E:46$ My idea: maybe this $(x,y)$…
math110
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Another olympiad problem

This problem is a problem in the last selection phase of the math olympiads in my country. If $\alpha, \beta,\gamma$ are angles $\in[0,\frac\pi2]$ such that $\sin^2(\alpha)+\sin^2(\beta)+\sin^2(\gamma)=1$.Minimize…
chubakueno
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Minimum number of operations to make a positive integer 1

The problem is Given a positive integer $n$, what is the minimum number of operations to make the number 1. There are 3 options to choose from (1) if the number is even you can divide by 2. (2) for any number you can add 1. (3) for any number you…
24n8
  • 1,455
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Pigeon groupings

Question:There are $51$ pigeons in a flock. The flock is divided into $n$ groups so that each pigeon is in exactly $1$ group. However, every pigeon dislikes exactly $3$ pigeons and thus does not want to be in the same group as the $3$. Find the…
Michael Li
  • 502
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Recursive Question in AMC 2009 (12A)

The first two terms of a sequence are $a_1 = 1$ and $a_2 = \frac {1}{\sqrt3}$. For $n\ge1$, $$a_{n + 2} = \frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}}.$$ What is $|a_{2009}|$? The simplest solution for this question was to just work out the sequence…
Hector Lombard
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find the value $\frac{49b^2+39bc+9c^2}{a^2}=52$

Let $a,b.c$ be real numbers such that $$\begin{cases}a^2+ab+b^2=9\\ b^2+bc+c^2=52\\ c^2+ca+a^2=49 \end{cases}$$ show that $$\dfrac{49b^2+39bc+9c^2}{a^2}=52$$ I have found this problem solution by geomtry methods.solution 1,can you someone have…
math110
  • 93,304
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Simple solution to Question 6 from the 1988 Math Olympiad

Recall Question 6 of the 1988 Math Olympiad Question 6 is as follows: Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^2 + b^2$. Show that $$\frac{a^2+b^2}{ab+1}$$ is the square of an integer. My proof: Proof: Let us denote the…
Pascal
  • 179
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Floyd's Triangle

The numbers are arranged in a interesting way: $$ \begin{array}{cccccccc} 1 \\ 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 & 10 \\ 11 & 12 & 13 & 14 & 15 \\ \vdots \end{array} $$ As you can see, every row has one extra number then the previous row. Using…
Soha Farhin Pine
  • 663
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3 answers

Olympiad - sequence of sum of complex numbers

This is the other problem I couldn't solve in the olympiad test I took today. Let $c_1, ... , c_n$ be complex numbers with unitary norm, and $S_k=\sum_{i=1}^n c_i^k$, $k\in \mathbb{N}$. Suppose $S_1, S_2, ...$ converges. Prove that $c_i=1$ for every…
kevincuadros
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