# Tagged Questions

Problems from or inspired by mathematics competitions. Questions regarding mathematics competitions.

2answers
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### Maximum distance between points in a triangle

An equilateral triangle has sides of unit length. a)Show that if five points lie in/on the triangle, then at least two of the points lie no farther than 0.5 units apart. b)Show that 0.5 cannot be ...
1answer
68 views

### Finding max perimeter of triangle of three circulating points

I'm thinking a plane geometry problem, and it seems quite puzzling. Here it is. Question: Consider three concentric circles with radius 3, 5 and 7 each. and construct a triangle by picking one ...
0answers
124 views

### 2016 AIME #7 - the controversy?

Here's the problem For integers $a$ and $b$ consider the complex number $$\frac{\sqrt{ab+2016}}{ab+100} - \left(\frac{\sqrt{|a+b|}}{ab+100}\right)i$$ Find the number of ordered pairs of ...
0answers
59 views

### Maximal dimension of a vector space of square matrices in which every nonzero matrix is invertible

I'm interested in the maximal dimension of a subspace $V\leq\mathbb R^{n\times n}$ in which every nonzero matrix is invertible. Odd $n$: For odd $n$ the maximum is $1$: if $A$ and $B$ would be ...
1answer
95 views

### 2009 Benelux Math Olympiad (BxMO) number theory problem

The following problem is taken from the first Benelux Mathematical Olympiad which occurred in 2009. Let $n$ be a positive integer and let $k$ be an odd positive integer. Moreover, let $a$, $b$ and ...
1answer
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### n∈ℕ, p∈ℂ[x], ∀z∈ℂ* show $p(z+\frac{1}{z})=(z^n +\frac{1}{z^n})$

With $n∈ℕ$, Show that there exist a unique polynomial $p∈ℂ[x]$such that $∀z∈ℂ^*$, $p(z+1∕z)=(z^n +1/z^n)$.
1answer
38 views

### There exist three consecutive vertices A, B, C in every convex n-gon with n≥3, such that the circumcircle of triangle ABC covers the whole n-gon

From Problem Solving Strategies by Arthur Engel: Problem to prove: There exist three consecutive vertices $A$, $B$, $C$ in every convex $n$-gon with $n \ge 3$, such that the circumcircle of triangle ...
2answers
58 views

### Find the parameters given $p(r)=s$ and $p(s)=r$

Problem: Find all values of the parameters $a$ and $b$ for which the polynomial $x^4+(2a+1)x^3+(a-1)^2x^2+bx+4$ can be factored into two quadratic monic polynomials $p(x)$ and $q(x)$ such that $q(x)$...
2answers
115 views

### There is no sequence such that $a_{a_n}=a_{n+1}a_{n-1}-a_{n}^2$

Prove that there is no infinite sequence of natural numbers such that $a_{a_n}=a_{n+1}a_{n-1}-a_{n}^2$ for all $n\geq 2$. This question is from a Belarusian math contest and any help is appreciated.
1answer
34 views

### What is the maximum possible number of distinct colors used?

To each element of set S={1,2,..,1000} a color is assigned.Suppose that for any two elements $a,b$ of S , if 15 divides $a+b$ then they are assigned both same color.What is the maximum possible number ...
0answers
68 views

### Putnam 2015 and Ravi Substitution

Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c$. Express $$\sum_{(a,b,c)\in T}\frac{2^a}{3^b5^c}$$ as a rational number in ...
4answers
116 views

### Factorize $(x^2+y^2+z^2)(x+y+z)(x+y-z)(y+z-x)(z+x-y)-8x^2y^2z^2$

I am unable to factorize this over $\mathbb{Z}:$ $$(x^2+y^2+z^2)(x+y+z)(x+y-z)(y+z-x)(z+x-y)-8x^2y^2z^2$$ Since, this from an exercise of a book (E. J. Barbeau, polynomials) it must have a neat ...
0answers
37 views

### A generalization of a geometry Olympiad problem involving $kn$ colored lines and a circle.

Let $n$ and $k$ be positive integers. Let $L$ be any set of $kn$ lines in the plane, no two of which are parallel. Each line in $L$ is colored one of $k$ colors, and there are $n$ lines of each color. ...
2answers
150 views

### Find all polynomials $p$ such that $p(x^2)=p(x)p(x+1)$

Find all polynomials $p$ such that $$p(x^2)=p(x)p(x+1).$$ The goal is to find a general formula for polynomials that satisfy the above equation.
2answers
82 views

### Solve in positive integers $\frac{x^{2}}{y}+\frac{y^{2}}{x}=9$

Solve in positive integers $\frac{x^{2}}{y}+\frac{y^{2}}{x}=9$ By inspection we see $x=4$ and $y=2$ is a solution. But are there any more solutions? I have tried to convert the equation to inequality ...
1answer
113 views

### How to find the no of Questions?

Liz and Mary compete in solving problems. Each of them is given the same list of 100 problems. For any problem, the first of them to solve it gets 4 points, while the second to solve it gets 1 point. ...
1answer
39 views

### Counters on a Chessboard (BMO 2010/11)

Isaac has a large supply of counters, and places one in each of the $1 \times 1$ squares of an $8 \times 8$ chessboard. Each counter is either red, white, or blue. A particular pattern of colored ...
2answers
84 views

### Inequality with large exponents, RMM 2016

Let $x,y$ be positive reals, with $x+y^{2016} \ge 1$. Prove that $x^{2016}+y > 1-\frac{1}{100}$. Wolframalpha gives that the minimum possible value for $x^{2016}+y$ is about $0.997415$. How would ...
1answer
58 views

### Show that $x^2 + xy^2+ xyz^2\geq 4xyz-4$ for positive real $x,y,z$

Let $x$,$y$ and $z$ are three positive real numbers.Show that $x^2 + xy^2+ xyz^2\geq 4xyz-4$. I have tried to attack the problem by order relationship $x \geq y \geq z$ and then converting them into ...
1answer
12 views

### consider the following statements regarding the smallest interior angle of a n sided polygon with perimeter n units and with maximum area?

let(f) be the relation defined by f(n) = The smallest interior angle value of the n sided polygon with perimeter n units with maximum area, for each positive integer n(>2).which of the following are ...
1answer
38 views

### How to find two square roots whose difference is greater than one.

How do you find the greatest $n$ such that the difference of its square root from some other integer is greater than or equal to one? For example : $$2011^{1/2} - n^{1/2} \ge1$$ What should be the ...
2answers
303 views

1answer
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### Bingo-like Game

In one board game, each player has a unique 4 x 4 grid with squares randomly labeled with each integer from 1 to 16. As the integers 1 to 16 are randomly called, each player puts an "X" in the square ...
0answers
39 views

### How many rational values of x are not integers and satisfy the following equation?

How many rational values of x are not integers and satisfy the following equation: $$x^7 - 6x^6 + 5x^5 - 4x^4 + 3x^3 - 2x^2 + 1 = 0 ?$$ Well, I got this question from one of the Mathcounts ...
1answer
93 views

### Tetrahedron packing in Cube

I'm thinking about following solid geometry problem. Q: Suppose you have a box of "cube" shape with edge length 1. Then, How many regular tetrahedrons(with edge length 1) can be in the box? So, this ...
3answers
39 views

1answer
36 views

### Finding a path on a coordinate plane

On a coordinate plane, a path consists of a series of moves in the positive $x$- or $y$- direction. If the first move is 1 unit in length, the second move is 2 units, the third move is 3 units, and so ...
2answers
40 views

### “Stairstep Numbers”

I've been preparing for Mathcounts competition, but this one question confused me a bit. If a stairstep number is defined as a number whose digits are strictly increasing in value from left to ...
1answer
76 views

### What is the largest six-digit number with an odd number of positive factors?

What is the largest six-digit number with an odd number of positive factors? So I know the number must be a perfect square, but how do I know six-digit number perfect squares? I'm pretty sure there's ...
2answers
56 views

### How many cubes must be randomly selected to ensure that at least one pair of each color has been removed from the bag?

There are $15$ red, $11$ blue and $13$ green cubes in a bag. All cubes are identical, except for color. How many cubes must be randomly selected to ensure that at least one pair of each color has been ...
4answers
1k views

### Factorial question: number of trailing zeroes in 125! [duplicate]

How many zeros are after the last nonzero digit of 125! ? The answer is 31, but how do you solve it?
1answer
35 views

### Example of $2$ nonisomorphic simple graphs satisfying conditions.

What is an example of $2$ simple graphs which: have the same degree sequence; for any $n \ge 2$ have the same number of copies of $K_n$; for any given $k \ge 3$ have the same number of induced ...
3answers
107 views

### Proving that $1^n+2^n+3^n+4^n$ $(n\in \Bbb N)$ is divisible by 10 when $n$ is not divisible by 4

I was solving some math problems to prepare for math contests and came across this one: Prove that $1^n+2^n+3^n+4^n$ $(n\in N)$ is divisible by 10 if and only if $n$ is not divisible by 4. So, from ...
2answers
59 views

### 2015 AIME #3: Where did I go wrong?

This is a question conerning 2015 AIME #3. The problem goes as follows: There is a prime number $p$ such that $\displaystyle 16p+1$ is the cube of a positive integer. Find $p$. Here is my ...
3answers
68 views

### Do the lengths of all three segments necessarily have the same distribution?

Let $A$ and $B$ be independent $U(0, 1)$ random variables. Divide $(0, 1)$ into three line segments, where $A$ and $B$ are the dividing points. Do the lengths of all three segments necessarily have ...
4answers
93 views

### Among the following, which is closest to $\sqrt{0.016}$?

Among the following, which is closest in value to $\sqrt{0.016}$? A. $0.4$ B. $0.04$ C. $0.2$ D. $0.02$ E. $0.13$ My Approach: \$(\frac{16}{1000})^\frac{1}{2} = (\frac{4}{250})^\frac{1}{2} = \...
0answers
51 views

### No eight digit perfect fourth powers with distinct digits and not containing 3.

Problem: Prove that there are no perfect fourth powers that have eight distinct digits in their base 10 representation and also don't contain 3 as a digit. My attempt: Since the problem is about ...