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

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1answer
107 views

Shortest distance between two circles

What is the shortest distance, in units, between the circles $(x - 9)^2 + (y - 5)^2 = 6.25$ and $(x + 6)^2 + (y + 3)^2 = 49$? Express your answer as a decimal to the nearest tenth. So I know that ...
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1answer
35 views

heart rate problem [closed]

The average heart rate of a shrew is 800 beats per minute, while an elephant has a heart rate of 25 beats per minute. If 1 billion heartbeats is a natural life span for each animal, on average, how ...
4
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3answers
87 views

How many different paths from top to bottom spell ALGEBRA?

Starting with the A on top and only moving one letter at a time down to the left or down to the right, how many different paths from top to bottom spell ALGEBRA? ...
3
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3answers
42 views

How many tokens would person A have under these conditions?

Persons A and B each have a positive integer number of tokens, and the number of tokens B has is a square number less than 100. B says to A, "If you give me all of your tokens, my total number of ...
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3answers
67 views

Three-digit numbers whose digits and digit sum are all prime

How many 3$$-digit numbers are there such that each of the digits is prime, and the sum of the digits is prime? Shouldn't it be $0$, because the only one digit primes are $2,3,5,7$, and so the ...
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4answers
110 views

Find the minimum value of $\frac{4}{4-x^2} + \frac{9}{9-y^2} $

Let $x, y ∈ (−2, 2)$ and $xy = −1$. Find the minimum value of $\frac{4}{4-x^2} + \frac{9}{9-y^2} $ ? My Attempt let $t=\frac{4}{4-x^2} + \frac{9}{9-y^2} $ , replacing $y$ by $- \frac{1}{x}$ we get ...
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2answers
54 views

Perimeter of Quadrilateral

The lengths of two sides of a quadrilateral are equal to 1 and 4. One of the diagonals has a lengths of 2 and divide the quadrilateral into two isosceles triangles. What is the perimeter of the ...
4
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2answers
124 views

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 ...
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1answer
59 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 ...
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0answers
116 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 ...
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0answers
56 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 ...
7
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1answer
92 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 ...
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1answer
50 views

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 $∀z∈ℂ^*$, $p(z+1∕z)=(z^n +1/z^n)$.
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1answer
37 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 ...
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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 ...
7
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2answers
112 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 ...
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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 ...
3
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0answers
66 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 ...
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4answers
114 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 ...
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0answers
36 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. ...
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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.
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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 ...
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1answer
103 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. ...
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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 ...
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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 ...
3
votes
1answer
55 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 ...
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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 ...
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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 ...
9
votes
2answers
254 views

Factorization game, can we find winning strategy?

I'm thinking about a game theory problem related to factorization. Here it is, Q: two players A and B are playing this factorization game. At very first, we have a natural number $270000=2^4\times ...
6
votes
0answers
95 views

$a,b,c$ are positive real numbers. How can we show this inequality? [closed]

How can we show ? $$\left(\frac{2a}{b+c}\right)^{2/3}+\left(\frac{2b}{c+a}\right)^{2/3}+\left(\frac{2c}{a+b}\right)^{2/3}\geq 3$$
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2answers
66 views

Prove that all the roots of $p(x)=F_{n}x^{n}+..+F_{1}x+F_{0}$ can't be real

Last night I have created this problem. Let $p(x)=F_{n}x^{n}+..+F_{1}x+F_{0}$ where $F_{n}$ is $n$th Fibonacci number. Prove that all the roots of $p(x)$ can't be real. Edit 1: $n>1$.
3
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4answers
57 views

How many integer pairs (x, y) satisfy $x^2 + 4y^2 − 2xy − 2x − 4y − 8 = 0$?

How many integer pairs (x, y) satisfy $x^2 + 4y^2 − 2xy − 2x − 4y − 8 = 0$? My Attempt Let $f(x,y)=x^2 + 4y^2 − 2xy − 2x − 4y − 8$ . So $f(x,0)=x^2 − 2x − 8$ . $f(x,0)$ has two roots $x=4 , -2$ . ...
7
votes
1answer
83 views

Polynomial game problem: do we have winning strategy for this game?

I'm thinking about some game theory problem. Here it is, Problem: Consider the polynomial equation $x^3+Ax^2+Bx+C=0$. A priori, $A$,$B$ and $C$ are "undecided", yet and two players "Boy" and ...
2
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1answer
44 views

if $A,B,C$ are real numbers such that ,${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1 $ and $A+B+C = 0 $ find the maximum value of $(ABC )^2$ [duplicate]

$$A,B,C$$ are real numbers such that ,$${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1 $$ and $$A+B+C = 0 $$ find the maximum value of ${ (ABC) }^{ 2 }$ I don't know how can I start to solve this ...
2
votes
2answers
41 views

Find the maximum value of $4x − 3y − 2z$ subject to $2x^2 + 3y^2 + 4z^2 = 1.$

Find the maximum value of $4x − 3y − 2z$ subject to $2x^2 + 3y^2 + 4z^2 = 1.$ My Attempt let $S=4x − 3y − 2z$ and $ t=2x^2 + 3y^2 + 4z^2$. Then $t-s =2x^2 + 3y^2 + 4z^2 -(4x − 3y − 2z)= 2(x-1)^2 + ...
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1answer
25 views

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 ...
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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 ...
7
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1answer
87 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 ...
2
votes
3answers
38 views

Find all possible values of $c^2$ in a system of equations.

Numbers $x,y,z,c\in \Bbb R$ satisfy the following system of equations: $$x(y+z)=20$$ $$y(z+x)=13$$ $$z(x+y)=c^2$$ Find all possible values of $c^2$. To try to solve this, I expanded the equations: ...
7
votes
2answers
62 views

Solve system of simulataneous equations in $3$ variables

Solve the following equation system: $$x+y+xy=19$$ $$y+z+yz=11$$ $$z+x+zx=14$$ I've tried substituting, adding, subtracting, multiplying... Nothing works. Could anyone drop me a few hints without ...
1
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1answer
40 views

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 $f$.
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0answers
40 views

A theorem about binomial coefficient module prime

For any integer $r$ and prime $p$, there is a integer $n$ which $\binom{2n}{n}\equiv r \pmod{p}$. I tried Lucas's theorem, but I was stuck. Suppose $r\neq 0$, otherwise we can let $n=p$. Let ...
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3answers
81 views

The total amount Edgar paid for a slice of pizza and a tip of exactly $24\%$ was between $\$2.50$ and $\$3.00$. What was the price of the pizza slice?

The total amount Edgar paid for a slice of pizza and a tip of exactly 24% was between $\$2.50$ and $\$3.00$. What was the price of the pizza slice? Well, I did the trial and error method and ...
1
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4answers
94 views

Problem about simple probability

I guess that this will be really simple for you guys, but i have no foundation in probability. Please, help me to find not only the answer but also what i need to learn in order to be able to solve ...
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2answers
96 views

Problem from Olympiads of mathematics about elementary number theory

Can you please help me with this problem from the Italian selection of the Olympiads of mathematics? Let $p(x)$ be a polynomial with integer coefficients and let $p(0)=6$. Exactly $40$ $p(n)$ with ...
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1answer
42 views

Quadratic recurrence relation (from a math-contest)

It's given the following quadratic relation: $$a_n = \frac{a_{n-1}^2+61}{a_{n-2}}$$ Find $a_{10}$. Note that I can't use a calculator or a computer, instead I was wondering if there's a trick to find ...
2
votes
1answer
35 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 ...
2
votes
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 ...
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1answer
61 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 ...
-1
votes
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 ...