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

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0
votes
8answers
164 views

The number of real roots of $x^5 + 2x^3 + x^2 + 2 = 0 $ is

The number of real roots of $x^5 + 2x^3 + x^2 + 2 = 0 $ is A. 0; B. 3; C. 5; D. 1. I don't know how to solve this.
3
votes
2answers
56 views

Number of common roots of $x^3 + 2 x^2 +2x +1 = 0$ and $x^{200} + x^{130} + 1 = 0 $

The equations $x^3 + 2 x^2 +2x +1 = 0$ and $x^{200} + x^{130} + 1 = 0 $ have exactly one common root; no common root; exactly three common roots; exactly two common roots. I factored the first ...
-2
votes
1answer
45 views

Let $a_1 = 2$ and for all natural number n, define $a_{n+1}= a_{n}(a_{n}+1)$. Then as $n\rightarrow \infty$, the number of prime factors of $a_{n}$ [closed]

Let $a_1 = 2$ and for all natural number n, define $a_{n+1}= a_{n}(a_{n}+1)$. Then as $n\rightarrow \infty$, the number of prime factors of $a_{n}$: goes to infinity. goes to a finite limit. ...
3
votes
0answers
116 views

(2016 China team selection Test) with a complex inequality

Let $z_{1},z_{2},z_{3}$ be complex numbers, such that: $z_{1}+z_{2}+z_{3}=0,|z_{i}|<1,i=1,2,3$. Find the minimum of the positive $A$ such that: ...
1
vote
1answer
66 views

Clever way to sum these angles? [closed]

In the image, is there a nice way to write down the sum of a+b+c?
6
votes
5answers
228 views

On equations $m^2+1=5^n$

I am looking for integer solutions of Diophantine equation $m^2+1=5^n$. I found that $m=0,n=0$ and $m=2,n=1$. I could not find any other solutions. I try to prove this but I could not. Could anyone ...
5
votes
4answers
146 views

How to simplify the nested radical $\sqrt{1 - \frac{\sqrt{3}}{2}}$ by hand?

I was solving a Mock Mathcounts Contest Mock contest (.pdf) written by a user on the Art of Problem Solving Forums. In problem #24 the only thing I couldn't do by hand was simplify the radical ...
21
votes
2answers
400 views

Prove that $f(1999)=1999$

A function $f$ maps from the positive integers to the positive integers, with the following properties: $$f(ab)=f(a)f(b)$$ where $a$ and $b$ are coprime, and $$f(p+q)=f(p)+f(q)$$ for all prime numbers ...
0
votes
6answers
68 views

How many digits are in the integer representation of 2 to the 30th power?

How many digits are in the integer representation of 2 to the 30th power? Since I didn't really know any 'expert' way to approach this, I just started out by listing the powers of 2, like 2, 4, 8, ...
0
votes
1answer
23 views

Centre of Invariant Circle under Inversion

Given an inversion of the plane, and a circle invariant under this inversion, what information do we know about the inverse of the centre this circle? (I know that an invariant circle must be ...
5
votes
1answer
97 views

If $a^{2}+84a+2008=b^{2}$ what is $a+b$

let $a, b$ are two positive integer satisfy the condition $a^{2}+84a+2008=b^{2}$. Find out $a+b$ My Solution $a^{2}+84a+2008=b^{2} \implies (a+42)^{2}+244=b^{2} \implies (b+a+42)(b-a-42)=2^{2}61$. ...
1
vote
0answers
26 views

Show lines through circumcenters of triangles concurrent with complex numbers

Cyclic quadrilateral $ABCD$ has circumcenter $O$. Point $O$ does not lie on any of the sides of the quadrilateral. Let $O_1,O_2,O_3,O_4$ denote the circumcenters of $\triangle OAB, \triangle OBC, ...
0
votes
1answer
20 views

Squares, midpoints and heights

Let $ABC$ be a traingle, we draw squares on the sides $AB$ and $AC$, now we draw a segment from the vertexes of the square which are closer and then it forms a triangle, so prove that the line throw A ...
5
votes
5answers
549 views

Find a polynomial with integer coefficients

Find a polynomial $p$ with integer coefficients for which $a = \sqrt{2} + \sqrt[3]{2}$ is a root. That is find $p$ such that for some non-negative integer $n$, and integers $a_0$, $a_1$, $a_2$, ..., ...
2
votes
0answers
59 views

2015 Mathcounts State Sprint #30 [duplicate]

NOTE: This question is not a duplicate. It is actually the other way around. This question was posted before the other question that this question was marked as a duplicate of. Please mark this ...
6
votes
0answers
63 views

Game, stealing edges in a graph.

I was inventing a problem for a math contest, I was really pleased with it, but then I found a mistake in my solution and have not been able to solve it. It is as follows: Alice and Bob play a game. ...
1
vote
1answer
14 views

Areal Co-ordinate Geometry Question

Let $P$ be an internal point of triangle $ABC$. The line through $P$ parallel to $AB$ meets $BC$ at $L$, the line through $P$ parallel to $BC$ meets $CA$ at $M$, and the line through $P$ parallel to ...
1
vote
0answers
24 views

Filling a grid square with 0,1,2 [duplicate]

Each of the 25 cells in a five-by-five grid square is filled with a 0, 1, or 2 in such a way that the numbers written in neighboring cells differ from the number in that cell by 1. Two cells are ...
2
votes
0answers
40 views

Absolute difference and probability [closed]

Fifty tickets numbered with consecutive integers are in a jar. Two are drawn at random and without replacement. What is the probability that the absolute difference between the two numbers is 10 or ...
0
votes
1answer
31 views

Pentagon Problem

In a regular pentagon ABCDE, point M is the midpoint of side AE, and segments AC and BM intersect at point Z. If ZA = 3, what is the value of AB? (The answer is supposed to be in simplest radical ...
1
vote
1answer
80 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 ...
-2
votes
1answer
34 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
votes
3answers
80 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
votes
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 ...
1
vote
3answers
57 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 ...
2
votes
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 ...
0
votes
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
votes
2answers
113 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 ...
2
votes
1answer
54 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 ...
1
vote
0answers
113 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 ...
2
votes
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
votes
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 ...
1
vote
1answer
49 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 $∀z∈ℂ^*$, $p(z+1∕z)=(z^n +1/z^n)$.
0
votes
1answer
33 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 ...
2
votes
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
votes
2answers
111 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 ...
1
vote
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
votes
0answers
65 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 ...
1
vote
4answers
113 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 ...
1
vote
0answers
35 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. ...
1
vote
2answers
146 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.
7
votes
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 ...
1
vote
1answer
102 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. ...
1
vote
1answer
36 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 ...
2
votes
2answers
82 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 ...
0
votes
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 ...
1
vote
1answer
37 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
206 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 ? $$\left(\frac{2a}{b+c}\right)^{2/3}+\left(\frac{2b}{c+a}\right)^{2/3}+\left(\frac{2c}{a+b}\right)^{2/3}\geq 3$$