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

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Let $p(x) = 1+a_1x+a_2x^2+\cdots+a_nx^n$ be a polynomial where $a_1,\ldots,a_n$ are integers, and $a_1 + … + a_n$ is even.

Let $p(x) = 1+a_1x+a_2x^2+\cdots+a_nx^n$ be a polynomial where $a_1,\ldots,a_n$ are integers, and $a_1 + \cdots + a_n$ is even. Prove that there is no integer x such that $p(x) =0$ I have started ...
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4answers
48 views

A function $f$ has the property that $f(x+y)=f(x)+f(y)+3xy$. If $f(1)=2$, what is $f(8)$?

A function $f$ has the property that $f(x+y)=f(x)+f(y)+3xy$. If f(1)=2, what is f(8)? I would like to try to tackle this problem but I need somewhere to start as I really have no idea at all on how ...
1
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3answers
43 views

If $x^2+bx+a=0$ and $x^2+ax+b=0$ have a common root $c$, Then what values of $(a,b)$ would work?

Let $a$ and $b$ be distinct integers. If $x^2+bx+a=0$ and $x^2+ax+b=0$ have a common root $c$, Then which of the following statements are true? 1) $c*(a+c)=-b$ 2) $a+b=-1$ 3) $a+b+c=0$ 4) $c=0$ ...
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0answers
60 views

Problem 6 - IMO 1985

For every real number $x_1$ construct the sequence $x_1,x_2,x_3,\ldots$ by setting $x_{n+1}=x_n(x_n+\frac{1}{n})$ for each $n \ge 1$. Prove that there exists exactly one value of $x_1$ for which $0 ...
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21 views

Convergence of series (putnam training) [on hold]

Does the series $\sum_{n=1}^{\infty} \frac {|\sin n|}{n} $ converge?
2
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2answers
33 views

Maximum of a function from integers to integers

Suppose $f$ is a function form positive integers to positive integers satisfying $f(1)=1$, $f(2n)=f(n)$, and $f(2n+1)=f(2n)+1$ for all positive integers $n$. Job: Find the maximum of $f(n)$ when $n$ ...
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1answer
18 views

1995 MathCounts State Team #8

In professional football, it is possible to score 2 points (for a safety), 3 points (for a field goal), or 6 points (for a touchdown). If a touchdown is scored, it is possible to score 1 additional ...
10
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0answers
85 views

Curious number theory problem

$k,m,n\in\mathbb{N}$ satisfy $k^{m+n}=nm^n$. How can I show that $m=k$ and $n=k^k?$
2
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1answer
38 views

Knot Theory: Mutations

Show that if we have three tangles as in Figure 2.33a, we can mutate several times in order to permute the tangles. Note that we can then permute n tangles in a row. This is from Colin Adams; The ...
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0answers
22 views

Triangles packed into a unit circle

2014 triangles have non-overlapping interiors contained in a unit circle.What is the largest possible value of the sum of their areas? What are some ideas that might help me start this?
7
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2answers
99 views

$a^3+3a^2+a$ is never a perfect square.

Prove that no number of the form $ a^3+3a^2+a $, for a positive integer $a$, is a perfect square. This problem was published in the Italian national competition (Cesenatico 1991). I've been ...
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1answer
40 views

Knot theory: pretzel knot

Prove that pretzel knot $K(p_1,p_2,p_3,\dots,p_n)$ with all $p_i >0$ is an alternating knot or link? I think since all $p_i$'s are positive, the sign has a lot to do with it but how to prove it is ...
2
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2answers
76 views

For any positive integers $a$ and $b$, the number $(36a+b)(a+36b)$ can never be a power of $2$.

APMO 1998: Show that for any positive integers $a$ and $b$, the number $(36a+b)(a+36b)$ can never be a power of $2$. The solution I've read substitutes $a=2^Ap,b=2^Bq$ where $p$ and $q$ are ...
0
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1answer
79 views

If $x + \frac{1}{x} = k$, what's the value of this sum?

Friends, if $x + \frac{1}{x} = k$, with $k$ positive real number, what's the value of $$1+ x^6 + x^{12} + x^{18} + x^{24}+x^{30}$$ I tried with the substitution $u= x^6$: $$a= 1+ u + u^2 + u^3 + u^4 ...
4
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1answer
80 views

Math Olympiads: GCD of terms in a sequence equals GCD of terms in other sequence

Recently, someone asked for a proof of a problem from the Russian Mathematical Olympiad, 1995. Math Olympiads: GCD of terms in a sequence equals GCD of their indices. The problem was to show that if ...
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3answers
40 views

Ratio Math Problem

This is from a competition math problem I had recently that I just couldn't figure out. If $ (x+y):(y+z):(x+z) = 1:2:4$ and $x+y+z=35$ compute the value of x. I can tell that $7*(x+y)=2x+2y+2z$ ...
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2answers
58 views

Maths brain teaser. Fifty minutes ago it was four times as many minutes past three o'clock

Fifty minutes ago it was four times as many minutes past three o'clock. How many minutes is it to six o'clock..? I have got the solution online but have doubts in it : ...
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2answers
134 views

Math Olympiads: GCD of terms in a sequence equals GCD of their indices.

The sequence $a_1 ,a_2 ,a_3 ,...$ of positive integers satisfies $\text{gcd}(a_i ,a_j ) = \text{gcd} (i, j)$ for $i \neq j$. Prove that $a_i = i$ for all $i$. Source: Russian Mathematical Olympiad, ...
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2answers
34 views

Maximize the inradius given the base and the area of the triangle

BdMO 2013 Secondary: A triangle has base of length 8 and area 12. What is the radius of the largest circle that can be inscribed in this triangle? Let $A,r,s$ denote the area,inradius and ...
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3answers
45 views

Prove that $(2m+1)^2 - 4(2n+1)$ can never be a perfect square where m, n are integers

I could prove it hit and trial method. But I was thinking to come up with a general and a more 'mathematically' correct method, but I did not reach anywhere. Thanks a lot for any help.
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3answers
41 views

What's the probability that $y\ge x+1$?

"Two numbers, $x$ and $y$ are selected at random from the interval $(0,3)$. What is the probability that $y\ge x+1$?" The answer key says $\frac 29$, but I can't figure out how to get to that answer. ...
2
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1answer
38 views

Find the sum of the maximum and minimum

For a real number $x$ find the sum of the maximum and minimum. $$y=\frac{x^2-2x-3}{2x^2+2x+1}$$ This is a sample question for a math competition. It seems like calculus would be used to solve this, ...
3
votes
1answer
54 views

Locus of the centres of equilateral triangles (contest problem)

Given a triangle $A_0A_1A_2$ determine the locus of the centres of the equilateral triangles $X_0X_1X_2$ satisfying the condition that each of the lines $X_kX_{k+1}$, $k=0,1,2$ passes through ...
2
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2answers
123 views

Cool Integral $\int_0^{\pi/2}dx\ln \sinh x$

$$ I_1=\int_0^{\pi/2}dx\ln \sinh x,\quad I_2=\int_0^{\pi/2}dx\ln \cosh x, \quad I_1\neq I_2. $$ I am trying to calculate these integrals. We know the similar looking integrals $$ \int_0^{\pi/2}dx\ln ...
0
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1answer
17 views

Bound the Number of Acute-angled Triangles

I encounter the following problem with solution. But I do not quite understand the argument for 5, 10 points and eventually 100 points. Can someone elucidate the details? Problem In a plane there ...
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1answer
58 views

How many zeros does this expression end in?

How many zeroes does $$\frac{50!}{2^95^5}$$ end in?
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68 views

Integrate $\int_0^\infty \frac{Li_n(-\sigma x)Li_m(-\omega x^2)}{x^3}dx$

I am trying to solve this integral $$ \int_{0}^{\infty} {{\rm Li}_{n}\left(-\sigma x\right){\rm Li}_m\left(-\omega x^{2}\right) \over x^{3}}\,{\rm d}x $$ which is from some high school IMO training ...
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2answers
94 views

Integral $ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $

I am trying to solve this integral $$ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $$ A closed form does exist despite the looks of the integrand. ...
1
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1answer
85 views

A geometry problem on power of points

An acute triangle $ABC$ is inscribed in a circumference of center $O$. Its heights are $AD$, $BE$ and $CF$. The line $EF$ intersects the circumference at two points, $P$ and $Q$. (a) Prove ...
2
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4answers
75 views

Find the number of elements in the range$ f(x) =[x] + [2x] +[2x/3] +[3x] +[4x] +[5x]$ for $0\le x \le3$.

Find the number of elements in the range $f(x) =[x] + [2x] +[2x/3] +[3x] +[4x] +[5x]$ for $0\le x \le3.$ I cant understand...It will go very long if i keep breaking them into small intervals .
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1answer
31 views

a spider has 1 sock and 1 shoe for each leg. then find out the the total possibilities.

a spider has one sock and one shoe for each of its 8 legs.in how many different orders can the spider put on its shocks and shoes; assuming that on each leg ;the shock must be put on before the shoe? ...
2
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0answers
64 views

The name of a game from the 2013 Putnam

Does the following game from the 2013 Putnam have an official name? Based on my searches, it seems to have been created just for the exam. Let $n\geq 1$ be an odd integer. Alice and Bob play the ...
3
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1answer
54 views

How prove $G,H,T $ are collinear.

Question: Circle $O_{1}$ and $O_{2}$ are internally tangent at point $T$. $AB$ and $CD$ are tangents of circle $O_{1}$, the angle bisectors of Angle $\angle ADB$ and $\angle CBD$ intersects at ...
3
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1answer
64 views

A geometry problem on cyclic quadrilaterals

The problem: Let $M$ be the point of intersection between the diagonals of a cyclic quadrilateral $ABCD$, where $\angle AMB$ is acute. The isosceles triangle $BCK$, whose base is $BC$, is ...
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3answers
122 views

$1992$ IMO Functional Equation problem

The problem states: Let $\Bbb R$ denote the set of all real numbers. Find all functions $f : \Bbb R \rightarrow \Bbb R$ such that $$f(x^{2}+f(y))=y+(f(x))^{2} \space \space \space \forall x, y \in ...
4
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3answers
137 views

How find this function $f(x)$ such $f(a+f(b))=f(a+b)+f(b)$

let function $f:R_{+}\to R_{+}$,and such $$f(a+f(b))=f(a+b)+f(b),\forall a,b\in R_{+}$$ Find $f(x)$. my try: let $a=b=1$,then $$f(1+f(1))=f(2)+f(1)$$ $a=1,b=2$,then $$f(1+f(2))=f(3)+f(2)$$ then I ...
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3answers
80 views

$1-x+x^2-x^3+. . .-x^{17}=a_0+a_1y+a_2y^2+. . .+a_{17}y^{17},y=x+1$

This is a previous AIME question. $1-x+x^2-x^3+. . .-x^{17}=a_0+a_1y+a_2y^2+. . .+a_{17}y^{17},y=x+1$. Then what is $a_{17}$? Is anything wrong with the following method? $1-x+x^2-x^3+. . ...
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Short list of the IMO 2003

Let $b$ a integer greater than $5$. For each positive integer $n$, consider the number $$x_n=\underbrace{11\ldots1}_{n-1}\ \underbrace{22\ldots 2}_{n}\ 5$$ written in base $b$. Prove that the ...
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2answers
80 views

Show the sum is equal to a product of six primes

On a set of math challenges, one of them is to prove that 145678+456781+567814+678145+781456+814567 is the product of six different primes. This sounds like number theory to me, but I have no ...
8
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2answers
242 views

Integral $I=\int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx$

Hi I am stuck on showing that $$ \int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx=\pi G-\frac{3\zeta(3)}{8} $$ where G is the Catalan constant and $\zeta(3)$ is the Riemann zeta function. Explictly ...
2
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1answer
43 views

Integral $6\int_{x=0}^{x=1}\int_{y=x}^{y=1}\int_{z=x}^{z=y} f(x) f(y) f(z)dxdydz=\bigg(\int_0^1 f(t) dt\bigg)^3$

Prove that $$ 6\int_{x=0}^{x=1}\int_{y=x}^{y=1}\int_{z=x}^{z=y} f(x) f(y) f(z)dxdydz=\bigg(\int_0^1 f(t) dt\bigg)^3 $$ assuming $f(x)$ is continuous on [0,1]. This is from an old Putnam exam. I am ...
4
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1answer
65 views

Integral $I=\int_0^\infty \frac{e^{\alpha x}-e^{\beta x}}{x(e^{\alpha x}+1)(e^{\beta x}+1)}dx, \ \ \alpha>\beta>0. $

$$ I(\alpha,\beta)=\int_0^\infty \frac{e^{\alpha x}-e^{\beta x}}{x(e^{\alpha x}+1)(e^{\beta x}+1)}dx, \ \ \alpha>\beta>0. $$ I am trying to solve this integral. This is from the old high school ...
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1answer
16 views

Help with distance question points A and B

Ok. I had no idea how to do the question but I tried fiddling with the triangles to see if I can get any value but only managed to get $MN$. I read the solution to this question, and it said that I ...
1
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1answer
30 views

Explain the following part of the exercise…

I've been looking at this IMO 2006 contest exercise and found a thing I cannot understand: The exercise is: Now what I do not understand is , why ??? Can you explain me this? Thank you in ...
12
votes
4answers
277 views

Integral $\lim_{n\to \infty} \int_0^1 \int_0^1…\int_0^1 \cos^2\big(\frac{\pi}{2n}(x_1+x_2+…x_n)\big)dx_1 dx_2…dx_n$

I am trying to evaluate$$ \lim_{n\to \infty} \int_0^1 \int_0^1...\int_0^1 \cos^2\big(\frac{\pi}{2n}(x_1+x_2+...x_n)\big)dx_1 dx_2...dx_n. $$ This is from an old Putnam mathematics competition. Either ...
6
votes
2answers
109 views

Integrating $ \int_2^4 \frac{ \sqrt{\ln(9-x)} }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)} } dx. $

Compute $$ \int_2^4 \frac{ \sqrt{\ln(9-x)} }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)} } dx. $$ I am not sure how to start this one...I am thinking of a substitution to get started.
4
votes
1answer
83 views

determine all polynomials $P(x)$ such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial. clearly we have to show $(x+1)P(x-1)-(x-1)P(x)=c$ for all values of $x$ ($c$ is a ...
3
votes
2answers
144 views

Find the number of series

Find the number of series $(a_1,..., a_{2n})$ that have terms from ${\{0,...9\}}$ so that: $$ 11|\sum_{i=1}^{n}a_i-\sum_{i=n+1}^{2n}a_i $$ (this is not a homework) There is a similar problem ...
1
vote
2answers
135 views

Analysis problem from Romanian Contest - 2 sequences which forms another one

Let $a,b$ be 2 real numbers, and the sequences $(a_n)_{n \geq 1}, (b_n)_{n \geq 1}$ defined by $a_{1}=a$, $b_{1}=b$, $a^2+b^2 <1$ and \begin{cases} ...
2
votes
1answer
52 views

Distance between two points in the plane

my teacher asked in the class today the following question: There exists an infinite set M of points in the plane with the property that any three points are non-collinear and such that the distance ...