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

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I'm confused about something.

Alex walks from village A and Bob walks from village B towards each other starting at the same time. They meet in 8 hours. If each person increases the speed by 2 km per hour, they will meet at the ...
2
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0answers
18 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
19 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?
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2answers
92 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
35 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
75 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 ...
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1answer
79 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
36 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
54 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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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
33 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
40 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
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1answer
52 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 ...
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2answers
122 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 ...
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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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66 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. ...
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1answer
84 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 ...
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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
63 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
53 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
120 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 ...
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3answers
136 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
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$1-x+x^2-x^3+. . .-x^{17}=a_0+a_1y+a_2y^2+. . .+a_{17}y^{17},y=x+1$

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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79 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 ...
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2answers
240 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 ...
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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
64 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
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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 ...
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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
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4answers
272 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
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2answers
108 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
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1answer
82 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
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2answers
141 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 ...
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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
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1answer
51 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 ...
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2answers
49 views

Chinese remainder theorem?

In the 2014 AIME 1, number 8 says: The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit ...
6
votes
2answers
256 views

A generalization of IMO 1977 problem 2

Here is the IMO 1977 problem 2: In a finite sequence of real numbers the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Determine the ...
0
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1answer
48 views

Competition math geometry question

The perimeter of triangle ABC is $36$, and its area is $36$. Compute $\tan\frac{A}2 \tan\frac{B}2 \tan\frac{C}2$. I found that the answer is $1/9$, but I was not able to find a reason for this. Could ...
13
votes
4answers
398 views

prove Diophantine equation has no solution $\prod_{i=1}^{2014}(x+i)=\prod_{i=1}^{4028}(y+i)$

show that this equation $$(x+1)(x+2)(x+3)\cdots(x+2014)=(y+1)(y+2)(y+3)\cdots(y+4028)$$ have no positive integer solution. This problem is china TST (2014),I remember a famous result? maybe ...
2
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0answers
38 views

An algorithm for solving linear diophantine equations?

I am entering an interesting team based math contest called the purple comet, and quite a lot of questions on this contest involve Diophantine equations. For this contest, you are given a computer, ...
2
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2answers
295 views

Putnam $\bf 2001$ problem A$\bf 1$ (Binary operation)

Let $*$ be a binary operation acting on a non-empty set $S$ such that $(a*b)*a=b$ for all $a,b\in S$. Prove that $a*(b*a)=b$ for all $a,b \in S$.
4
votes
2answers
74 views

Finding the value of $(bc-ad)(ac-bd)(ab-cd)$

Let $a,b,c,d$ be $4$ distinct non-zero integers such that $a+b+c+d = 0$. It is know that the number $$M = (bc - ad)(ac - bd)(ab-cd)$$ lies strictly between $96100$ and $98000$. Determine the value ...