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

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3
votes
1answer
44 views

Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \ge \sqrt{(bc)^a(ac)^b(ab)^c}$

Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \ge \sqrt{(bc)^a(ac)^b(ab)^c}$ I tried it to do using $AM \ge GM$ but don't know how to proceed. Please help.
8
votes
1answer
119 views

How to fill up $(0,1)$ with disjoint closed intervals all total measure one

This is a problem which was proposed, but not chosen, for a mathematics competition for University students not long ago, and its solution is missing: Let $\sum_{n=1}^\infty a_n=1$, where ...
11
votes
10answers
2k views

Find five consecutive odd integers such that their sum is $55$.

So my professor asked us to do an Olympiad exercise which says that the sum of five consecutive odd integers is $55$, find those integers. But I've never seen such an exercise so it is quite new and ...
5
votes
3answers
194 views

What is the smallest natural number n?

What is the smallest natural number n for which there is a natural k, such that, the lasts 2012 digit in the representation decimal of $n^k$ are equal to 1? I don't even know how to start with it ... ...
1
vote
2answers
66 views

Posssible pentagons in 3D

A non-planar pentagon in $\mathbb{R}^3$ has equal sides and four right angles. What are the possible values for the fifth angle? My attempt It was quite easy to find an example for 60 $^\circ$: ...
1
vote
2answers
46 views

Finding the minimum of a function of two variables

Find the smallest value of $\displaystyle \sqrt{49+a^2-7{\sqrt{2}}\ a}+\sqrt{a^2+b^2-{\sqrt{2}}\ ab}+\sqrt{50+b^2-10b}\quad \quad$ for $a,b$ real and positive. What I've done so far: Let ...
7
votes
3answers
277 views

A problem with 26 distinct positive integers

I am trying to solve the following problem. Assume that we are given 26 distinct positive integers. Show that either there exist 6 of them $x_1<x_2<x_3<x_4<x_5<x_6$, with $x_1$ ...
3
votes
0answers
152 views

How many times is the digit $3$ repeated in $9^{666}$? [closed]

How many times is the digit $3$ repeated in the number $9^{666}$ ? Thanks.
4
votes
0answers
53 views

Smallest value that a certain variable can take in a system of equations.

Consider the solutions $(x,y,z,u)$ of the system of equations: $$\begin{cases} x+y=3(z+u)\\ x+z=4(y+u)\\ x+u=5(y+z)\\ \end{cases}$$ where $x,y,z \text{ and } u$ are positive integers. What ...
9
votes
4answers
647 views

Find coefficient of $x^8$ in $(1-2x+3x^2-4x^3+5x^4-6x^5+7x^6)^6$

Find coefficient of $x^8$ in $(1-2x+3x^2-4x^3+5x^4-6x^5+7x^6)^6$ how to do it? I think it should be $3^6$ since $(3x^2)^6=3^6x^8$. (this is false) Is this true?
1
vote
1answer
100 views

international mathematical competition for college students

I randomly came across with the following problems: Let $A,B \in M_n (\mathbb{C})$ such that $A^2B+B^2A=2ABA.$ Prove that $(AB-BA)^k=0$ for some positive integer $k$. The proof is as follows: Let ...
4
votes
2answers
88 views

Finding the ratio of areas produced by perpendiculars from the $3$ sides of an equilateral triangle.

A point O is inside an equilateral triangle $PQR$ and the perpendiculars $OL,OM,\text{and } ON$ are drawn to the sides $PQ,QR,\text{and } RP$ respectively. The ratios of lengths of the ...
2
votes
2answers
122 views

AIME number theory problem (unique factorization domains)

I'd greatly appreciate some help with the following problem, from a mock AIME I took. Compute the largest squarefree positive integer $n$ such that $\mathbb{Q}(\sqrt{-n})\cap \overline{\mathbb{Z}}$ ...
7
votes
1answer
170 views

A functional relation which is satisfied by $\cos x$ and $\sin x$

Assume that the functions $f,g : \mathbb R\to \mathbb R$ satisfy the relations \begin{align} \left\{ \begin{array}{ll} f(x+y) &=& f(x)f(y)-g(x)g(y), \\ g(x+y) &=& f(x)g(y)+f(y)g(x), ...
15
votes
2answers
491 views

$\cos x\,$ is the only function satisfying $\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y.$

I need to find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$. I can prove that ...
2
votes
1answer
125 views

Characterization of arithmetic mean

Let $f_m$: $\mathbb{R}_{\geq 0}^m \to \mathbb{R}_{\geq 0}$ be a series of functions that satisfy symmetry (when permuting indices), strong monotonicity (in every entry), homogeneity of degree 1, ...
2
votes
1answer
99 views

How prove this convex quadrilateral $ABCD$ is rhombus.

In a convex quadrilateral $ABCD$ ,$AB\cap CD=O$, let $r_{1},r_{2},r_{3},r_{4}$ is the radius of inscribed circle in triangle $\Delta OAB,\Delta OAD,\Delta OBC,\Delta ODC$ respectively,such ...
10
votes
1answer
130 views

A Fantabulous integer is an integer which has another fantabulous integer smaller than it

BdMO 2013 problem-7: A positive integer is called “Fantabulous” if there is another fantabulous positive integer smaller than it. Find the number of fantabulous integers. I am bamboozled at ...
4
votes
0answers
51 views

Minimizing the distance between points in two sets

Given two sets $A, B\subset \mathbb{N}^2$, each with finite cardinality, what's the most efficient algorithm to compute $\min_{u\in A, v\in B}d(u, v)$ where $d(u,v)$ is the (Euclidean) distance ...
4
votes
1answer
76 views

Elegant proof of icosohedron property

This problem was question A1 on the 2013 Putnam contest. Is there a better way to solve this problem than just using pigeonhole principle? Specifically, is there a group theoretic way to interpret ...
4
votes
1answer
94 views

Question from Putnam '08: Given $F_n(x)$, find $\lim_{n\to\infty}\frac{n!F_n(1)}{\ln(n)}$

Problem Statement: Let $F_0(x) = \ln(x)$. For $n\ge0$ and $x\gt0$, let $F_{n+1}(x) = \int_0^xF_n(t)dt$. Evaluate $$\lim_{n\to\infty}\frac{n!F_n(1)}{\ln(n)}$$ Source: Putnam 2008, Problem B2. ...
0
votes
1answer
127 views

Let $f(x)=\exp(-a|x|)$ and $a>0$. Show that there exists $C$ and $\alpha$ such that $|f(x)-f(y)|\le\frac{C|x-y|}{1+x^2}$ for $|x-y|\le\alpha$.

Let $f(x)=\exp(-a|x|)$ and $a>0$. Show that there exists $C$ and $\alpha$ such that $$|f(x)-f(y)|\le\frac{C|x-y|}{1+x^2}$$ for $|x-y|\le\alpha$. From the mean value theorem, given any $x,y$ with ...
3
votes
1answer
77 views

Assuming on the AIME?

Is it OK to assume on the AIME competition? In geometry problems, could you assume that a trapezoid is isosceles or something like that? Could you give some examples, too? Thanks.
3
votes
1answer
134 views

Find a number leaving a particular remainder with 3 different numbers

I have the following question: Let $N$ be the greatest number that will divide $1305, 4665$ and $6905$, leaving the same remainder in each case. What is the sum of digits of $N$. My approach ...
0
votes
2answers
77 views

Inequality regarding areas of triangles

BdMO Nationals 2013: There is a point O inside ∆ABC. After joining A,O; B,O and C,O extend those line and they will intersect BC, AC and AB at points D, E and F respectively. ...
2
votes
1answer
53 views

Travelling to the point of origin without using the same road twice

BdMO 2013 Secondary: There are $n$ cities in a country. Between any two cities there is at most one road. Suppose that the total number of roads is $n$. Prove that there is a city such that ...
0
votes
4answers
70 views

How to prove this ineqality

prove that $1 \leq \frac{1}{1001} + \frac{1}{1002} + ......+\frac{1}{3001} \leq \frac{4}{3} $ it seems from some Olympiad. i tried using sum of series etc. but could not get it.
1
vote
1answer
46 views

Deducing a weight function from a set function.

I was working on this problem for which I think I have almost the solution, but if you could help me finish it, I would be so grateful. ${\bf{ Problem:}}$ So, $P$ is a nonempty collection of subsets ...
1
vote
1answer
65 views

Sum of an infinite geometric series?

BdMO Nationals 12: Each room of the Magic Castle has exactly one exit door.The rooms are designed such that when you can go from one room to the next one through a door, the second room's ...
1
vote
1answer
71 views

What kind of methods there are to solve a Diophantine equation from IMO longlist?

Namely, in IMO longlist 1987 were given the equation $3z^2=2x^3+385x^2+256x-58195$ and asked to find its integer points. How can I find those? I tried to substitute $z=12k,x=6t$ to get ...
3
votes
3answers
100 views

A Question regarding radius of circumcircle and sides of a triangle

NOTE: I am looking for a hint,not the whole solution. A question from BdMO Nationals 2012 Given triangle $ABC$, the square $PQRS$ is drawn such that $P$,$Q$ are on BC, $R$ is on $CA$ and $S$ on ...
7
votes
1answer
174 views

Korean Math Olympiad (Construct rectangle)

Prove that an $m$ × $n$ rectangle can be constructed using copies of the following shape if and only if $mn$ is a multiple of 8 where $m$ > 1 and $n$ > 1. My solution: starting from 2 × 4 and 3 × 8 ...
4
votes
2answers
65 views

finding the value of $f(\frac{1}{7})$

$f$ is a function mapping positive reals between $0$ and $1$ to reals. Let $f$ be given by, $f( \frac{x+y}{2} ) = (1-a)f(x)+af(y)$ where $y > x$ and $a$ being a constant. Also,$f(0) = 0$ and $f(1) ...
3
votes
2answers
151 views

How find this $a^3+b^3+c^3-20(a+3)(b+3)(c+3)=2013$ equation integer solution

if $a,b,c\in Z$,and $a\le b\le c$ and such $$\begin{cases} a+b+c=-3\\ a^3+b^3+c^3-20(a+3)(b+3)(c+3)=2013 \end{cases}$$ Find the value $3a+b+2c=?$ my try $$a+b+c=-3\Longrightarrow ...
3
votes
1answer
145 views

IMO Hong Kong TST 2014

Let $m,n$ be distinct positive integer not exceeding 2013 and $d$ be their gcd. Suppose $d^2|3(m-n)$. Find the greatest possible value of $d(m+n)$. I only know $m-n$ should be a perfect square, but ...
6
votes
2answers
92 views

Functional Equation f(x) = f(x/2)

Find all functions $f$ satisfying the property that $$ f(x) = f(x/2) $$ for all $x \in \mathbb{R}$ So far I've come up with the following assumptions: -$f$ is periodic, i.e of form $f(x) = A ...
1
vote
1answer
44 views

Checking whether the number is composite

Prove that $5^{125}-1$/ ($5^{25} - 1$) is composite I have written $5^{125}-1$ as $(5^{25}-1)(5^{100}+5^{75}+5^{50}+5^{25}+1)$ but what should I do after this? Sorry about earlier mistake in ...
2
votes
0answers
58 views

Is the number a perfect square? [closed]

Prove that for any positive integer $n$, $n^7 + 7$ is never equal to a perfect square.
2
votes
1answer
62 views

sums of squares of integers

We have to prove that there exists infinitely many integers $a,b,c$ such that $a^2 + b^2 = c^2 + 3$ . This looked like a very straight-forward question . I did some algebraic manipulations but ...
12
votes
2answers
444 views

If 1 boy knows r girls and 1 girl knows r boys ,then number of boys=girls

Yet another question from BdMO 2013 Nationals: In a class,every boy knows $r$ number of girls and every girl knows $r$ number of boys.Show that there are equal number of boys and girls[Assume that ...
2
votes
1answer
67 views

Highest $n$ such that $2^n|a^{2012}+a^{2013}+a^{2014}+\cdots +a^{3012}$,$a=4k+2$

A question from BdMO 2013 Nationals: Let $a$ be an integer divisible by 2 but not divisible by 4. What is the largest positive integer n such that ...
2
votes
3answers
69 views

Proving $4(a^3 + b^3) \ge (a + b)^3$ and $9(a^3 + b^3 + c^3) \ge (a + b + c)^3$

Let $a$, $b$ and $c$ be positive real numbers. $(\mathrm{i})$ Prove that $4(a^3 + b^3) \ge (a + b)^3$. $(\mathrm{ii})$Prove that $9(a^3 + b^3 + c^3) \ge (a + b + c)^3.$ For the first one I ...
3
votes
2answers
89 views

2 is a primitive root mod $3^h$ for any positive integer $h$

It's easy to verify that 2 is a primitive root mod $3^2$. But then why does it follow that 2 is a primitive root mod $3^h$ for any positive integer $h$? This was used in the solution of 2009 Putnam ...
0
votes
2answers
74 views

How to prove $(F,+)$ and $(F\setminus \{0\},\cdot)$ aren't isomorphic, where $(F,+,\cdot)$ is an arbitrary field .

Assume $(F,+,\cdot)$ is an arbitrary field. How to prove $(F,+)$ and $(F\setminus \{0\},\cdot)$ aren't isomorphic? Thanks in advance.
2
votes
0answers
144 views

Korean Math Olympiad 2005 (trapezoid & tangent circles)

In a trapezoid $ABCD$ with $AD||BC$, $O_1$, $O_2$, $O_3$, $O_4$ denote the circles with diameters AB, BC, CD, DA, respectively. Show that there exists a circle with center inside the trapezoid which ...
0
votes
1answer
77 views

Figuring out an angle in an isosceles triangle

A problem from BdMO 2013: Let $ABC$ be an isoscles triangle with $AB=AC$.The bisector of $\angle B$ meets $AC$ at $D$.Given that $BC=BD+AD$,we need to figure out $\angle A$. If we consider ...
10
votes
2answers
107 views

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $\sum_{k=1}^n\frac{a_k-b_k}k$

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $$\sum_{k=1}^n\frac{a_k-b_k}k$$ By computing some partial sums, the answers are 0. It seems an inductive argument is possible.
0
votes
0answers
46 views

Find all rational solutions to $x^3 - y^2 = 2$. [duplicate]

Find all rational solutions to $x^3 - y^2 = 2$. The only integers solutions are $(3,\pm5)$: http://mathforum.org/library/drmath/view/51569.html
5
votes
2answers
84 views

Proof that b is not divisible by 6

$$b=\left \lfloor (\sqrt[3]{28}-3)^{-n} \right \rfloor$$ The brackets mean that the number is the largest integer smaller than $(\sqrt[3]{28}-3)^{-n} $ Proof that b is never divisible by 6. I have ...
14
votes
1answer
130 views

Closed form for $\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}}$

Let $$S=\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}},\tag1$$ where $\operatorname{Li}_a(z)$ is the polylogarithm. For $a=1/2$ it can be represented as ...