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

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2
votes
1answer
35 views

Sports competition team gaming

Let $A,B,C,D,E,F$ be six teams in a sports competition, and each team will play exactly once with another team. Now we know that Team $A,B,C,D,E$ had already played $5,4,3,2,1$ games, ...
1
vote
1answer
124 views

find the value of 1/(2+1/(4+1/(4+1/(…))))

the question is to find the value of this ugly non-stopping fraction $$\frac{1}{2+\frac{1}{4+\frac{1}{4+\frac{1}{\ldots}}}}$$. I have totally no clue; thanks for the help! How am I suppose to solve ...
1
vote
2answers
314 views

$\frac1a+\frac1b+\frac1c=0 \implies a^2+b^2+c^2=(a+b+c)^2$? [closed]

How to prove that $a^2+b^2+c^2=(a+b+c)^2$ given that $\frac1a+\frac1b+\frac1c=0$?
2
votes
3answers
107 views

How many isosceles triangles with total side length $100$ are there?

Let the sum of the three sides of a triangle be $100,$ and all the sides are positive integers length, how many possible isosceles triangles are there?
0
votes
1answer
66 views

Find the value of $\sqrt{(b-a-4)^2}- \sqrt{(a-b+1)^2}$ if a>0 and b<0

Find the value of $\sqrt{(b-a-4)^2}- \sqrt{(a-b+1)^2}$ if $a>0$ and $b<0$. How do i find the value? This doesn't make any sense.
1
vote
0answers
197 views

Modular arithematic Equation

We have an equation: $a^x+b^x+c^x \equiv m \pmod n $ also given $a,b,c < y $ what are the total number of solutions of this equation?
3
votes
2answers
202 views

seeming ugly limit

i want to compute the limit $$\lim_{x \rightarrow 0} \frac{e^x-1-x-\frac{x^2}{2}-\frac{x^3}{6}-\frac{x^4}{24}-\frac{x^5}{120}-\frac{x^6}{720}}{x^7}$$ Instead of doing some messy calculation, I think ...
1
vote
2answers
64 views

$P(x)\in\mathbb Z$ iff $Q(x)\in\mathbb Z$

Well I have a problem on polynomial, it said like that: Let $P,Q$ be polynomials with real coefficients (that is $P,Q\in\mathbb R[x]$). We assume that for every $x\in\mathbb R$ then $P(x)\in \mathbb ...
1
vote
1answer
80 views

divisibility problem of unknown positive integer

If $a,b \in \mathbb{Z}^+$ such that $b^2+ab+1|a^2+ab+1$, prove that $a=b$. I don't have any clue on solving this problem, can anyone give me some hints? I know $a \geq b$ and $b^2+ab+1|a^2-b^2$. ...
8
votes
2answers
3k views

IMO 2013 Problem 6

Let $n\geq 3$ be an integer, and consider a circle with $n+1$ equally spaced points marked on it. Consider all labelings of these points with the numbers $0,1,\dots, n$ such that each label is used ...
1
vote
2answers
415 views

Lattice Paths from $(1, 1) \to (x, y)$ [on hold]

Moderator Note: This is a current contest question on Brilliant.org. Let $S$ be the set of $\{(1,1), (1,−1), (−1,1), (1,0), (0,1)\}$-lattice paths which begin at $(1,1),$ do not use the same ...
6
votes
1answer
280 views

Prove that: $\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}\leq1$

Let $a$, $b$, $c$ and $d$ are non-negative numbers such that $abc+abd+acd+bcd=4.$ Prove that: $\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}\leq1$ I simplified it and it turns out that ...
16
votes
1answer
507 views

How find all positive $a^3=b^2+2000000$

Find all positive integer $a$ and $b$,such $$a^3=b^2+2000000$$ This problem is from china Math competition(2013,7.10) So I think this problm have nice methods,because is from competition. Thank you ...
0
votes
1answer
103 views

Infinite Series (Telescoping?)

$$\sum_{n=0}^\infty \frac{\tan(a/2^n)}{2^n},$$ where $a$ isn't a multiple of $\pi$. I've been going through several telescoping questions, and It seems I have hit a brick wall with this one, any ...
5
votes
1answer
111 views

Existence of $j$ with strange sequence.

I define a sequence $(a_n)$ $$a_n= \begin{cases} 0 &\text{if $\cos{\left ( \dfrac{2^n\pi}{q}\right )}<-\dfrac12$} \\\\ 1 &\text{if $\cos{\left ( \dfrac{2^n\pi}{q}\right )}>-\dfrac12$} ...
3
votes
2answers
89 views

cyclic sum of primes

Suppose $p,q$ and $r$ are primes such that $pq+qr+rp-p-q-r=357$. If $p<q<r$, find the sum of all possible value of $r$. I can't identify the kernel of the problem, can anyone give me a hint, ...
1
vote
1answer
131 views

Sums and products involving Fibonacci

In summary, if $\phi$ is the golden ratio, I want to show: \begin{align} \sum_{n=1}^\infty \frac1{F_n} &= 4-\phi \\ \sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{F_nF_{n+1}} &= \phi-1 \\ ...
0
votes
2answers
48 views

Given $a > b+c$, $e>d+f$, and $i>g+h$, can the quantity $a(ei-hf) + b(-di+fg) - c(dh+eg)$ ever be zero?

Given positive reals $a > b+c$, $e>d+f$, and $i>g+h$, can the quantity $a(ei-hf) + b(-di+fg) - c(dh+eg)$ ever be zero?
10
votes
4answers
253 views

Fibonacci Cubes: $F_n^3 + F_{n+1}^3 - F_{n-1}^3 =F_{3n}$

Prove $$F_n^3 + F_{n+1}^3 - F_{n-1}^3 =F_{3n}$$ I've tried induction, either its just very long or a neat trick is required in the inductive step but for some odd reason its not working out. ...
20
votes
1answer
844 views

2013 USAMO problem 5

This is currently unsolved in the AoPS site, the problem says: Given postive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same ...
1
vote
3answers
100 views

Solving for $(x,y): 2+\frac1{x+\frac1{y+\frac15}}=\frac{478}{221}$

Solving for $x,y\in\mathbb{N}$: $$2+\dfrac1{x+\dfrac1{y+\dfrac15}}=\frac{478}{221}$$ This doesn't make any sense; I made $y+\frac15=\frac{5y+1}5$, and so on, but it turns out to be a very ...
1
vote
4answers
137 views

Simplifying the expression $(1+\sqrt[4]3)/(1-\sqrt[4]3)+1/(1+\sqrt[4]3)+2/(1+\sqrt{3})$

Can anyone give provide me some help to simplify this expression? The three denominators are pretty much different, and I can't find a common denominator. ...
1
vote
1answer
133 views

Simplifying the expression $(\sqrt{5}+\sqrt{7})/(\sqrt{10}+\sqrt{14}+\sqrt{15}+\sqrt{21})$

Alrite guys, this question might sound stupid, but I can't find a way to simplify this complicated expression: $$\frac{\sqrt{5}+\sqrt{7}}{\sqrt{10}+\sqrt{14}+\sqrt{15}+\sqrt{21}}$$ I can't take the ...
-3
votes
3answers
148 views

Let $f$ be a function satisfying such that $f(xy)=\frac{f(x)}{y}$, what is $f(600)$? [closed]

Let $f$ be a function satisfying such that $f(xy)=\frac{f(x)}{y}$ for all positive real numbers $x$ and $y$. Given that $f(500)=3$, what is $f(600)$?
9
votes
2answers
2k views

Preparing for Mathematics Olympiad

Friends, Please don't take it as an off-topic question. I actually want to learn different concepts of maths and physics as well as of chemistry. I am preparing for Mathematics Olympiad , can any ...
5
votes
1answer
215 views

Summation of weighted squares of binomial coefficients

Show that $$\sum_{k=0}^n \left[ \frac{n-2k}{n} {n\choose k}\right]^2=\frac{2}{n}{2n-2 \choose n-1}.$$
5
votes
4answers
370 views

Solutions to $p+1=2n^2$ and $p^2+1=2m^2$ in Natural numbers.

$$p+1=2n^2$$$$p^2+1=2m^2$$ Find positive integers $m,n$ and prime $p$ satisfying the above two equations. What would people commonly do? Subtracting both the equations. You get: ...
1
vote
2answers
468 views

Sums $\sum_{k=1}^n \sin(2k-1)\theta$, $\sum_{k=1}^n \sin^2(2k-1)\theta $

To prove: $1.$ $$\sum_{k=1}^n \sin(2k-1)\theta = \frac{\sin^2 n\theta}{\sin \theta}.$$ $2.$ $$\sum_{k=1}^n \sin^2(2k-1)\theta = \frac{n}{2} - \frac{\sin 4n\theta}{4\sin 2\theta}.$$
1
vote
1answer
268 views

Math Olympiad problem

A year is peculiar if the sum of the first two digits and the last two digits is equal to the middle two digits. For example, 1978. When was the last peculiar year and is there an algorithm to find ...
0
votes
1answer
278 views

Given a polynomial $P$ find $Q$ such that $Q(x)-Q(x-1)-Q(x-2)=P(x)$ for all $x$

Let $P\in\mathbb{Z}[x]$ be a given polynomial of degree $d$. I want to find the unique polynomial $Q\in\mathbb{Z}[x]$ of degree $d$ such that $Q(x)-Q(x-1)-Q(x-2)=P(x)$. It is possible to construct the ...
3
votes
1answer
462 views

Throw a die three times, and get maximum number of different sums.

The IBM Ponder This problem for July 2013 throws an 8 sided die 3 times, and can get 120 possible different positive integer sums. If all the faces have positive integer sides, what is the lowest ...
3
votes
0answers
87 views

Find integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers? [duplicate]

Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers? $a,b$ are distinct integers. P.S.: I think trying to find some special cases would not be helpful.
0
votes
3answers
467 views

Prove that $a^3+b^3+c^3 \geq a^2b+b^2c+c^2a$

Let $a,b,c$ be positive real numbers. Prove that $a^3+b^3+c^3\geq a^2b+b^2c+c^2a$. My (strange) proof: $$ \begin{align*} a^3+b^3+c^3 &\geq a^2b+b^2c+c^2a\\ \sum\limits_{a,b,c} a^3 &\geq ...
7
votes
1answer
137 views

How find this equation all solution $2a^2-1=b^{2013}$

let $a,b\in \mathbb Z$, and such $$2a^2-1=b^{2013}$$ find all value $a,b$ I think $(a,b)=(0,-1),(1,1),(-1,1)$ is solution, and have other solution. Thank you everyone
0
votes
1answer
131 views

Optimizing $x^2+y^2$ from two given equations? [duplicate]

What is the maximum value of $x^2+y^2$, where $(x,y)$ are solutions to: $$2x^2+5xy+3y^2=2$$ and $$6x^2+8xy+4y^2=3$$ Note: Calculus is not allowed. I tried everything I could but whenever I got for ...
4
votes
1answer
167 views

Given the norm of a Gaussian integer, how to find the original Gaussian integer?

For $p= a + bi\in\mathbb{Z}[i]$, its norm is $$N(p) = (a + bi)(a - bi) = a^2 + b^2.$$ For example, $N(2+7i) = 2^2+7^2 = 4+49 = 53$. How to find $2+7i$ from $53$? Is there any method?
10
votes
2answers
169 views

Show that there are infinitely many solutions of distinct natural numbers $m,n$ such that $n^3+m^2\mid m^3+n^2$

Show that there are infinitely many solutions of distinct natural numbers $m,n$ such that $n^3+m^2\mid m^3+n^2$. This question appeared in Round $2$ of British Math Olympiad $2007-08$. I have ...
1
vote
1answer
119 views

prove that there are infinite positive numbers $\overline{a_{1}a_{2}a_{3}\cdots a_{n}a_{1}a_{2}a_{3}\cdots a_{n}}$ is square number

Show that there are infinitely many positive integer number $A=\overline{a_{1}a_{2}a_{3}\cdots a_{n}}$,and $0\le a_{i}\le 9$,such that $\overline{a_{1}a_{2}a_{3}\cdots a_{n}a_{1}a_{2}a_{3}\cdots ...
13
votes
1answer
189 views

integral inequality $\int_0^a \left(\frac{f(x)}{2x}\right)^2 dx \le \int_0^a (f'(x))^2 dx$

Let $f:[0,a]\rightarrow\mathbb{R}$ be continuous differentiable function satisfying $f(0)=0$. Prove the following inequality $$\int_0^a \left(\frac{f(x)}{2x}\right)^2 dx \le \int_0^a (f'(x))^2 dx$$ ...
3
votes
3answers
101 views

Manipulating Algebraic Expression

$a + b + c = 7$ and $\dfrac{1}{a+b} + \dfrac{1}{b+c} + \dfrac{1}{c+a} = \dfrac{7}{10}$. Find the value of $\dfrac{a}{b+c} + \dfrac{b}{c+a} + \dfrac{c}{a+b}$. I algebraically manipulated the ...
6
votes
4answers
303 views

Polynomials Question: Proving $a=b=c$.

Question: Let $P_1(x)=ax^2-bx-c, P_2(x)=bx^2-cx-a \text{ and } P_3=cx^2-ax-b$ , where $a,b,c$ are non zero reals. There exists a real $\alpha$ such that ...
-2
votes
1answer
1k views

Solve: $T(n) = T(n-1) +(1/n)$ by iteration

Use iteration method to solve: $1.$ $T(n) = T(n-1) + \frac{1}{n},\,(T(0)=1)$ $ 2.$ $T(n) = 3T\left(\dfrac{n}{3}\right) +1,\,(T(3)=1)$
4
votes
4answers
222 views

Mathcounts 2013 state sprint round #14

How many ways can all six numbers in the set $\{4, 3, 2, 12, 1, 6\}$ be ordered so that $a$ comes before $b$ whenever $a$ is a divisor of $b$?
2
votes
2answers
160 views

competition math references?

i want to prepare for competition math next years I am searching reference and some book or website about it What resources are available to prepare me for the competition math? if anyone had any ...
3
votes
2answers
191 views

choosing $5$ non consecutive books from a shelve of $12$

In how many ways can you pick five books from a shelve with twelve books, such that no two books you pick are consecutive? This is a problem that I have encountered in several different forms ...
3
votes
3answers
378 views

There are infinitely many $N$ such that $\frac{N}{2}$ is a perfect square, $\frac{N}{3}$ is a perfect cube, and $\frac{N}{5}$ is a perfect fifth power

Show that there are infinitely many $N$ such that $\frac{N}{2}$ is a perfect square, $\frac{N}{3}$ is a perfect cube, and $\frac{N}{5}$ is a perfect fifth power. A hint is given with this ...
3
votes
2answers
250 views

Proving there are no integer solutions for $3x^2=9+y^3$

Prove there are no $x,y\in\mathbb{Z}$ such that $3x^2=9+y^3$. Initial proof Let us assume there are $x,y\in\mathbb{Z}$ that satisfy the equation, which can be rewritten as $$3(x^2-3)=y^3.$$ So, ...
4
votes
1answer
139 views

Probability that the first digit of $2^{n}$ is 1

Let $a_{n}$ be the number of terms in the sequence $2^{1},2^{2},\cdots ,2^{n}$ which begins with digit 1. Prove that $$\log2 -\frac{1}{n}<\frac{a_{n}}{n}<\log2\text{ (log base is 10)}$$ ...
3
votes
2answers
126 views

Berkeley exam summer '79, sequence of continuous functions, integral, convergence

I've recently been browsing some Berkeley exams and I'm particularly interested in Problem 19 here. Let ${f_n}$ be a sequence of continuous real functions defined on $[0,1]$ such that $\int_0^1 ...
7
votes
1answer
132 views

Subgroup of elements of order at most $2^{m}$

The problem A5 in Putnam 2009 reads as follows: Is there a finite abelian group $G$ such that the product of the orders of all its elements is $2^{2009}$? The answer is No. I am reading the ...