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

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3
votes
2answers
89 views

For what $n$ is it true that $(1+\sum_{k=0}^{\infty}x^{2^k})^n+(\sum_{k=0}^{\infty}x^{2^k})^n\equiv1\mod2$

Let $A:=\sum_{k=0}^{\infty}x^{2^k}$. For what $n$ is it true that $(A+1)^n+A^n\equiv1\mod2$ (here we are basically working in $\mathbb{F}_2$.) The answer is all powers of 2, and it's fairly simple ...
2
votes
1answer
95 views

Determine all positive integers $n$ for which $B_n=\{0\}$.

Let $A_1,A_2,...,A_n,...$ and $B_1,B_2,...,B_n,...$ be sequences of sets defined by $a_1=\emptyset$, $B_1=\{0\}$, $A_{n+1}=\{x+1|x\in B_n\},B_{n+1}=(A_n\cup B_n)\setminus(A_n\cap B_n)$. Determine all ...
4
votes
0answers
84 views

Pairwise sums are equal

The distinct positive integers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ with $n\ge2$ have the property that the $\binom{n}2$ sums $a_i+a_j$ are the same as the $\binom{n}2$ sums $b_i+b_j$ (in some order). ...
1
vote
1answer
70 views

Minimum comparisons to identify the heaviest weights.

How many times I would have to make comparisons (between 2 weights) at minimum in order to identify the heaviest weight and also the second heaviest weight out of 128 weights? I'm not sure how to do ...
8
votes
1answer
223 views

How to prove there exists a polynomial with degree at most $100\sqrt{nk}$ satisfying this condition

Show that for arbitrary positive integers $n,k$, there exists a polynomial $p(x)$, with degree at most $100\sqrt{nk}$, such that ...
4
votes
1answer
151 views

Prove that $f$ is a linear combination of $f_1,f_2,\dots,f_n$.

Let $V$ be a vector space and let $f, f_1,f_2,\dots,f_n$ be linear maps from $V$ to $\mathbb{R}$. Suppose that $f(x)=0$ whenever $f_1(x)=f_2(x)=\cdots=f_n(x)=0$. Prove that $f$ is a linear combination ...
3
votes
2answers
232 views

Find the value of $\sin 2013^\circ$

How do I find the value of $\sin 2013^\circ$? A precise decimal is not required, but must be expressed with $\sin 30^\circ,$ $\sin 45^\circ,$ and $\sin 60^\circ$ (cosine is also fine). Hint: Use ...
0
votes
1answer
63 views

On a infinite series problem of IMC

In the solution 2 of problem of 2 of IMC 1999 I want to ask why $$\sum_{n=1}^{\infty}\frac{\pi (n)}{n^2}= \sum_{n=1}^{\infty}(\pi (1)+ \pi(2)+\cdots + \pi(n))\left( ...
0
votes
3answers
253 views

geometry problem on circles from a competition

Triangle $\triangle ABC$ is an equilateral triangle whose side is $16$. A circle meets the sides of the triangle at $6$ points: it intersects $AC$ at $G$ and $F$ and $|AG|=2$, $|GF|=13$, $|FC|=1$. ...
2
votes
4answers
192 views

How do I show that $\gcd(a^2, b^2) = 1$ when $\gcd(a,b)=1$? [duplicate]

How do I show that $\gcd(a^2, b^2) = 1$ when $\gcd(a,b)=1$? I can show that $\gcd(a,b)=1$ implies $\gcd(a^2,b)=1$ and $\gcd(a,b^2)=1$. But what do I do here?
2
votes
3answers
253 views

Show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ [duplicate]

How to show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ for coprime $a$ and $b$? I know the fact that $\gcd(a,b)=1$ implies $\gcd(a,b^2)=1$ and $\gcd(a^2,b)=1$, but how do I apply this to that?
2
votes
3answers
139 views

Find the $\gcd(6, 14, 21)$ and express it in the form $6r+14s+21t$ for $r,s, t\in\mathbb{Z}$

Find the $\gcd(6, 14, 21)$ and express it in the form $6r+14s+21t$ for $r,s,t\in \mathbb{Z}$. I'm trying to learn some number theory, which starts with this gcd thing. But I ran into a problem: I ...
1
vote
3answers
88 views

Choosing $n$ objects in $2^{2n}$ ways

Of $3n+1$ objects, $n$ are indistinguishable, and the remaining ones are distinct. How that one can choose from them $n$ objects in $2^{2n}$ ways.
1
vote
1answer
67 views

If n=(sin^2(2x))/4cos^2(x))+1/(sec^2(x)) and x=2.01307, find 2013n^2013

If $n=\dfrac{sin^2(2x)}{4cos^2(x)+\dfrac{1}{sec^2(x)}}$ and $x=2.01307$, find 2013n^2013 Your edits are wrong! These are two separate fractions not together!anymore!
1
vote
3answers
323 views

Find the remainder when $11^{2013}$ is divided by $ 61$ [duplicate]

How do I find the remainder when $11^{2013}$ is divided by $61$? Brute force? Without a calculator? How did people do that?
25
votes
8answers
3k views

Reasoning that $ \sin2x=2 \sin x \cos x$

In mathcounts teacher told us to use the formula $ \sin2x=2 \sin x \cos x$. What's the math behind this formula that made it true? Can someone explain?
3
votes
4answers
1k views

$\sum_{k=0}^{n/2} {n\choose{2k}}=\sum_{k=1}^{n/2} {n\choose{2k-1}}$, Combinatorial Proof:

How am I supposed to prove combinatorially: $$\sum_{k=0}^{n/2} {n\choose{2k}}=\sum_{k=1}^{n/2} {n\choose{2k-1}}$$ ...
5
votes
2answers
194 views

Combinatorial Proof of ${n\choose{m}}=\frac{n}{m}{{n-1}\choose{m-1}}$

How do prove the following identity combinatorially? $${n\choose{m}}=\frac{n}{m}{{n-1}\choose{m-1}}$$ Any help or hints would be great!
2
votes
1answer
34 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
304 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
106 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
201 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
79 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 ...
0
votes
2answers
402 views

Lattice Paths from $(1, 1) \to (x, y)$

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
279 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
505 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
102 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
128 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
250 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
831 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
99 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
140 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
214 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
369 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
453 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
261 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
273 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
458 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.