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

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4
votes
2answers
515 views

how to prove following matrix is invertible? [duplicate]

how to prove A is invertible or $\ detA\neq 0$ $$A=\begin{pmatrix} \frac11 & \frac12 & \frac13 & \cdots & \frac1n \\ \frac12 & \frac13 & \frac14 & \cdots & ...
5
votes
4answers
6k views

Expected Number of Coin Tosses to Get Five Consecutive Heads

A fair coin is tossed repeatedly until 5 consecutive heads occurs. What is the expected number of coin tosses?
0
votes
1answer
386 views

What is the most number of regions that 9 lines can cut the plane into [duplicate]

0 lines cuts the plane into at most 1 region. A line cuts the plane into at most 2 regions. 2 lines cut the plane into at most 4 regions. What is the most number of regions that 9 lines can cut ...
0
votes
3answers
251 views

A circle has diameter $AD$ of length $400$

A circle has diameter $AD$ of length $400$. $B$ and $C$ are points on the same arc of AD such that $|AB|=|BC|=60$. What is the length $|CD|$?
5
votes
2answers
80 views

Follow on from previous question: Functional Equation - a little tricky

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f[f(x)^2+f(y)]=xf(x)+y$ for all real numbers $x$ and $y$. The answer to this has already been posted, but it doesn't explain why ...
2
votes
2answers
111 views

Minimum Value of expression

Given that $x$, $y$ and $z$ are positive real numbers satisfying $xyz=32$, find the minimum value of: $$x^2+4xy+4y^2+2z^2$$ Perhaps AM-GM and manipulation but I'm not quite sure how? Source BMO.
1
vote
0answers
196 views

What is Putnam exam/competition?

Just heard of Putnam competition. There is not a lot of info about it on the net. Could you tell me a little about it? What is its purpose? What kind of math levels does it test? Also, I found a ...
5
votes
2answers
129 views

Showing $\{x\} + \{\frac{1}{x}\} \lt 1.5$ and other problems.

For any real number $x$, let $[x]$ be the greatest integer not exceeding $x$. We also define $\{x\}=x-[x]$. We now define the function: $f(x)=\{x\}+\{\frac{1}{x}\}$. (a) Prove that $f(x)<1.5$ for ...
-1
votes
3answers
135 views

Find the smallest natural number that satisfy $13^N = 1 \pmod {2013}$

Moderator Note: This is a current contest question on Brilliant.org. Find the smallest natural number that satisfy: $$13^N = 1 \pmod {2013}$$ My idea is to use the Fermat's little theorem ...
8
votes
2answers
225 views

Functional Equation: a little tricky

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f[f(x)^2+f(y)]=xf(x)+y$ for all real numbers $x$ and $y$. Clearly $f(x)=x$ is a solution, check by substitution. I'm at a loss as ...
5
votes
2answers
324 views

Find all positive integers $a, b, c$ such that $1/a + 1/b + 1/c = 4/3999$

Find all positive integers $a, b, c$ such that $1/a + 1/b + 1/c = 4/3999$. The contest is just ended, so you may freely answer. (I did not attend the contest: it is an Italian contest for schools and ...
1
vote
0answers
66 views

Matheamatics competition for university students

Currently I am studying mathematics as my major in university and I would like to join International Mathematics Competition next year or next next year. So what can I do and what books should I read ...
9
votes
5answers
306 views

How to find $\int_0^\pi (\log(1 - 2a \cos(x) + a^2))^2 \mathrm{d}x, \quad a>1$?

Integration by parts is of no success. What else to try? $$\int_0^\pi (\log(1 - 2a \cos(x) + a^2))^2 \mathrm{d}x, \quad a>1$$
1
vote
1answer
88 views

3 complex-variable equation

Moderator Note: This is a current contest question on Brilliant.org. $x,y,z$ are complex numbers satisfying $$ \begin{align} x+y+z & =1\\ x^2+y^2+z^2 & =2\\ x^3+y^3+z^3 & =3 ...
-3
votes
2answers
135 views

Lucky Lattice Points

How many lattice points lie on the sphere given by following equation ? $$x^2+y^2+z^2=2013$$ Hint: A lattice point has integer coordinates.
1
vote
2answers
151 views

Ordered triples solution to system of equations

How many ordered triples $(x,y,z)$ of integer solutions are there to the following system of equations? $$ \begin{align} x^2+y^2+z^2&=194 \\ x^2z^2+y^2z^2&=4225 \end{align} $$
0
votes
0answers
177 views

Sum of squares function

The function rk(x) denotes the number of ways an integer x can be expressed as the sum of squares of k integers [the integers can be positive, can be negative, can be zero]. What is the value of ...
-2
votes
1answer
150 views

From a Generating Function find $R(x)$ as an infinite product of Quotients

Let $r(n)$ be the number of partitions of $n$ so that no multiple of $3$ appears as a part. For example, $r(8) = 13$. Let $R(x) =\sum_0^\infty r(n)x^ n $ be the generating function for $r(n)$. Find ...
-2
votes
1answer
200 views

Find a form for $Q(x)$ as an infinite product of polynomials

Let $q(n)$ be the number of partitions of $n$ so that no part appears three or more times. For example, $q(8) = 13$ Let $Q(x) = \sum\limits_{n=0}^\infty q(n) x^n$ be the generating function for ...
-6
votes
1answer
751 views

How to find a Recurrence Relation from a word problem?

Suppose you have 5 kinds of wooden blocks: red blocks which are 2 inches high, blue blocks which are 2 inches high, green blocks which are 2 inches high, yellow blocks which are 3 inches high, and ...
-2
votes
1answer
494 views

Find a closed form for a generating function and recurrence

Find a closed form for the generating function $R(x) = \sum_{n=0}^\infty r_nx^n$, where $r_n$ is given by the recurrence $r_n = 3r_{n-1} + 5r_{n-2} + 6n$ for $n \geq 2$ and initial conditions $r_0 = ...
1
vote
2answers
146 views

Finding the distance between the $x$-intercepts of two lines

A line with slope $4$ intersects a line with slope $7$ at the point $(10,28)$. What is the distance between the $x$-intercepts of these two lines? This question was asked in a Math Competition in ...
3
votes
2answers
150 views

Finding the number of different ordered quadruples $(a,b,c,d)$ of complex numbers

Find the number of different ordered quadruples $(a,b,c,d)$ of complex numbers such that: $$a^2=1$$ $$b^3=1$$ $$c^4=1$$ $$d^6=1$$ $$a+b+c+d=0$$
2
votes
1answer
78 views

functional equation problem in competition

Find all $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that $f(1)=2$ and $f(xy)=f(x)f(y)−f(x+y)+1$. for all $x,y \in \mathbb{Q}$. thank you very much!
17
votes
3answers
891 views

Finding all integer solutions of $5^x+7^y=2^z$

Find all integers $x,y,z$ such that $5^x+7^y=2^z$. This one comes from an online contest that I arranged some years ago, and I can assure that a completely elementary solution exists.
4
votes
2answers
290 views

Solving $x+2y+5z=100$ in nonnegative integers

I have not done combinatorics since high school, so this is an embarrassingly simple question. We can solve the diophantine equation $x+y+z=100$ in nonnegative integers using the "bars and boxes" ...
7
votes
1answer
103 views

How to find a limit of this sequence?

I would appreciate if somebody could help me with the following problem: Let the sequences $\{a_n\}$, $\{b_n\}$ be defined as $$a_n=\int\limits_0^{\pi/2}\sin^{3n} x \;dx,\qquad ...
1
vote
2answers
91 views

What are the number of greatest/least possible maxima for $f(x) = \frac{x^n}{x-1}$?

I have a math contest question that I found in my textbook and I have no idea where to start. Please provide some hints as to how to go about solving this: Consider the function $f(x) = ...
2
votes
1answer
64 views

Tennis Tournament - Olympic training

A tennis tournament is played between two teams. Each member of a team plays with one or more members of the other team, so that i) Two members of the same team have exactly one opponent in common. ...
2
votes
1answer
60 views

If $H$ has an $a\times b$ submatrix of all $1$s, please prove that $ab\le n$.

Let $H$ be an $n\times n$ matrix with entries $\pm1$. Its rows are mutually orthogonal. If $H$ has an $a\times b$ submatrix of all $1$s, please prove that $ab\le n$.
4
votes
1answer
118 views

Prove that the polynomial $P(x_1,x_2…,x_n)=0$ given a set of conditions.

Let $P(x_1,...,x_n)\in\mathbb{R}[x_1,...,x_n]$ (i.e. $P$ is a polynomial of real coefficients in $x_1,..,x_n$). We are given that $\left(\frac{\partial^2}{\partial ...
5
votes
1answer
71 views

increasing sequence specific properties

I am doing some olympiad exercises and have difficulties with the following one: Consider a sequence $a_1,a_2,...$ which is strictly monotonically increasing and $a_1,a_2,...\in\mathbb N$ Now I know ...
0
votes
4answers
292 views

How to verify method used to solve integral was actually the fastest?

Is there any way to verify if the method I chose to integrate (by hand) was indeed fastest, or if there exists some better technique? Can a computer tell me or show me what the fastest method was, ...
2
votes
1answer
56 views

Trigonometric Sums - URSS

Calculate the value of the sums: (a) $\cos x+\binom{n}{1}\cos 2x +\cdots+\binom{n}{n} \cos (n+1)x $; (b) $\sin x+\binom{n}{1}\sin 2x +\cdots+\binom{n}{n} \sin (n+1)x $.
4
votes
1answer
195 views

An integral involving the error function

I have in my notes the following problem. I recall it being quite difficult and needing a change of variables into polar or spherical coordinates. Assuming I have not made a typo, there is a nice ...
6
votes
2answers
140 views

Trying to recall an integration trick

In my notes, I have the following problem. Find the volume of (a) $x^2+y^2 \le 1$, $x^2+z^2\le 1$ in $\mathbb R^3$ (b) $x^2+y^2 \le 1$, $x^2+z^2\le 1$, $y^2+z^2\le 1$ in $\mathbb R^3$ ...
1
vote
1answer
318 views

Let $N$ = $11^2 \times 13^4 \times 17^6$. How many positive factors of $N^2$ are less than $N$ but not a factor of $N$?

Let $N$ = $11^2 \times 13^4 \times 17^6$. How many positive factors of $N^2$ are less than $N$ but not a factor of $N$? $Approach$: $N$=$11^2$.$13^4$.$17^6$ $N^2$=$11^4$.$13^8$.$17^{12}$ This ...
1
vote
0answers
73 views

Pi identity with sum and product

Please prove this identity $$\sum_{ n=1 }^{\infty }\left({\left(-1\right) }^{ n }\frac{\prod_{ j=1 }^{ n }{\left(\frac{ 3 }{ 2 }-j\right) }}{\left( 2n+1\right)\left( n!\right) }\right) =\frac{\pi }{ ...
2
votes
2answers
91 views

Combinatorial Sum

I am trying to prove $$0^2 \binom{n}{0}+3^2\binom{n}{3}+6^2\binom{n}{6}+ \cdots + \left[\dfrac{n}{3}\right]^2 \binom{n}{\left[\dfrac{n}{3}\right]},$$ where $[x]$ is the greatest integer not exceeding ...
3
votes
1answer
51 views

$\lim_{x\to\infty} x^{1+1/x}-x-\log x$ and $\lim_{x\to\infty}\frac{x^{1+1/x}-x}{\log x}$

Evaluate $$\lim_{x\to\infty} x^{1+1/x}-x-\log x$$ and $$\lim_{x\to\infty}\frac{x^{1+1/x}-x}{\log x}$$ Would knowing one necessarily give the other?
2
votes
1answer
33 views

Prove that $\lim_{t\to0}(\log t)(1-(2t)^{t/2})=0$

Please prove that $(\log t)(1-(2t)^{t/2})$ tends to 0 as t tends to 0. http://www.wolframalpha.com/input/?i=lim+t-%3E0+%281-%282t%29^%28t%2F2%29%29logt It seems the limit converges to 0 pretty ...
1
vote
1answer
70 views

Vertex of a pentagon-Does the algorithm always stop?

To each vertex of a pentagon,we assign an integer $x_{i}$ with sum $s=\sum x_{i}>0$. If x,y,z are the numbers assigned to 3 successive vertices and if $y<0$,then we replace $(x,y,z)$ by ...
7
votes
4answers
389 views

Given that $f(1)= 2013,$ find the value of $f(2013)$?

Suppose that $f$ is a function defined on the set of natural numbers such that $$f(1)+ 2^2f(2)+ 3^2f(3)+...+n^2f(n) = n^3f(n)$$ for all positive integers $n$. Given that $f(1)= 2013$, find the value ...
1
vote
1answer
52 views

how to prove this question about limit and derivative

Suppose $f:(a,b)\to\mathbb R$ that $ f $ satisfies: $$f\in C^1$$ $$\lim_{x\to a ^ +}f^2(x)=0$$ $$\lim_{x\to b ^ -}f^2(x)=e-1$$ if $\forall x \in(a,b) : 2f(x)f '(x)-f^2(x)\ge1 $, then how to ...
0
votes
1answer
55 views

Equation with fractional parts

How many solutions has the equation: $\{20 \cdot \{ 13 \cdot\{20 \cdot\{ 13\cdot x\}\} \}\}=x^{2013}$ -? Here $\{z\}=z-[z]$, where $[z]=m \in \mathbb{Z}, \ m \le z < m+1$.
10
votes
2answers
124 views

how prove $n! \mid \prod_{k=0}^{n-1}(2^n-2^k) :\forall n \in \mathbb N$?

how prove $$n! \mid \prod_{k=0}^{n-1}(2^n-2^k) :\forall n\in \mathbb N $$ Thanks in advance
2
votes
3answers
76 views

Help with inequality please

Once again I have come across an olympiad-type problem which probably requires some sort of insight even though it looks simple. The question is as follows: Let $a$, $b$ and $c$ be positive real ...
11
votes
2answers
231 views

Functions satisfying $f(m+f(n)) = f(m) + n$

I am a real newbie when it comes to funtions, and I don't understand what is supposed to happen or what I'm supposed to find when I get given an olympiad type question concerning functions. Could you ...
6
votes
1answer
181 views

how to prove $\sum _{|k|\lt\sqrt m}\binom{2m}{m+k}\ge2^{2m-1}$

how to prove $$\sum _{|k|\lt\sqrt m}\binom{2m}{m+k}\ge2^{2m-1},\forall m\ge1$$ Thanks in advance .
8
votes
1answer
334 views

Probem proposed for IMO 26

I have come across this problem and I really don't know how to construct this. Any ideas would be very much apreciated. Given 3 concentric circles, construct an equilateral triangle with a vertex on ...