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

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4
votes
2answers
134 views

I have used Cauchy and Jensen. It is not helping me very much. Advice on solving this problem.

Let $a$, $b$ and $c$ be positive real numbers with $abc=1$. Prove that $$ \frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $$ for each ...
-4
votes
1answer
205 views

Count good numbers in between L and R

Let length(A) denote the count of digits of a number A in its decimal representation. All non-negative numbers of length 1 are Good. Further, a number X with length(X) $≥ 1$ can also be considered ...
0
votes
1answer
57 views

Any thoughts on how to solve this problem? [closed]

How many numbers do there exist having 2013 digits, in which every two-digit number composed of two consecutive digits is a multiple of either 17 or 23? (Taken from Singapore and Asian Schools Math ...
2
votes
2answers
285 views

Interpolation polynomial Challenge

suppose $p(x)=x^k-x^t, k \neq t $ (k,t is a positive integer). function q(x) be a Interpolation polynomial from degree lower or equal n, to data $i=1,...,n+1, (x_i ,p(x_i))$. if ----------- then ...
4
votes
1answer
156 views

Polynomials with coefficients $1$ or $2014$ [closed]

Let $$ P(x)=a_mx^m+a_{m-1}x^{m-1}+ \cdots+a_1x+a_0$$ and $$\quad Q(x) =b_nx^n+b_{n-1}x^{n-1}+ \cdots+b_1x+b_0 $$ be two polynomials where $a_i,b_j \in \{1,2014\}$ for all $i,j$. Suppose that ...
3
votes
1answer
87 views

Functions for non-negative integers [duplicate]

Let $\Bbb{Z}^+$be the set of all non-negative integers where $n$ and $k$ are given natural numbers. We consider the following non-decreasing function, $$f:\Bbb{Z}^+ \to \Bbb{Z}^+$$ such that ...
1
vote
1answer
52 views

Remainder of a summation divided by $2^{12}$

For a positive integer $n$, let $f(n)$ be equal to $n$ if there is an integer $x$ such that $x^2-n$ is divisible by $2^{12}$, and let $f(n)$ be $0$ otherwise. Determine the remainder when ...
0
votes
0answers
13 views

Integer multiplication in 5T(n/3) [duplicate]

x and y has n bits x=x0*(10^2n/3)+x1*10^n/3+x2 y=y0*(10^2n/3)+y1*10^n/3+y2 x*y=x2y2+(x2y1+x1y2)10^n/3+(x2y0+x1y1+x0y2)10^2n/3+(x1y0+x0y1)10^n+x0y0*10^4n/3 now 9 multiplication of n/3 bit numbers ...
2
votes
3answers
91 views

Given $x+y+z=3$ prove that $4 \geq x^2y+y^2z+z^2x$

Given positive reals $x,y,z$, and $x+y+z=3$ prove that $$4 \geq x^2y+y^2z+z^2x$$. This problem was homogenized so I set $x+y+z=3$ to cancel stuff. Now I'm stuck. I have noticed a funny equality ...
1
vote
2answers
99 views

Mind Teasers : Difficult Brain Twister (Today Challenge)

Question can be found in the link below Source: http://gpuzzles.com/mind-teasers/difficult-brain-twister/
8
votes
1answer
201 views

How to prove that $\frac{1}{x_1}+\frac{1}{x_2}+…+\frac{1}{x_n}-\frac{1}{x_1x_2…x_n}\in \mathbb{N}\cup \{0\}$

Question: Show that for every natural number $n$ there exist $n$ natural numbers $ x_1 < x_2 < ... < x_n ,$ such that $$ ...
11
votes
1answer
526 views

A Putnam Integral $\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)} + \sqrt{\ln(x+3)}}.$

This is a Putnam Problem that I have been trying to solve (on and off) for two years, but I have failed. I am in Calculus BC. This problem comes from the book "Calculus Eighth Edition by Larson, ...
2
votes
3answers
152 views

How many strings of $8$ Hs and $8$ Ts are there such that there are at most $2$ consecutive Hs?

How many strings of 8 Hs and 8 Ts are there such that there are at most 2 consecutive Hs? I don't really understand how to approach this question. What would be the quickest way to solve it? Thanks ...
3
votes
1answer
84 views

If $f,g$ are $2$ onto homomorphisms, $\exists$ $y \neq e \in M$ such that $f(y)=g(y)$.

The below problem appeared in Schweitzer contest. Let $M$ and $N$ be two groups of finite order and let $f,g : M \to N$ be $2$ onto but not injective group homomorphisms. Then prove that there ...
7
votes
0answers
81 views

How are inequalities from IMO built?

I notice that there are lots of apparently difficult inequalities in IMO. Are there some techniques to manipulate well-known inequalities in order to built a difficult exercise? What are the main ...
2
votes
2answers
80 views

Functional equation - Understading an easy step in my solution.

I am trying to solve the equation and find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that: $f(m+f(n))=f(f(m))+f(n)$ for all $n, m \in \mathbb{N_{0}} $. A reasonable approach to begin with ...
1
vote
0answers
24 views

Maximum number of non-zero entries ,such that no two non-zero entries are on the same row or column.

In an M x N matrix such that all non-zero entries are covered in "a" rows and " b" columns. Then the maximum number of non-zero entries ,such that No two non-zero entries are on the same row or column ...
5
votes
2answers
254 views

Prove that there are infinitely many integer solutions to a diophantine equation

Prove that there are infinitely many integer solutions to the diophantine equation: $(x-y)^7 = x^3y^3$
3
votes
2answers
49 views

Minimum sum of set whose average of subsets is positive integer

A finite set of positive integers $A$ is called meanly if for each of its nonempty subsets the arithmetic mean of its elements is also a positive integer. In other words, $A$ is meanly if ...
11
votes
5answers
207 views

Show that $({\sqrt{2}\!+\!1})^{1/n} \!+ ({\sqrt{2}\!-\!1})^{1/n}\!\not\in\mathbb Q$

How could we prove that for every positive integer $n$, the number $$({\sqrt{2}+1})^{1/n} + ({\sqrt{2}-1})^{1/n}$$ is irrational? I think it could be done inductively from a more general ...
1
vote
1answer
38 views

Functional equation- solving techniques

I'm basically a total novice with functional equations and have some questions regarding the solving technuiqes of them. Although, i'm adware of the lack of general solving methods, I have noticed ...
0
votes
1answer
61 views

Looking for mathematics contests

I want to hone my problem solving skills. I have been working on the Putnam exam, but I have found that the easy problems are a bit too easy and the hard problems often require advanced number theory ...
1
vote
4answers
103 views

Example of a non-trivial function such that $f(2x)=f(x)$

Could you give an example of a non-constant function $f$ such that $$ f(x) = f(2x). $$ The one that I can think of is the trivial one, namely $\chi_{\mathbb{Q}}$, the characteristic function on the ...
3
votes
2answers
97 views

Nice Question in Mathmatics about Times

I ran into a nice question from one book in Discrete Mathematics. I want to someone lean me how solve such a problem, because I prepare for entrance exam. if the time is "Wednesday 4 ...
4
votes
2answers
162 views

Polynomial satisfying $ P \big(P (x)\big)=P (x)+ P\big(x^2\big)$

If $P(x)$ is a polynomial with integer coefficients such that for all integer $x$, $$P (P (x)) = P (x)+P (x^2).$$ I've tried solving it putting it as a function instead. Not much though. How do you ...
0
votes
1answer
26 views

Extend Metric Space Challenge

Let $(E, D)$ be a metric space. Consider $D_1: E\times E \to \mathbb{R}$ where $$ D_1(x,y)=\frac{D(x,y)}{1+ D(x,y)}. $$ I read some note about it but I want to find why $D_1$ is also a metric and ...
3
votes
3answers
66 views

Prove the following fraction is irreducible

Prove $\frac{21n + 4}{14n + 3}$ is irreducible for every natural number $n$. I was thinking of taking a number-theory based approach. Can you suggest the following method Calculus/Number theory ...
0
votes
1answer
64 views

A large number divisible by 4 [closed]

Let $S=\displaystyle x! + \sum_{k=0}^{2013} k!$, where $x$ is a one-digit non-negative integer. How many possible values of $x$ are there so that $S$ is divisible by 4?
0
votes
1answer
46 views

Find the minimum value of an algebraic expression using simple algebra

Find the minimum value $2a^8+2b^6+a^4-b^3-2a^2-2$, where $a$ and $b$ are real numbers. I was told to use Lagrange multipliers but I found out this belongs to Calculus department. I tried factoring the ...
1
vote
1answer
37 views

simplifying expression of roots of cubic equation

I came across with this question about roots of polynomials. Suppose $a$, $b$ and $c$ are the roots of $x^3-4x+1=0$. Find the value of $ \frac{a^2bc}{a^3+1}+\frac{ab^2c}{b^3+1}+\frac{abc^2}{c^3+1}. $ ...
2
votes
0answers
48 views

Prove that $a,b,c$ are the sides of a triangle

$a,b,c\in\mathbb R_{>0}$ are such that $$\begin{cases}a^2x+b^2y+c^2z=1\\xy+yz+zx=1\end{cases}$$ has a unique solution $(x,y,z)\in\mathbb R^{3}$. Prove that $a,b,c$ are the sides of a ...
3
votes
1answer
69 views

Find all functions $f:\Bbb Q\rightarrow\Bbb Q$ satisfying $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y\in\Bbb Q$

Find all functions $f:\Bbb Q\rightarrow\Bbb Q$ satisfying $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y\in\Bbb Q$ I don't know how to proceed, any help would be really appreciated..
0
votes
2answers
107 views

partitions and their generating functions and Partitions of n

A partition of an integer, n, is one way of writing n as the sum of positive integers where the order of the addends (terms being added) does not matter. p(n, k) = number of partitions of n with k ...
1
vote
1answer
51 views

Competencia Iberoamericana Interuniversitaria

Let $f$ a rational function with complex coeficients and without mutiple roots in the denominator. Let $u_0,u_1,...,u_n$ ($n \ge 1$) complex roots of $f$ and $w_1,w_2,...,w_n$ roots of $f'$ (each root ...
2
votes
0answers
26 views

How prove that there are $a,b,c$ such that $a \in A, b \in B, c \in C$ and $a,b,c$ (with approriate order) is a arithmetic sequence?

Let $N=\{ 1,2,3,..., 3n \}$ with $n$ is a positive integer and $A,B,C$ are three arbitrary sets such that $A \cup B \cup C = N, A \cap B = B \cap C = C \cap A = \varnothing, |A| = |B| = |C| = n $. How ...
3
votes
1answer
72 views

How to prove that $r\geq\frac {1}{2(1+\sqrt 3)}$?

Each interior point of an equilateral triangle of side $1$ lies in one of six congruent circles of radius $r$. How to prove that $r\geq\frac {1}{2(1+\sqrt 3)}$?
1
vote
2answers
60 views

$m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $

Prove that $m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $ for every $m, n, l >0$.
0
votes
1answer
70 views

Two circles are tangent to each other, find the ratio of line that splits the area into $1:2$

There is one circle with radius $1$. There is another circle with radius $2$. They are tangent to each other and touch each other at point $c$. A line through $c$ splits the area formed by the ...
3
votes
0answers
37 views

placing chess knights in a numbered chessboard.

Suppose you have a square board where the number on the square in column $i$ and row $j$ is $(j-1)8+i$ you have to place knights on the board so no two knights threaten each other and the sum of the ...
0
votes
1answer
441 views

Birdwatching question [closed]

Brent, Corrigan and Bieber went out to birdwatching. Each one saw a bird that no one else did. Each pair between them saw a bird that the third didn't. Finally, one bird was seen by everyone. From the ...
1
vote
1answer
51 views

Prove for relatively prime numbers.

Prove that for relatively prime positive integers $a$ and $b$, the equation $ax+by=c$ must have non-negative integer solution if $c>ab-a-b$.
2
votes
3answers
78 views

Maximum value of $a+b$ given that $\frac{1}{a} + \frac{1}{b} = \frac{1}{20}$

What is the maximum value of $a+b$ given that $\frac{1}{a} + \frac{1}{b} = \frac{1}{20}$ here $a,b \in \mathbb{Z^+}$? What I have gotten so far: From the above, $\frac{a+b}{ab} = ...
4
votes
2answers
82 views

Inequality $\frac{\sqrt a+\sqrt b+\sqrt c}{2}\ge\frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}$ with weird condition

I want to prove the following inequality: $$\frac{\sqrt a+\sqrt b+\sqrt c}{2}\ge\frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}$$ Where $a,b,c$ are positive reals and with the horrible ...
2
votes
1answer
68 views

How $\frac{\cos \alpha_1}{\sin \alpha}+\frac{\cos \beta_1}{\sin \beta}+\frac{\cos \gamma_1}{\sin \gamma}\leq\cot \alpha+\cot \beta+\cot \gamma$

Let are any two triangles with angles $\alpha, \beta, \gamma$ and $\alpha_1, \beta_1, \gamma_1$. How prove that $$\frac{\cos \alpha_1}{\sin \alpha} + \frac{\cos \beta_1}{\sin \beta}+ \frac{\cos ...
-1
votes
2answers
61 views

Cube root equations 1

$$E_{1} : \sqrt[3]{1+z}-\sqrt[3]{1-z}=\sqrt[6]{1-z^{2}} $$ Let $a=\sqrt[3]{1+z}$ and $b=\sqrt[3]{1-z}$ $E_1$ is equivalent to $E_2$ : $$ E_2:\ ...
0
votes
1answer
21 views

Do the functions have monotone on $\mathbb{R}$ a vector space?

Denote by $E$ the $\mathbb{R}$-vector space of all mappings from $\mathbb{R}$ to $\mathbb{R}$. Rigorously justifying your answer in each case, indicate whether the following subsets of $E$ are ...
0
votes
2answers
12 views

Do the functions have zero on the interval $[-n_{f},+n_{f}]$ } a vector space?

Denote by $E$ the $\mathbb{R}$-vector space of all mappings from $\mathbb{R}$ to $\mathbb{R}$. Rigorously justifying your answer in each case, indicate whether the following subsets of $E$ are ...
0
votes
0answers
43 views

Do the functions periodic with period $1$ a vector space?

Denote by $E$ the $\mathbb{R}$-vector space of all mappings from $\mathbb{R}$ to $\mathbb{R}$. Rigorously justifying your answer in each case, indicate whether the following subsets of $E$ are ...
0
votes
2answers
59 views

Do the functions with infinitely many zeros form a vector space?

Denote by $E$ the $\mathbb{R}$-vector space of all mappings from $\mathbb{R}$ to $\mathbb{R}$. Rigorously justifying your answer in each case, indicate whether the following subsets of $E$ are ...
0
votes
1answer
35 views

Help understanding example in Engel's *Problem Solving Strategies*

I've spent a lot of time trying to follow the chain of reasoning, but to no avail. I lose track of how it works at the "Adding (1) and (2)" part. Could someone help me understand this, please?