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

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0answers
10 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 ...
-2
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1answer
33 views

I do not know how to slove it [on hold]

If a child can run 10 meters while a car travels 30 meters, how many meters can the child run while the car travels 66 meters
5
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2answers
191 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: $(x-y)^7 = x^3y^3$
0
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0answers
12 views

Applying Lovász local lemma, how to fix this solution

We have U - a "small" graph, which is fixed. The goal is to find a coloring(with d colors) of the edges of the complete graph with n vertices($K_n$, n-fixed), without monochromatic copy of U. We also ...
3
votes
2answers
34 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 ...
9
votes
3answers
139 views

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

How could we prove that there does not exist a positive integer $n$, so that $$({\sqrt{2}+1})^{1/n} + ({\sqrt{2}-1})^{1/n}$$ is rational? I think it could be done with induction from a more general ...
1
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1answer
23 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
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1answer
52 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
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4answers
69 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 ...
4
votes
2answers
90 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 ...
2
votes
1answer
17 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
54 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
61 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
38 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
35 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
46 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
66 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..
3
votes
2answers
67 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
45 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
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0answers
23 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 ...
2
votes
1answer
69 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 $r\geq\frac {1}{2(1+\sqrt 3)}$?
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2answers
49 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
50 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
31 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 ...
2
votes
1answer
143 views

Birdwatching question

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
27 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 $c>ab-a-b$.
1
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3answers
74 views

Probability question [closed]

Consider two urns. Urn A has 4 black balls and 3 white balls, urn B has 4 black balls and 6 white balls. From urn A we draw 2 balls and transfer in urn B. Then from urn B we pick a ball and its known ...
2
votes
3answers
73 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
78 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
62 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
55 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
16 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
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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
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0answers
36 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
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2answers
56 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
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1answer
31 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?
2
votes
1answer
115 views

Unsolved/Least Solved IMO Questions

I recently read this article http://blog.mathfights.com/once-upon-a-time-on-imo/ where the author discusses an IMO problem from 2006 that only about 20 participants out of 600 were able to solve. So ...
0
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0answers
613 views

Make beautiful Arithmetic progression [closed]

Moderator Note: This is a current contest question on codechef.com. Given an array that consists of $n$ integers, we have to change at most $k$ elements so that the resulting array will be a most ...
1
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2answers
170 views

If the sum of $n$ cubes is zero, then the sum must be no larger than $\frac n3$.

Assume that $a_1,...a_n$ are real numbers and $-1 \leq a_i\leq 1$ for $1\leq i\leq n$. If $$a_1^3+\ldots +a_n^3=0$$ Then show that $$a_1+a_2+\ldots+a_n\le n/3$$ I just came cross this problem the ...
9
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1answer
262 views

How can this technique be applied to a different problem?

Here is the problem (copy and pasted if you don't want to click on the link). Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. ...
3
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0answers
32 views

The Basic Example and Output of Algorithms [closed]

if exg(x,y) swap the x,y, and array A contains integer numbers, the following algorithm how modify the $A[1]$ and what is the operation of the following algorithm? i confused to trace this code. any ...
3
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1answer
22 views

statistics basic question on covariance

anyone would help me in a basic example? a fair coin is tossed, n times. X is the number of Head and Y is the number of Tails. what is the COV(X,Y).
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0answers
22 views

Statistics and Some Information Challenge

relation between two attribute x,y is $y=\alpha\beta^{-x}$. According to 8 experiments these information were gained. what is the estimation of ( $\alpha, \beta$) using Least Square Error? it's 2010 ...
2
votes
0answers
38 views

Add two a,b bits number Algorithm

Suppose we want add two numbers that has a and b bits. we do such operation in O(max{a,b}). we want to add n, 1 bit numbers (0 or 1). what is the best and worst case of this algorithm? i ran into ...
4
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0answers
57 views

local informatics Olympiad and Algorithm

I see one of recent local informatics Olympiad question. i have a trouble to solve it. any idea? hint? or solutions? thanks to all creative man. We have two function $P_1, P_2$ and input an array $n$ ...
2
votes
0answers
41 views

Diophantine equations which are easier to solve using $\mathbb{Z}[i]$ compared to $\mathbb{Z}$

I wanted to know applications of arithmetic in $\mathbb{Z}[i]$ that helps in some problems of $\mathbb{Z}$. I found a wonderful set of notes by Keith Conrad. Now I want to read more on a similar ...
4
votes
1answer
50 views

infinitely many primes $p$ such that $p$ divides $a_{1}^k+a_{2}^k+…+a_{n}^k$

Consider the positive integers $a_{1},a_{2},...,a_{n}$, not all identical ($n>1$). Prove that there are infinitely many primes $p$ such that $p$ divides $a_{1}^k+a_{2}^k+...+a_{n}^k$ for some ...
2
votes
1answer
112 views

Water Box with n Liter

I ran into a basic challenging problem. I see an high school local math Olympiad question. we have a box that keep n Liter water. each time we extract 1/k Water from box. how many times (minimum) we ...
12
votes
0answers
135 views

Fractional Part of $ a^n $

Prove that there exists a real number $ a>1 $, such that $ \{a^n\} $ belongs to $[\frac{1}{3},\frac{2}{3}]$ for all positive integers $n$ and $\lfloor a^n\rfloor$ is even iff $n$ is a prime. ...
1
vote
0answers
17 views

How solve the equation $a^x+\left(2a+1\right)^y=\left(a+1\right)^z$ for $a\in N - \{1\}$ and $x,y,z\in N\cup\{0\}$?

How solve the equation in natural numbers $a^x+\left(2a+1\right)^y=\left(a+1\right)^z$ for $a\in N - \{1\}$ and $x,y,z\in N\cup\{0\}$?