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

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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
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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
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
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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?
2
votes
1answer
160 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 ...
1
vote
2answers
171 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
votes
1answer
278 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. ...
1
vote
1answer
36 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).
0
votes
0answers
29 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 ...
3
votes
2answers
172 views

Algorithm for adding n 1-bit numbers

suppose adding two numbers, (that first number has a bits and second number has b bits) can be done in ...
2
votes
0answers
69 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
42 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 ...
5
votes
1answer
56 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 ...
0
votes
1answer
115 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 ...
14
votes
1answer
211 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. ...
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0answers
18 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 $x,y,z\in N\cup\{0\}$?
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0answers
55 views

Proof Verification: Putnam 1995 A4

PROBLEM: Suppose we have a necklace of $n$ beads. Each bead is labelled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive ...
7
votes
1answer
133 views

Sum over all non-evil numbers

I'm working on the following contest math problem: Define an evil number to be any positive integer that contains the digit $9$. Show that $$ \sum_{x} \frac{1}{x} < 80 $$ where the ...
1
vote
1answer
32 views

Need an Algorithm Such that $\sum_{k-i}^{j}{A[k]}$

I need an algorithm for real application. Suppose we have array A (positive & negative ) numbers. we want to find index i, j such that $\sum_{k-i}^{j}{A[k]}$ has the lowest difference to zero. ...
1
vote
1answer
57 views

Binomial Congruence (mod 5) Identity

I've got a (hard?) Putnam-style problem that I've been given to look at . . . I've never worked any problem even vaguely like this, but my director thinks I should be able to do it. I doubt it (100% ...
1
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0answers
38 views

Can we sort 6 numbers with at most 9 comparison? [duplicate]

i know there is an algorithm to sort 5 numbers with 7 comparison. Can we sort 6 numbers with at most 9 comparison? thanks to all.
2
votes
3answers
103 views

2000 Olympiad in Informatics Question on Box

I have an old Olympiad question on informatics. There are 31 boxes. In each box there is one number. We know the number if and only if we open the box. We want to calculate the minimum number of ...
7
votes
1answer
119 views

How prove $ y^2=x^3+x+1370^{1370}$ has at least 6 answers in $ \mathbb{Q}$?

How prove that $ y^2=x^3+x+1370^{1370}$ has at least 6 answers in $ \mathbb{Q}$?
1
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0answers
34 views

How find a triangle ABC minimizing $\frac{\sqrt{1 + 2\cos^2 A}}{\sin B} + \frac{\sqrt{1 + 2\cos^2 B}}{\sin C} + \frac{\sqrt{1 + 2\cos^2 C}}{\sin A}$?

How find in triangle $ABC$ the minimum value of : $$\frac{\sqrt{1 + 2\cos^2 A}}{\sin B} + \frac{\sqrt{1 + 2\cos^2 B}}{\sin C} + \frac{\sqrt{1 + 2\cos^2 C}}{\sin A}\text{ ?}$$
-1
votes
1answer
65 views

Suppose f(x) + 2f(1/x) = x . Evaluate f(5) in simplest form. [closed]

If f(x) + 2(f(1/x)) = x, evaluate f(5). How can I go about solving this problem?
6
votes
3answers
146 views

An uncanny inequality with Gamma function

Prove for $x>0$ that $$ \frac{\Gamma^{\prime}(x+1)}{\Gamma(x+1)}>\log x$$ How to prove this inequality? thanks. This is a problem from Miklos Schweitzer Competition.
2
votes
1answer
75 views

What is wrong with the following induction argument?

I found a problem on a note on induction. The problem went like this: "Let $n$ be a non-negative integer. Suppose we are given a triangle and n points inside it, with no three of the given $n + 3$ ...
2
votes
1answer
48 views

Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$
4
votes
1answer
119 views

After how many steps can compositions of $x\mapsto x+1$ and $x\mapsto x^2+1$ produce the same result starting from $1$ and $3$?

This problem is from a Turkish contest: Let $P(x)=x+1$ and $Q(x)=x^2+1$. Consider all sequences $(x_k,y_k)$ such that $(x_1,y_1)=(1,3)$ and $(x_{k+1},y_{k+1})$ is either $(P(x_k),Q(y_k))$ ...
4
votes
2answers
82 views

Find the 1005th digit after the decimal point expansion of the square root of N.

Let $N$ be the positive integer with $2008$ decimal digits, all of them $1$. That is, $N=1111...1111$, with $2008$ occurrences of the digit $1$. Find the $1005th$ digit after the decimal point ...
1
vote
1answer
159 views

lifting the exponent lemma for $p=2$.

I am trying to understand the proof of theorem 3 (in p.4) of http://www.artofproblemsolving.com/Resources/Papers/LTE.pdf However, I dont understand the last sentence "This means the power of $2$ in ...
1
vote
3answers
83 views

Finding the sum of $3+4\cdot 3+4^2\cdot 3+\dots +4^{\log n-1} \cdot 3$

I see this: $$A=3+4\cdot 3+4^2\cdot 3+\dots +4^{\log n-1} \cdot 3=3\cdot ([4^{\log n}-1]/3)=n^2-1$$ The base of logarithm is $2$, and $n$ is $2,4,8,\dots$ Anyone could describe me how this sum was ...
4
votes
1answer
276 views

Prove or disprove that there exists a unique positive integer sequence $\{a_{n}\}$ satisfying a condition

Question: Prove or disprove: there exists a unique positive integer sequence $\{a_{n}\}$ satisfying the following condition: $\forall m\in N^{+}$, there exists a unique integer sequence ...
4
votes
3answers
76 views

how find $\sum_{k \in A} \frac{1}{k-1} $ for $ A = \{ m^n| \text{ } m, n \in Z \text { and } m, n \ge 2 \} $

If $ A = \{ m^n| \text{ } m, n \in Z \text { and } m, n \ge 2 \} $, then how find $\sum_{k \in A} \frac{1}{k-1} $?
4
votes
2answers
63 views

Find the smallest constant K satisfying the inequality

Find the smallest constant $K$satisfying the inequality $$x^{1\over 3}+y^{1\over 3} \le K(x+y)^{1\over 3}$$ The official proof makes the substitution $a=x^{1\over 3}$ and $b=y^{1\over 3}$, which does ...
1
vote
2answers
49 views

How to show that $\frac {q + \frac {1}{2}}{p - \frac {1}{2}} > \sum_{i = p}^q\frac {1}{i}$ if $q\ge p > 0?$

How to show that : $$\frac{2q+1}{2p-1}>\sum_{i=p}^q\frac{1}{i}$$ if $q\ge p>0$
8
votes
1answer
124 views

Math competitions for hobbyists?

Are there any math competitions for hobbyist / amateur mathematicians? Something like the Putnam or the International Mathematical Olympiad, but open to regular people who are not full-time students?
-2
votes
1answer
271 views

Modulo of a large sequence of $1$s

Given two numbers $N$ and $M$, we need to find the remainder when $111 \cdots1$ ($N$ times) is divided by $M$. Here $N$ can go up to $10^{16}$ and $M$ up to $10^9$. How to solve this problem? ...
3
votes
1answer
50 views

How prove that $q \geq b+d$ for $ad-bc = 1$ and $\frac{a}{b} > \frac{p}{q} > \frac{c}{d}$?

Let $a,b,c,d,p$, and $q$ be natural numbers such that $ad-bc = 1$ and $\frac{a}{b} > \frac{p}{q} > \frac{c}{d}$. How prove that $q \geq b+d$?
5
votes
1answer
66 views

How prove that $ x+y+z>4$ for $ a+b+c=4$ and $ ax+by+cz=xyz$?

Given positive reals $ a,b,c,x,y,z$ such that $ a+b+c=4$ and $ ax+by+cz=xyz$. How prove that $ x+y+z>4$?
2
votes
0answers
38 views

Speed dating/networking challenge

I am trying to organise an event with 54 participants. I want them to participate in 9 different activities at stations around a hall. Obviously this will require 9 sessions to allow the participants ...
3
votes
2answers
72 views

Prove that $\frac{a^3}{x} + \frac{b^3}{y} + \frac{c^3}{z} \ge \frac{(a+b+c)^3}{3(x+y+z)}$ a,b,c,x,y,z are positive real numbers.

I stumbled upon it on some olympiad papers. Tried to AM>GM but didn't get any idea to move forward.
1
vote
0answers
97 views

For which real numbers $c$ is there a straight line that intersects the curve $y = x^4 + 9x^3 + c x^2 + 9x + 4$ in four distinct points?

For which real numbers $c$ is there a straight line that intersects the curve $y = x^4 + 9x^3 + c x^2 + 9x + 4$ in four distinct points? I don't quite the understand the solution which is in ...
4
votes
3answers
109 views

$ 1987 \mid \left( n^n + (n+1)^n \right) $

Problem from the 1987 Leningrad Math Olympiad: Is there a positive integer $n$ such that $ n^n + \left( n + 1 \right)^n $ is divisible by $ 1987 $? The provided solution: The answer is ...