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

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3
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0answers
72 views

A Tricky Quetiones on Creative Algorithm in Graph

an agent is works between n producer and m consumers. i'th producer, generate $s_i$ candy and j'th consumer, consumes $b_j$ candy, in this year. for each candy that sales, agent get 1 dollar payoff. ...
2
votes
0answers
57 views

Graph Algorithm and Cycle Detection

In $O(|V|+|E|)$, we can detect whether a Directed Graph has a cycle or not. ---> True In depth-first seach on DAG, there is no Back Edge. ---> True With known Number of Edges, in $O(|V|)$ and not ...
0
votes
1answer
50 views

Give an example of a function that is not strictly increasing. Draw its graph and prove that the function is not strictly increasing

I picked x^4 to be a function which is not strictly increasing for all real numbers. Since to be not strictly increasing means that for the function y=f(x) x1 < x2 then f(x1)< f(x2) but ...
0
votes
1answer
22 views

Computation Operation in one Recurrence Relation

We want to calculate $T(n)$ from recurrence relation $ T(n)= \Sigma_{i=1}^{n-1} T(i) \times T(i-1)$` and we know $T(0)=T(1)=2$. How many computation operation, an Efficient Algorithm needs for ...
0
votes
0answers
37 views

A room contains n people. Everybody wants to shake everyone else’s hand (but not their own).

(a) Suppose that n people require hn handshakes. If an (n + 1)th person enters the room, how many additional handshakes are required? Obtain a recurrence relation for hn+1 in terms of hn. (b) ...
1
vote
1answer
25 views

How does the sequence have $n$ possibilities?

From: Solution 2003 A1 and problem: Problem 2003 A1 The first part of the solution is fine: The problem is here: The issue is this: (1) They say "once $a_1$ is fixed..." there are $k$ different ...
6
votes
3answers
160 views

When is $2^x+3^y$ a perfect square?

If $x$ and $y$ are positive integers, then when is $2^x+3^y$ a perfect square? I tried this question a lot but failed. I tried dividing cases into when $x,y$ are even/odd, but still have no idea ...
11
votes
1answer
220 views

if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $\color{red}n+\color{red}c$ divides ...
2
votes
0answers
19 views

Can you recommend me a book about integration? [duplicate]

I'm new to this site. I'm an university student in korea and I major in engineering. Recently, I've been quite interested in calculating integration. So nowadays as my hobby, I seek many interesting ...
3
votes
3answers
713 views

Number of six-digit integers with increasing digits [duplicate]

How many six-digit positive integers can you write, if each number must have strictly increasing digits from left to right? How is it allowed to use: $$ \binom{9}{3}$$ Because this says out of $9$ ...
1
vote
3answers
92 views

What does $\binom{a}{b}$ represent?

Problem: How many six-digit positive integers can you write, if each number must have strictly increasing digits from left to right From the other link, How do I know if I use $\binom{a}{b}$ or ...
1
vote
1answer
18 views

Set Theory problem with unique numbers

Let $A_0$ be the set {$1, 2, 3, 4$}. Let $A_{i+1}$ be the set of all possible sums which can be obtained by adding two numbers in $A_i$ , where the two numbers do not have to be different. How many ...
4
votes
4answers
117 views

If $abc=1$ then $\sum_{cyc}^{}{\frac{1}{b(a+b)}}\ge \frac{3}{2}$

If $abc=1$ for positive $a,b,c$, then $\sum_{cyc}^{}{\dfrac{1}{b(a+b)}}\ge \dfrac{3}{2}$ I have tried the following,in decreasing order of success: 1)AM-GM:$a+b+c\ge 3$ and $ab+bc+ca\ge 3$ ...
0
votes
1answer
51 views

Two Place Position and Model Question

! i get trouble in one multiple choice question in logic course: any one could help me with some description ? if we have Two-place position predicate, like : 1) all models of $\varphi$ is ...
1
vote
1answer
32 views

First Order Logic and Some Validity Checking

I'm sorry for put an image insted of typing it... infact this is an 2012-exam on Logic. i found the solution of this quiz that wrote by one TA. he wrote just the second line is not valid logically in ...
0
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1answer
100 views

If $\Gamma\cup\{\sim(A\land B)\}$ is consistent, what can be said about $\Gamma\cup\{\sim(A\lor B)\},\Gamma\cup\{\sim A\},\Gamma\cup\{\sim B\}$?

The following question arose in the NOI of India Section taken a few days back: Let $\Gamma$ be a set of predicate formulas, and let $A, B$ be two predicate formulas; if the theory $\Gamma \cup \{ ...
1
vote
2answers
102 views

How many 6 digit numbers can you write?

How many six-digit positive integers can you write, if each number must have strictly increasing digits from left to right I thought it was 6! Butt hat is wrong. How do you do these sort of problems ...
0
votes
1answer
71 views

AoPS putnam 2003 A1 solution issue

I am having problems transferring and copying the data here. Take a look at AOPS KENT MERRYFIELD ANSWER. I don't understand. He gets to: $$a(k-r) + (a+1)r = n$$ But how does this show that there ...
0
votes
2answers
55 views

Nice proof of a polynomial root $x \in [0, 1]$

Prove that if $$\sum_{k=0}^{n} \frac{a_k}{k+1} = 0$$ then $$\sum_{k=0}^{n} a_k\cdot x^n = 0$$ for some $x$ in $[0,1]$. (original image) So: $$\sum_{k=0}^{n} \frac{a_k}{k+1} = 0$$ We ...
3
votes
2answers
193 views

Determine whether $712! + 1$ is a prime number or not

Let $n = 712! + 1$ If $n$ was a prime number then, by Wilson's theorem: $ (712!)! \equiv -1 \pmod{712}$ The double factorial makes it seriously more difficult... But We can require: $$712!! + 1 ...
4
votes
1answer
675 views

1000 numbers on a blackboard

The numbers $1, 2, …,1000$ are written on a blackboard, in some order. Between every pair of consecutive terms, the absolute difference of the two terms is written between them, and then all the ...
0
votes
1answer
37 views

For how many distinct triangles ABC, with AB = $2011$, are both $\cos(2\angle{A} + 3\angle{C})$ and $\sin(2\angle{B} + \angle{C})$ integers?

I came across a tough geometry question: For how many distinct triangles $\text{ABC}$, with $\text{AB}$ = $2011$, are both $\cos(2\angle{A} + 3\angle{C})$ and $\sin(2\angle{B} + \angle{C})$ ...
1
vote
1answer
40 views

The quadratic $x^2-4kx+3k = 0$ has two distinct roots $m$ and $n$, where $m > n$ and $m - n = m^2+n^2$. What is the sum of all possible values of k?

I was trying to solve the following question: The quadratic $x^2-4kx+3k = 0$ has two distinct roots $m$ and $n$, where $m > n$ and $m - n = m^2+n^2$. What is the sum of all possible values of ...
3
votes
2answers
56 views

How many positive integers less than $2011$ cannot be expressed in the form $4a + 5b$, where $a$ and $b$ are positive integers?

How would I solve the following question: How many positive integers less than $2011$ cannot be expressed in the form $4a + 5b$, where $a$ and $b$ are positive integers? I was trying to apply ...
0
votes
1answer
44 views

Proof of an alternate form of the triangle inequality

Since it is all positive squaring does not change anything. So: $$ (a_1^2 + \cdots + a_n^2) + 2\sqrt{(a_1^2 + \cdots + a_n^2)(b_1^2 + \cdots b_n^2)} + (b_1^2 + \cdots + b_n^2) \ge (a_1 + b_1)^2 + ...
0
votes
1answer
48 views

Trying to simplify the expression

Can anybody simplify it? Show me the way of simplification. The expression is as follows: $$F(x) = 1 *(1!+x)+2*(2!+x)+ ..+x*(x!+x)$$ for a positive integer $x$ I've tried but nothing got.
1
vote
1answer
22 views

Finding the rank of a particular number in a sequence of the sum of numbers and their highest prime factor

This question comes from a maths contest (infer no calculators or other electronic calculating aids) for 14-16 year olds (infer no use of complicated theorems, but those accessible to high-school ...
5
votes
1answer
65 views

prove the inequality with fractional parts

$$ \frac{n^k-n}{2} \leq \left\{\sqrt[k]{1}\right\} + \left\{\sqrt[k]{2}\right\} + \dots + \left\{\sqrt[k]{n^k}\right\} \leq \frac{n^k-1}{2} $$ how it can be proven?
3
votes
2answers
51 views

Numbers $a$ that are the sum of the fractional parts $\{x^2\} + \{x\}$ for some $x$

Are there infinitely many rational numbers $a\in\mathbb{Q}$ with the following property: $\{x^2\}+\{x\}=a$ for infinitely many $x\in\mathbb{Q}^+$
0
votes
1answer
50 views

Find all functions satisfying $(1+y)\,f(x) - (1+x)\,f(y) = y \, f(x/y) - x \, f(y/x)$

Find all functions which satisfy: $$(1+y)\,f(x) - (1+x)\,f(y) = y \, f(x/y) - x \, f(y/x)$$ for all real, $x,y \ne 0$ and which takes the values $f(1) = 32$ and $f(-1) = -4$ I am not sure, which ...
-3
votes
2answers
45 views

If a machine takes 3min to process 1 byte, how many machines are required to process 1000 bytes in 30min?

We have a machine that takes 3 minutes to process a byte. Now if I send 1000 bytes together the machine will take 3000 minutes to process them serially. If we want to do that in 3 minutes only we need ...
1
vote
1answer
151 views

Prove that if x is irrational, then sqrt(x) is irrational.

I believe the contrapositive method should be correct but i get, The contrapositive of this statement should be, (If $\sqrt{x}$ is rational, then $x$ is rational) Then I end up with ...
1
vote
1answer
96 views

Putnam A4 2010, proving an expression is not prime.

Prove that each positive integer $n$: $ x = \displaystyle 10^{10^{10^n}}+10^{10^n}+10^n-1 $ is not prime. This seems like a very difficult problem, any ideas at all? I would like to use modular ...
0
votes
1answer
63 views

what is required for a person to do well on imo

What kind of skill is required to solve IMO or Putnam sort of problems. Does one have to be a genius or just learn some tricks.
3
votes
2answers
56 views

A grandmother is giving out apples to her grandchildren.

A grandmother has 7 grandchildren, and 14 apples to give. How many ways can she give apples to her grandchildren so that each grandchild gets aT LEAST one? (but she has to get rid of hers). This ...
0
votes
1answer
54 views

Consider the set $Q=\{p+q \sqrt2 : p,q \in\Bbb Q\}$. Prove that if $a\in Q\setminus\{0\}$ then $1/a\in Q$

Given (For all $a,b\in Q$, $a+b\in Q$ and $ab\in Q$) This was a two part question. Part a) is to prove that $Q$ is closed under addition and multiplication. Part b) is prove that if $a\in Q$ and ...
1
vote
0answers
52 views

Number theory - equation

I´m preparing for math contests and found the following problem from this pdf: http://www.fmf.uni-lj.si/~lavric/Santos%20-%20Number%20Theory%20for%20Mathematical%20Contests.pdf Find all integers $a, ...
0
votes
1answer
42 views

In how many ways can we distribute 6 identical pears?

In how many ways can we distribute 6 identical pears between 3 children so that each child receives at least one pear? I am not too sure. I thought, 6 ways to distribute to first, 5 ways to second, ...
4
votes
1answer
62 views

Does performance in math competitions accurately reflect natural aptitude in mathematics? [closed]

Many great and respected mathematicians have won accolades in math (ex: IMO), does that necessarily mean that these competitions reflect one's potential to be a great mathematician?
7
votes
1answer
103 views

Using two coins to select a person fairly.

Good evening, I would like to know if the solution to this problem, I know it can be solved because it is from a Hungarian Olympiad. The problem is as follows: You need to fairly select a person ...
0
votes
2answers
30 views

Given $A \subseteq \mathbf{Z}$ and $x\in \mathbf{Z}$, we say that $x$ is $A$-mirrored if and only if $−x\in A$. We also define…

Sorry if this question seems kind of long but I am confused for part C. My proof for part C that $M_a$ is closed under addition is as follows: The set $M_a$ is closed. Let $x$ be in $M_a$ and ...
1
vote
1answer
30 views

Number of paths in 3D coordinates

A cute problem which is an extension of a well-known counting problem: Find the number of paths of length $12$ from $(0,0,0)$ to $(4,4,4)$ passing through adjacent lattice points (for two ajacent ...
3
votes
1answer
95 views

Functional equation defined over non-negative real numbers

I'm new to this forum and I don't know how to write mathematical symbols. I have the following functional equation: $f$ defined on $[0, +\infty)$ with values in $[0, +\infty)$ $f$ is bijective and ...
9
votes
5answers
1k views

How to derive this infinite product formula?

Show: $$\prod_{n=0}^{\infty}\left(1 + x^{2^n}\right) = \frac{1}{1-x}$$ I tried numerous things, multiplying by $x$, dividing, but none of that worked. Also, I realized that: $$\prod_{n=0}^{\infty} ...
3
votes
1answer
96 views

Length of the non-periodic portion of the decimal expansion of $\frac 1n$

The following question was asked in the Indian National Mathematics Olympiad (INMO) 2015. For any natural number $n>1$,write the infinite decimal expansion of $\frac 1n$. Determine the length ...
1
vote
4answers
225 views

Reversing the digits with a subtraction [closed]

How many 3-digits numbers possess the following property: After subtracting $297$ from such a number, we get a $3$-digit number consisting of the same digits in the reverse order.
0
votes
1answer
45 views

Solving the Sequence of this question on Putnam Exam

Problem: Solution: Solution for 2003 A1 Putnam $ka_1 = a_1 + a_1 ... a_1 \le n \le a_1 + (a_1 + 1) + (a_1 + 1) ... (a_1 + 1)$ $= ka_1 + k - 1$ I know these then: What should I do next? Without ...
9
votes
3answers
720 views

How many 0's are in the end of this expansion?

How many $0's$ are in the end of: $$1^1 \cdot 2^2 \cdot 3^3 \cdot 4^4.... 99^{99}$$ The answer is supposed to be $1100$ but I have absolutely NO clue how to get there. Any advice?
2
votes
1answer
77 views

Prove that $ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2\ge\left( \frac{x_1+\dots+x_n}{n} \right)\left(\frac{y_1+\dots+y_n}{n} \right). $

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that ...
4
votes
0answers
53 views

Sequence of non-collinear integer points.

This is a question from a British Olympiad, I've completed the first 3 but this one had me rather stumped. Given two points $P$ and $Q$ with integer coordinates, we say that $P$ sees $Q$ if the ...