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

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3
votes
2answers
221 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 ...
0
votes
0answers
82 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
45 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
70 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
121 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
220 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. ...
0
votes
0answers
23 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 $x,y,z\in N\cup\{0\}$?
1
vote
0answers
67 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 ...
8
votes
1answer
185 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
37 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
66 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% ...
0
votes
0answers
44 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
118 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 ...
6
votes
1answer
124 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}$?
0
votes
0answers
42 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 : $$\frac{\sqrt{1 + 2\cos^2 A}}{\sin B} + \frac{\sqrt{1 + 2\cos^2 B}}{\sin C} + \frac{\sqrt{1 + 2\cos^2 C}}{\sin A}\text{ ?}$$
6
votes
3answers
153 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
92 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
54 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
125 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
117 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
366 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
85 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
287 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 ...
3
votes
3answers
85 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
76 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 ...
0
votes
2answers
60 views

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

How to show that : $$\frac{2q+1}{2p-1}>\sum_{i=p}^q\frac{1}{i}$$ if $q\ge p>0$
8
votes
1answer
167 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
282 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
65 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 $q \geq b+d$?
4
votes
1answer
77 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$?
3
votes
0answers
54 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
75 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
150 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
119 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 ...
3
votes
2answers
73 views

Elementary algebra problem

Consider the following problem (drawn from Stanford Math Competition 2014): "Find the minimum value of $\frac{1}{x-y}+\frac{1}{y-z}+ \frac{1}{x-z}$ for for reals $x > y > z$ given $(x − y)(y − ...
1
vote
1answer
51 views

combinatorics board with digits neat problem

this is a problem I wanted to share with you that I just saw today. There is a board square board (think of chess) with $10$ columns and $n$ rows. Each square contains a digit (an integer between $0$ ...
1
vote
1answer
69 views

If $AB+BA=0$ and $B=AX+XB$, then $B$ is nilpotent.

Suppose $A,B,X \in M_n(\mathbb{R})$ and that $AB+BA=0$ and $B=AX+XA$. Prove that $B$ is a nilpotent matrix.
3
votes
1answer
199 views

Chessboard problem in IMO2014

This is the second problem on the IMO2014 problem list: Let n $\ge 2$ be an integer. Consider an $n \times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this ...
0
votes
2answers
45 views

Suppose T(k) denotes the smallest number of steps needed to move from k to 100.Find y & z such that T(9)= 1+ min (T(y),T(z)).

Suppose you want to move from 0 to 100 on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut is simply a pre-specified pair of integers i , j ...
4
votes
1answer
66 views

Initial value for a sequence to become periodic.

The following is from the previous Proofathon contest: Let $a_{n}$ be the sequence defined by the recursion $ \sqrt{a_{n+1}}= (2(\sqrt[2014]{a_n})-1)^{2014}. $ Find all the values of $a_1$ ...
1
vote
2answers
209 views

maximum number of independent bishops on a nxn chessboard

So I came across this problem where we have to find the maximum number of independent bishops on a nxn chessboard such that no two bishops attack each other . So after drawing the cases for $3$x$3$ , ...
0
votes
1answer
120 views

How to write Putnam Examination proofs?

I am studying for the Putnam exam and I have learned that the graders are quite strict and will cut off points for a variety of reasons. I want to know exactly how to write a Putnam proof. How ...
3
votes
1answer
89 views

Solve in $\mathbb{R}$: $(x^2-3x-2)^2-3(x^2-3x-2)-2-x=0$

Solve in $\mathbb{R}$: $(x^2-3x-2)^2-3(x^2-3x-2)-2-x=0$ I'm supposed to solve this equation. It's from a math contest so solving it by hand would be preferable (no quartic formulas). I thought ...
2
votes
2answers
44 views

An arctan problem including a diophantine equation

This is a follow-up question to An equation of the form A + B + C = ABC . I totally messed up with making the equation from the question specification . Actually the question was $$ ...
5
votes
1answer
66 views

On a strange pigeonhole principle problem

Given distinct integers $a_1, a_2, \cdots, a_{63}$. Prove that there exists $a_i, a_j, a_m, a_n$ such that $(a_i - a_j)(a_m - a_n)$ is divisible by $1984$. I have no idea of how to create the ...
0
votes
1answer
316 views

Select elements from $N$ sets

$N$ sets are given which can have any number of elements from $1-100$ each. Now we need to count arrangements in which we select $1$ element from each set under the condition that we can not choose ...
0
votes
1answer
121 views

An equation of the form A + B + C = ABC

So I was on a SPOJ spree until I came across this question . The question says $$\tan(\frac{1}{A}) = \tan(\frac{1}{B}) + \tan(\frac{1}{C})$$ where we have to find the $\min(B+C)$ for a fix $A$ where ...
4
votes
3answers
2k views

Math Olympiads: Hard work or talent? [closed]

I have a question regarding Math Olympiads. I always asked myself if Math Olympiads need natural intelligence or rigorous hard work (or both) in order to reach levels such as the IMO. I always hear ...
0
votes
2answers
151 views

The number of functions $f: {\cal P}_n \to \{1, 2, \dots, m\}$ such that $f(A \cap B) = \min\{f(A), f(B)\}$ (Putnam 1993)

Let ${\cal P}_n$ be the set of subsets of $\{1, 2, \dots, n\}$. Let $c(n, m)$ be the number of functions $f: {\cal P}_n \to \{1, 2, \dots, m\}$ such that $f(A \cap B) = \min\{f(A), f(B)\}$. Prove that ...
1
vote
0answers
115 views

Arithmetic Mean and Geometric Mean Question, Guidance Needed

I am super new to olympiad-style math which focuses on a lot of inequalities, and tough problems which highschool students do not go over. I'm in 9th grade, and am trying to get into all of this stuff ...