# Tagged Questions

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### If one plays $132$ games in $77$ days, there is a span of consecutive days with exactly $21$ games

This is a high school contest question. Simple answers are required A chess player has $77$ days to prepare for a tournament. During this time he wants to have a match everyday and to have $132$ ...
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### Last 7 digits of 7th powers

Alice and Bob play the following game. They alternately select distinct nonzero digits from $1$ to $9$, until they have chosen seven such digits. Consider the resulting seven-digit number by joining ...
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### Trouble with inequalities

I'm a 9th grade student, going into 10th grade. Math has always been a subject I enjoyed and excelled in. I'm writing a schoolboard-wide math contest next year in mid-February I believe. To prepare ...
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### Solve for “lucky” numbers

A rational number is called "lucky" if it equals both $a+\frac{b}{c}$ and $a\times\frac{b}{c}$ for some positive integers $a,b,c$. How many lucky numbers are there between $5$ and $10$? Here's what I ...
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### Math Olympiads: GCD of terms in a sequence equals GCD of terms in other sequence

Recently, someone asked for a proof of a problem from the Russian Mathematical Olympiad, 1995. Math Olympiads: GCD of terms in a sequence equals GCD of their indices. The problem was to show that if ...
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### Math Olympiads: GCD of terms in a sequence equals GCD of their indices.

The sequence $a_1 ,a_2 ,a_3 ,...$ of positive integers satisfies $\text{gcd}(a_i ,a_j ) = \text{gcd} (i, j)$ for $i \neq j$. Prove that $a_i = i$ for all $i$. Source: Russian Mathematical Olympiad, ...
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### Some Questions regarding preparing for Math Olympiads (searched but didn't get answers)

Many questions have been asked on this site regarding preparation for olympiads like the Putnam. I've read those questions and accordingly decided to start with Engel's "Problem Solving" but I have a ...
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### Separating $3n$ points on the plane by a line

I am trying to solve a problem in geometry (a contest-type question), and I wondering if the following result is true. (If it is true, then it makes life much easier!) Suppose there are $3n$ ...
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### There exists an integer with alternating digits $1$ and $2$ which is divisible by $2013$

Could someone give me hints in how to solve the following (rather interesting) problem? Prove that there exists an integer consisting of an alternance of $1$s and $2$s with as many $1$s as $2$s ...
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### Putnam-Style Sequences Problem

Let $S_1$ denote the sequence of positive integers $1,2,3,4,5,6,\ldots,$ and define the sequence $S_{n+1}$ in terms of $S_n$ by adding $1$ to those integers in $S_n$ which are divisible by $n$. Thus, ...
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### In a party with 2000 persons, determine # of people who know everyone

In a party with 2000 persons, among any set of four there is at least one person who knows each of the other three. There are three people who are not mutually acquainted with each other. How many ...
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### Continuous functions on $\mathbb{R}^2$ with special property

The following problem is from Miklos Schweitzer competition (Year 1983, Problem 7): Prove that if the function $f: \mathbb{R}^{2}\to [0, 1]$ is continuous, and its average on every circle of ...
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### Combinatorial Proof of ${n\choose{m}}=\frac{n}{m}{{n-1}\choose{m-1}}$

How do prove the following identity combinatorially? $${n\choose{m}}=\frac{n}{m}{{n-1}\choose{m-1}}$$ Any help or hints would be great!
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### find the value of 1/(2+1/(4+1/(4+1/(…))))

the question is to find the value of this ugly non-stopping fraction $$\frac{1}{2+\frac{1}{4+\frac{1}{4+\frac{1}{\ldots}}}}$$. I have totally no clue; thanks for the help! How am I suppose to solve ...
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### $\frac1a+\frac1b+\frac1c=0 \implies a^2+b^2+c^2=(a+b+c)^2$? [closed]

How to prove that $a^2+b^2+c^2=(a+b+c)^2$ given that $\frac1a+\frac1b+\frac1c=0$?
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### How many isosceles triangles with total side length $100$ are there?

Let the sum of the three sides of a triangle be $100,$ and all the sides are positive integers length, how many possible isosceles triangles are there?
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### Find the value of $\sqrt{(b-a-4)^2}- \sqrt{(a-b+1)^2}$ if a>0 and b<0

Find the value of $\sqrt{(b-a-4)^2}- \sqrt{(a-b+1)^2}$ if $a>0$ and $b<0$. How do i find the value? This doesn't make any sense.
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### IMO 2013 Problem 6

Let $n\geq 3$ be an integer, and consider a circle with $n+1$ equally spaced points marked on it. Consider all labelings of these points with the numbers $0,1,\dots, n$ such that each label is used ...
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### Solving for $(x,y): 2+\frac1{x+\frac1{y+\frac15}}=\frac{478}{221}$

Solving for $x,y\in\mathbb{N}$: $$2+\dfrac1{x+\dfrac1{y+\dfrac15}}=\frac{478}{221}$$ This doesn't make any sense; I made $y+\frac15=\frac{5y+1}5$, and so on, but it turns out to be a very ...
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### Let $f$ be a function satisfying such that $f(xy)=\frac{f(x)}{y}$, what is $f(600)$? [closed]

Let $f$ be a function satisfying such that $f(xy)=\frac{f(x)}{y}$ for all positive real numbers $x$ and $y$. Given that $f(500)=3$, what is $f(600)$?
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### Suppose that $x > y > 0$. Which of the following is equivalent to $\dfrac{x^y y^x}{y^y x^x}$? [closed]

Suppose that $x > y > 0$. Which of the following is equivalent to $\large\dfrac{x^y y^x}{y^y x^x}$? $(x-y)^\dfrac{y}{x}$ $\left(\dfrac{x}{y}\right)^{\large x-y}$ $1$ ...
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### Solve: $T(n) = T(n-1) +(1/n)$ by iteration

Use iteration method to solve: $1.$ $T(n) = T(n-1) + \frac{1}{n},\,(T(0)=1)$ $2.$ $T(n) = 3T\left(\dfrac{n}{3}\right) +1,\,(T(3)=1)$
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### choosing $5$ non consecutive books from a shelve of $12$

In how many ways can you pick five books from a shelve with twelve books, such that no two books you pick are consecutive? This is a problem that I have encountered in several different forms ...
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### Proving there are no integer solutions for $3x^2=9+y^3$

Prove there are no $x,y\in\mathbb{Z}$ such that $3x^2=9+y^3$. Initial proof Let us assume there are $x,y\in\mathbb{Z}$ that satisfy the equation, which can be rewritten as $$3(x^2-3)=y^3.$$ So, ...
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### Maximizing an unusual function (Putnam 1996)

“Fish," he said, "I love you and respect you very much. But I will kill you dead before this day ends.” -- Ernest Hemingway, The Old Man and the Sea I have, with varying degrees of concentration, ...
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### Mathematical problem with square numbers in the decimal system

Moderator Note: this is a question from the Federal Mathematics Competition 2013. Good morning, here's another (pretty difficult) mathematical problem... The task may sound a little strange (I'm ...
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### Multiplication Table with a frame and picture of equal sum

Is there an $n \times n$ multiplication table such that if you form a border of width $k$ ("the frame") and sum its elements, the total will equal the sum of the remaining elements ("the picture")? ...
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### How many rationals of the form $\large \frac{2^n+1}{n^2}$ are integers?

This was Problem 3 (first day) of the 1990 IMO. A full solution can be found here. How many rationals of the form $\large \frac{2^n+1}{n^2},$ $(n \in \mathbb{N} )$ are integers? The possible ...
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### How many $N$ of the form $2^n$ are there such that no digit is a power of $2$?

How many $N$ of the form $2^n,\text{ with } n \in \mathbb{N}$ are there such that no digit is a power of $2$? For this one the answer given is the $2^{16}$, but how could we prove that that this ...
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I wish to ask today's Putnam problem B6: Suppose $p$ is an odd prime. Prove that for $n\in \{0,1,2...p-1\}$, at least $\frac{p+1}{2}$ number of $\sum^{p-1}_{k=0} k! n^{k}$ is not divisble by $p$. ...
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### Finding positive real numbers $x$,$y$ and $z$ IMO Shortlist 1995 A4

How can we find all of the positive real numbers like $x$,$y$,$z$, such that : 1.) $x + y + z = a + b + c$(here $a$,$b$ and $c>0)$ and 2.) $4xyz = a^2x + b^2y + c^2z + abc$ ?(Both the conditions ...
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### Counting ordered triples of non-negative integers not greater than 100

Can we find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying (i) $x \leq y \leq z$ (ii) $x + y + z \leq 100$? Source:Regional Mathematics Olympiad India (2003) Thank you.I ...
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### Average and minimum Values of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$, $\forall x \in \mathbb{R}$

A problem was asked at Putnam Competition in 2003 (Problem 3), about finding the minimum Value of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$ where $x$ is Real. the question paper and ...
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### Counting digits in an arithmetic sequence

Given $a, d, n, x$. Suggest me a suitable algorithm to compute the number of times the digit $x$ appearing in the arithmetic sequence $a, a + d, a + 2 \times d, \cdots, a + n \times d$. For ...