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

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Points on a unit circle

Let $P_1, P_2,..., P_n$ be points equally spaced on a unit circle. For how many integer $n \in \{2,3,...,2013\}$ is the product of all pairwise distances: $$\prod_{1\le i\lt j\le n} P_{i}P_{j}$$ a ...
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45 views

Issue with a right-angled triangle

The area of the right angle triangle is $18\text{ cm}^2$ and the ratio of its legs is $2:3$. What is the length of the hypotenuse? I assumed the lengths of two sides to be $2x$ and $3x$. I used ...
4
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1answer
52 views

Find the value of $\frac{w+1}{1-w}$ given that $w^2=-1$

Question There is a new real number $w$ such that $w^2 = -1$. If all the laws of arithmetic applies, find the value of $\dfrac{w+1}{1-w}$ . I tried the following: $$\frac{w+1}{1-w} = ...
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2answers
192 views

Volume of pyramid intersection

Suppose that there are two square pyramids on the $xyz$-plane. Both have base coordinates of $(0,0,0)$, $(30,0,0)$, $(0,30,0)$, and $(30,30,0)$. One pyramid has its apex at $(10,10,30)$, while the ...
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2answers
114 views

If $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty a_n^{n/({n+1})}$

This is the $1988$ Putnam $B4$ Problem: Prove that if $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty a_n^{n/({n+1})}$. My problem lies in ...
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40 views

Problem on multiplication formulae.

Given $a^3 + b^{3}+ c^{3}= (a+b+c)^{3} $. Prove that for any natural number $n$, $$a^{2n+1}+b^{2n+1}+c^{2n+1}=(a+b+c)^{2n+1}$$ I first tried mathematical induction but did not proceed anywhere. Can ...
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2answers
65 views

Proof That,all the perfect squares each of which is the product of four consecutive odd natural numbers.

It's a question from the Bangladesh Mathematical Olympiad. It still haunts me a lot. I want to find an answer to this question. Find, with proof, all the perfect squares each of which is the ...
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1answer
27 views

Explanation of Proof Using Viete

The problem is from Putnam and Beyond. If $x + y + z = 0$, prove that $\frac{x^2 + y^2 + z^2}{2}\frac{x^5 + y^5 + z^5}{5} = \frac{x^7 + y^7 + z^7}{7}.$ The solution is as follows. Consider the ...
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1answer
52 views

Explain proof of irreducibility of $x^{p-1} + 2x^{p-2} \dots (p-1)x + p$

This is a question from Putnam and Beyond, and I have a question about the proof. The question is: Show $x^{p-1} + 2x^{p-2} + 3x^{p-3} + \dots + (p-1)x + p$ is irreducible over $\mathbb{Z}[X]$. ...
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133 views

Put $2^{600}$, $3^{500}$, $4^{400}$, $5^{300}$, and $6^{200}$ in order from least to greatest

Put $2^{600}$, $3^{500}$, $4^{400}$, $5^{300}$, and $6^{200}$ in order. Problem I found while looking at old problems from math competitions. Clearly a simple solution would be to compare ...
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3answers
73 views

Express this sum of radicals as an integer?

I have read somewhere that the radical $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=1$ and I don't understand it. How do you solve this(when the RHS is unknown)?
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1answer
55 views

Contest Question

http://hmmt.mit.edu/static/archive/february/solutions/1998/advtop.pdf In the solution of Question 10 I'm unsure how they obtained the recurrence $F(2)=\frac{3}{4}+\frac{A(1)}{4}$ does anyone have ...
3
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1answer
66 views

Greatest number equals sum of remaining numbers

Is it possible to place positive integers in a $100\times 101$ array so that in each row/column, the greatest number is equal to the sum of the remaining integers in that row/column? [Source: Russian ...
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243 views

How many $n$-element subsets $A$ of $\{1,2,3,\cdots,2n\}$ with specified sum are there?

Question: Let $ n$ be an postive integer number.and let $A=\{x_{1},x_{2},\cdots,x_{n}\}$, How many $ n$-element subsets $ A$ of $ \{1,2,\dots,2n\}$ are there,such ...
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54 views

sum of integers of two exponents equal

For what values of n, such that $n \in \mathbb{Z}^+,$ does the sum of digits $(214)^n$ and $(2014)^n$ equal? So I found $1$, which is fairly obvious, there are supposed to be more?
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1answer
113 views

Putnam Training: “Crunch Time” Topic Selection

There is about a month left before the Putnam Exam, and it will be the last one I could take. I have looked over several problems from previous exams, and done several dozen problems from Paul Zeitz's ...
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1answer
53 views

Placing non-attacking $2\times 2$ squares

Given a $1000\times 1000$ board. We can place non-overlapping $2\times 2$ squares on the cells. Two $2\times 2$ squares are said to attack each other if they lie in the same pair of adjacent rows (or ...
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3answers
382 views

Three baskets and transferring apples

This is from a math contest, and I do not have the idea how to approach it: There are 6, 7, and 11 apples in three baskets. The goal is to make all basket contain equal number of apples, but ...
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1answer
162 views

How to Solve Problem Similar to IMO(1995) Problem

Question: Let $ n$ be an postive integer number. How many $ n$-element subsets $A$ of $ \{1,2,\dots,2n\}$ are there such that $1+2+\cdots+2n$ is divisible by the sum of the elements of $A$. I ...
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1answer
62 views

$\frac{a_n - a_{n+1}}{a_n} \approx \frac{1}{n}$? (part of 2010 Putnam exam)

Given a non-negative sequence $a_n$, strictly decreasing and tending to zero, can we show that (for large $n$) $$ \frac{a_n - a_{n+1}}{a_n} \approx \frac{a_n}{na_n} = \frac{1}{n} \text{ }?$$ ...
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2answers
88 views

Minimum of $\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\sqrt{\dfrac{c}{a+b}}$

What is the minimum of $$f(a,b,c):=\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\sqrt{\dfrac{c}{a+b}}$$ where $a,b,c$ are positive real numbers? When $a=b=c$, we have ...
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1answer
27 views

Game picking cards so that sum is divisible by $25$

Adele and Bryce play a game. There are $50$ cards, numbered $1,2,\ldots,50$. They take turns alternately picking a card, with Adele going first. If at the end, the sum of the numbers on Adele's cards ...
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1answer
68 views

Average question GRE

The average daily temperature from 9th to 16th January(both inclusive) was 38.6 C and that from the 10th to 17th January(inclusive) was 39.2 C. what was the temperature on 17th January? I am able ...
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2answers
76 views

$ab$ divides $3^a+1$ and $3^b+1$

Find all positive integers $a,b$ such that $ab$ divides $3^a+1$ and $3^b+1$. It is clear that $3$ cannot divide either $a$ or $b$, because $3$ doesn't divide $3^a+1$ or $3^b+1$. ...
7
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1answer
65 views

Blackboard operation $x,y,z\rightarrow x,y,1/(zx+zy)$

The three numbers $2,3,6$ are written on the blackboard. In each move, we can pick any two numbers, say $x,y$, and replace the third number $z$ by $1/(zx+zy)$. Using finitely many operations, is it ...
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34 views

On alternating sums of the elements of subsets.

Recently in a contest a question was asked as under. We define a lead element of a set $\{a_1,a_2,a_3, \cdots a_n\}$ as $$l(\{a_1,a_2,a_3, \cdots a_n\})=|a_1-a_2+a_3-a_4 \cdots(-1)^{n-1}a_n|$$ Now ...
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28 views

How do you calculate this

I know it converges, but i need to know the sum of this, i don't know the expression because i'm not English... I need it for my homework and I don't know how to do it, so please if somebody knows how ...
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1answer
121 views

I am having trouble with this integral from the 2012 MIT Integration Bee

$$\int\frac{dx}{(1+\sqrt{x})\sqrt{x-x^2}} $$ Could someone explain to me how to integrate this integral. Thank you and cheers.
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54 views

“Massaging” inequalities to prove them (esp. in contest math like the IMO/Putnam)?

What's the contest inequality solving technique where you do something like representing each side as the function of some sequence and replacing the max/min terms of the sequence with their average, ...
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1answer
54 views

Bit tricky plot on GRE

In a city 90% of the population own a car, 15% own a motor cycle, and everybody owns one or the other or both. Find the percentage of motorcycle owners to car owners. In order to solve it i ...
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1answer
56 views

Maximum number of squares with same number

Given a $1000\times 1000$ board. At the beginning, all cells have $0$ written on it. In an operation, we are allowed to choose any $130\times 130$ subboard and increase every number in this subboard ...
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54 views

Integers neither as sum nor difference of perfect powers

Are there infinitely many positive integers $n$ for which there do not exist integers $a,b\geq 1$ and $c,d\geq 2$ such that $n=a^c+b^d$ or $n=a^c-b^d$? [Source: Hungarian competition problem]
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1answer
163 views

Quadratic equation, math olympiad question

So this is a 9-10th grade, math olympiad problem I found. Define the parabola $y=ax^2+bx+c$ such that $a,b,c$ are positive integers. Suppose that the roots of the quadratic equation $ax^2+bx+c=0$ are ...
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2answers
28 views

$a^2\equiv 10\pmod b$ and $a^3\equiv 33\pmod b$

Let $a,b$ be positive integers such that $a^2\equiv 10\pmod b$ and $a^3\equiv 33\pmod b$. What are all possible values of $b$? We have that $10a\equiv 33\pmod b$, but how does that determine the ...
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2answers
129 views

Inequality $(1+x_1)(1+x_2)\ldots(1+x_n)\left(\dfrac{1}{x_1}+\dfrac{1}{x_2}+\cdots+\dfrac{1}{x_n}\right)\geq 2n^2.$

Let $n\geq 2$, and $x_1,x_2,\ldots,x_n>0$. Show that $$(1+x_1)(1+x_2)\ldots(1+x_n)\left(\dfrac{1}{x_1}+\dfrac{1}{x_2}+\cdots+\dfrac{1}{x_n}\right)\geq 2n^2.$$ For $n=2$, this reduces to ...
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3answers
206 views

likely open number theory problem: finite sum of $\zeta(2)$ equal to a square of rationals

Which $n$ can let $S=1+\frac14+\frac19+\cdots+\frac1{n^2}$ be a square of a rational number? Obviously, $1$ and $3$ work, but how to prove they are the only ones? I think this problem is really hard. ...
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1answer
64 views

Bounded area for any triangle formed by polygons

Let $P_1,P_2,P_3$ be closed polygons on the plane. Suppose that for any points $A\in P_1$ (meaning $A$ can be inside or on the boudary of $P_1$), $B\in P_2,C\in P_3$, we have $[ABC]\leq 1$. Is it ...
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41 views

How should I approach solving this floor function?

Prove that for all $n \in \Bbb Z, \lfloor\sqrt {(n)}+ \sqrt {(n+1)} \rfloor = \lfloor \sqrt{4n+2}\rfloor$. There must be some algebraic substitution?
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3answers
103 views

Every $3\times 3$ square has even number of painted cells

Given a $1000\times 1000$ board. We paint some cells (at least one) so that in every $3\times 3$ square, an even number of cells are painted. What is the minimum number of painted cells? One way to ...
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1answer
46 views

Condition $|x_1x_2+1|<x_1+x_2$ in quadratic polynomial

Let $x^2-ax+b$ be a polynomial with real coefficients having two nonzero roots. Given that $|b+1|<a$, and one of the roots have modulus $<1$, show that the other root has modulus $>1$. We ...
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1answer
38 views

GCD of adjacent pairs take on all possible values

Given a fixed positive integer $n$. Consider the numbers $1,2,\ldots,2n$. The GCD of any pair is one of $1,2,\ldots,n$. Suppose that all $2n$ numbers are placed around a circle. Is it possible that ...
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3answers
49 views

Finding the range of equation. Any tricks?

I m working on the following problem For real numbers $a,b$, if $a+ab+b=3$, then find the range of $m=a-ab+b$. Is there any inequalities here to use?
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111 views

Least possible number of squares with odd side length

An $n\times(n+3)$ rectangular grid ($n>10$) is cut into some squares, with all cuts being along the grid lines. What is the least possible number of squares with odd side length? [Source: Russian ...
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1answer
37 views

Quadratic polynomial with alternate negative value

Let $f(x)=-x^2+ax+b$, where $a,b\in\mathbb{R}$. Suppose there exist distinct integers $m,n$ such that $f(m)=-n^2$ and $f(n)=-m^2$. Prove that there are infinitely many pairs of integers $x,y$ such ...
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55 views

Maximum number of acute triangles

Given $n$ points on the plane, no three of which are collinear, what is the maximum number of acute triangles formed by them? [Source: Based on Hungarian competition problem]
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58 views

How solve the equation $x^{4}-y^{4}= 80z^{4}$ for x and y odd integers, and z integer

Let x and y be odd integers, and let z be an integer. How solve the equation $x^{4}-y^{4}= 80z^{4}$?
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94 views

Integer solutions of $a^3+2a+1=2^b$

What are the solutions in integers of $a^3+2a+1=2^b$? [Source: Serbian competition problem]
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365 views

IMO programs of different nations?

We in Albania have a good team in the IMO, and this year I will probably be part of it. Since Albania does not have a public training programme, I have to consult the training programmes of other ...
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0answers
49 views

Ratio of product from one point and minimum distance

Given points $A_0,A_1,\ldots,A_n$ in the plane, let $m$ denote the minimum distance among any two points. What is the minimum value of $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$ ...
7
votes
1answer
102 views

Dividing tournament into “equal” groups

In a tournament of $n$ teams, each team plays all other teams exactly once, with no draw. For which $n$ is it always possible to divide all teams into several groups so that each group of teams won ...