Tagged Questions

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

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0
votes
2answers
24 views

how can you solve this pipe question?

Two pipes, A and B can fill a tank in 24 and 35 minutes respectively. If both the pipes are opened simultaneously, after what time should A be closed so that the tank is filled in 18 minutes? Can you ...
1
vote
1answer
40 views

Triangle inscribed in an ellipse [on hold]

What is the maximum area of a triangle that can be inscribed in an ellipse with semi-axes $a$ and $b$?
0
votes
0answers
16 views

removable singularity and injective function

Let $U \subset \mathbb{C} $ a conected open subset, $ a \in U $ and $ f:U- \{a\} \to \mathbb{C}$ a holomorphic function such that $ V=f (U-\{a\}) $ is a open bounded subset. (A) Show that $ f $ has a ...
0
votes
2answers
51 views

Math Conundrum regarding Usain Bolt's 100m world record

Consider the suvat equation, S = ut + 1/2 at^2 Usain bolt ran 100 metres in 9.58 seconds for the world record, and going by the suvat equation above, his acceleration over a distance of 100 metres ...
4
votes
1answer
84 views

Different solution for MOSP(Mathematical Olympiad Summer Program) 2001 Test 9 Problem

Let $ABCD$ be a convex quadrilateral and let $O$ be the point of intersection of its diagonals. Prove that if the perimeters of $\triangle ABO$,$\triangle BCO$,$\triangle CDO$ and $\triangle ...
0
votes
3answers
66 views

Math is Cool: Probability

Kailash always has a $\frac{3}{4}$ chance of winning any game he plays. What is the probability that out of 5 games he plays, he wins $2$ and loses $3$? I know the answer is $\frac{45}{512}$, but ...
1
vote
1answer
44 views

Queens on a chessboard

What is the smallest number of queens that can be placed on a chessboard so that every square is either occupied or can be reached in one move?
1
vote
1answer
22 views

Obtuse triangles in a regular polygon

How many triangles formed by three vertices of a regular $17$-gon are obtuse? As an extension, how many triangles formed by three vertices of a regular $n$-gon are obtuse?
3
votes
4answers
191 views

Sum of fractions with square roots inequality

What is the greatest integer $n$ such that $n \leq 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \ldots + \frac{1}{\sqrt{2014}}$?
6
votes
2answers
469 views

Sum of digits raised to a power

Let $S$ equal the sum of the digits of $2014^{2014}$. Let $T$ equal the sum of the digits of $S$. Let $U$ equal the sum of the digits of $T$. What is $U$?
2
votes
0answers
34 views

Help in deriving a formula

Background I am working on a vocabulary building application under which I am trying to build an adaptive test for the student. The test would be adaptive to the user's response: When the student ...
2
votes
1answer
30 views

Remainder of a combination

Problem from a contest: What is the remainder when $\binom{169}{13}$ is divided by $13^5$? I thought that Wolstenholme's/Babbage's would help, but not entirely sure how.
-1
votes
3answers
64 views

Evaluating $ \int \frac{3}{x^2+8x+17}dx$ [closed]

How to find this integral: $$\displaystyle \int \dfrac{3}{x^2+8x+17}dx$$
2
votes
4answers
78 views

can have solution of $x^4-3x^3+2x^2-3x+1=0$ using only high school methods

can have solution of $x^4-3x^3+2x^2-3x+1=0$ using only high school methods??? i only know quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ i tried many algebraic manipulations and i get ...
0
votes
1answer
68 views

How prove $\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\leq\frac{3\sqrt{3}}{4}$ [closed]

How prove $\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\leq\frac{3\sqrt{3}}{4}$ for $ a,b,c>0: (a+b)(b+c)(c+a)=ab+bc+ca $?
5
votes
2answers
125 views

What is the coefficient of $x^{25}$ in $(x^3 + x + 1)^{10}$?

Working on some contest problems and came across this question. Here's what I have so far on the off chance that my thinking is correct... So using Vieta's the coefficient of the $x^{25}$ should be ...
-1
votes
1answer
35 views

Factorize a number into coprime numbers

I want to know if there is a way to factorize a number into coprime numbers; for example $N = a_1 \cdot a_2 \cdot a_3 \cdots a_i$ And $a_i$ and $a_j$ are coprime for any $i \ne j$ Thanks
0
votes
0answers
28 views

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 ...
1
vote
0answers
35 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
votes
1answer
51 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} = ...
0
votes
0answers
83 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 ...
5
votes
2answers
80 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 ...
0
votes
2answers
37 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 ...
2
votes
2answers
47 views

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

It's a question from $BdMO$.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 product of four consecutive odd ...
0
votes
1answer
19 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 ...
4
votes
1answer
46 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 $\mathbb{Z}[X]$. ...
7
votes
3answers
118 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 ...
1
vote
3answers
50 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)?
1
vote
0answers
47 views

A Strange Algorithm on Processor [closed]

We have n processes, each with a predetermined start and end time. We want to use the ...
0
votes
0answers
34 views

What's the solution set $S \subset \mathbb{R}^2$ of this equation?

I see that $(1,1)$, $(2,4)$ and $(4,2)$ are in $$S= \{(x,y) \in \mathbb{R}^2: \, x^y = y^x\}$$ My question is: The set $S$ contains many others elements? Thanks for any suggestions and helpful ...
1
vote
1answer
43 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
votes
1answer
47 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 ...
8
votes
0answers
165 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 ...
1
vote
0answers
49 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?
2
votes
1answer
65 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 ...
2
votes
1answer
42 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 ...
5
votes
3answers
355 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 ...
1
vote
1answer
121 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 ...
0
votes
1answer
55 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 $n$) $$ \frac{a_n - a_{n+1}}{a_n} \approx \frac{a_n}{na_n} = \frac{1}{n} \text{ }?$$ ...
7
votes
2answers
83 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 ...
3
votes
1answer
23 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 ...
2
votes
1answer
44 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 ...
7
votes
2answers
73 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
votes
1answer
61 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 ...
0
votes
0answers
25 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 ...
-1
votes
2answers
24 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 ...
0
votes
0answers
28 views

Ratio's concept in GRE question

If you can buy A apples for n nickels(five cent coin),how many apples can you buy for d times and q quarters? This Question is using Ratio's concept but the second part of the question is ...
5
votes
1answer
94 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.
1
vote
0answers
36 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, ...
0
votes
1answer
38 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 ...