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

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NIMO 16.8 Expected Value + Probability

Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is monotonically bounded if $1\leq a_i\leq i$ for all $1\leq i\leq 2^{16}$. We say that a term $a_k$ in the sequence ...
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0answers
30 views

Finding maximum value of a 3-variable function using inequality.

Let $a, b, c$ be positive real numbers satisfying $a^2 +b^2+c^2=14$. Find the maximum value of $f(a,b,c)=\frac{4(a+c)}{a^2+3c^2+28}+\frac{4a}{a^2+bc+7}+\frac{5}{(a+b)^2}-\frac{3}{a(b+c)}$
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3answers
69 views

Olympic elementary combinatorics problem

This is a problem taken from the regional selections of the Italian mathematical olympiads: A knight is placed on the bottom left corner of a $ 3\times3 $ chess board. In how many ways can you move ...
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17 views

Maximum number of highways

There are 20 cities in a country, some of which have highways connecting them. Each highways goes from one city to another, both ways. There is no way to start in a city, drive along the highways of ...
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3answers
37 views

Maximizing sin(a-b) given a trig relation

Suppose $a$, $b$ are acute angle measures such that $\tan a = 5\tan b$. Find the maximum value of $\sin(a-b)$. $\sin(a-b)=4\sin b \cos a$, but I don't know what to do from here.
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1answer
29 views

How many peas one can win

$A$ and $B$ plays the following game. In a table there are $n>1$ plates which are empty at the beginning. In the beginning of every round, $A$ moves some plates to the right hand side of the board, ...
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2answers
21 views

Maximizing Utility

A farmer learns that he will die at the end of the year (day 365, where today is day 0) and that he has a number of sheep. He decides that his utility is given by $ab$ where $a$ is the money he makes ...
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24 views

Number of sock combinations with limited information

Suppose in your sock drawer of 14 socks there are 5 different colors and 3 different lengths present. One day, you decide you want to wear two socks that have both different colors and different ...
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2answers
918 views

Placing 5 pieces on a 5x5 grid with no main diagonal

A 5x5 grid is missing one of its main diagonals. In how many ways can we place 5 pieces on the grid such that no two pieces share a row or column?
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1answer
61 views

Triangle, Circle Problem

What is the area $\triangle DEF$ ? I solved it using analityc geometry. I want to see if there is way to solve it using plane geometry. I did it: $x^2+y^2=400$ $(x+10)^2+y^2=100$ I found the ...
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3answers
38 views

Secant line and diameter of a circle

A secant line incident to a circle at points $A$ and $C$ intersects the circle's diameter at point $B$ with a $45^\circ$ angle. If the length of $AB$ is $1$ and the length of $BC$ is $7$, then what is ...
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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 ...
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1answer
42 views

Triangle inscribed in an ellipse [closed]

What is the maximum area of a triangle that can be inscribed in an ellipse with semi-axes $a$ and $b$?
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0answers
21 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 ...
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2answers
52 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 ...
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1answer
90 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 ...
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3answers
68 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 ...
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1answer
47 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?
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1answer
26 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?
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4answers
197 views

Sum of fractions with square roots inequality

What is the greatest integer $n$ such that $n \leq 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \ldots + \frac{1}{\sqrt{2014}}$?
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2answers
476 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 $U$?
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Help in deriving a formula [on hold]

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 ...
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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.
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4answers
79 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 ...
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2answers
139 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 ...
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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
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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 ...
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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 ...
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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} = ...
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0answers
88 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
87 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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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 ...
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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 ...
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1answer
24 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
47 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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3answers
120 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
53 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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53 views

A Strange Algorithm on Processor [closed]

We have n processes, each with a predetermined start and end time. We want to use the ...
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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 ...
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1answer
44 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 ...
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1answer
48 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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167 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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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?
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1answer
68 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
45 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
356 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
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 ...
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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 $n$) $$ \frac{a_n - a_{n+1}}{a_n} \approx \frac{a_n}{na_n} = \frac{1}{n} \text{ }?$$ ...
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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 ...
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1answer
25 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 ...