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

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2
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2answers
59 views

On $n \times n$ grids filled with $1$ and $-1$

The following question was asked in a contest which I have difficulty proving . Let $n$ be an odd positive integer and suppose that each square of an $n \times n$ grid is filled with either $1$ or ...
1
vote
1answer
121 views

How to prove this integral-inequality.

Suppose $f$ is twice differentiable and satisfies $f(0)=0$. Prove the inequality. $$\int_0^1 |f(x)f'(x)| dx \le\ \frac{1}{2} \int_0^1 |f'(x)|^2 dx $$ This is a problem from undergraduate math ...
1
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0answers
19 views

KPMT: Probability [duplicate]

The question, problem 38 from Knights of Pi Math Tournament: Dec. 12, 2009: Lord Voldemort is buying snakes. There are an infinite number of four varieties of snakes: garter snakes, king cobras, boa ...
0
votes
1answer
140 views

Prove that there are infinity many numbers you can't write in the form $a^{T(a)}+b^{T(b)}$.

Prove that there are infinity many numbers you can't write in the form $a^{T(a)}+b^{T(b)}$ where a and b are positive integers. T(a) represents the number of divisors number a has. Source: 3rd ...
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2answers
151 views

Find all real real functions that satisfy the following eqation $f(x^2)+f(2y^2)=[f(x+y)+f(y)][f(x-y)+f(y)]$

Find all real functions $f:\Bbb R\rightarrow\Bbb R$ so that $f(x^2)+f(2y^2)=[f(x+y)+f(y)][f(x-y)+f(y)]$, for all real numbers $x$ and $y$. $f(x)=x^2$ is the only solution I think. So far I have got: ...
4
votes
3answers
95 views

How many $2$'s are needed?

There is a positive integer $N$. $N$ is made up of only two distinct digits- $2$ and $3$. $N+18$ is divisible by $37$. What is the minunum amount of times the number $2$ can appear in $N$? I'm pretty ...
3
votes
4answers
103 views

Proof check for Putnam practice problem

I realize this is simply an A1 problem, but my proof seems way too simple, so I would like someone to point out whether or not it's correct (and most importantly, fix any flaws in it). Problem. ...
1
vote
1answer
54 views

3 balls in a box

We have a box with $3$ balls, that can be black or white. We extract a ball, and it's white. Then we put the ball in the box, we extract again a ball and it's white. What is the probability that in ...
1
vote
1answer
56 views

Probability and recurrence

One day, one alien has come to the earth. Every day, each alien does one of four things, each with a probability of $1/4$: 1) destroying himself, 2) splitting into 2 aliens, 3) splitting into 3 aliens ...
7
votes
1answer
178 views

What branches are these (contest) maths questions from?

The OP is studying for his local math competition (Australian), and when running through past papers I found some questions subtle to handle. I decide to buy some books to aid my study, but there are ...
2
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2answers
52 views

Number of length-n paths in a graph with a fixed start vertex

So I was looking at a few past-years' papers from the ZIO (an IOI qualifier held here in India), and I found this question: I think this is the same as finding the number of paths of (let's take (a)) ...
3
votes
1answer
49 views

The order of element in $\mathbb{Z} / 2^{2014}\mathbb{Z}$

Find the smallest integer $n$ such that $2^{2014}|17^n-1$. i.e. Find the order of $17$ in $(\mathbb{Z}/ 2^{2014} \mathbb{Z})^{\times}$. I think we have to use the lifting the exponent lemma: If ...
1
vote
1answer
66 views

For which positive integers $n$ does $P(n)$ fail to hold?

Let $n$ be a natural number and let $z$ be a complex number. Consider the following proposition: $P(n)$: If $\cos (nz)$ is bounded above by one in absolute value, then $\cos z$ ...
2
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0answers
30 views

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 ...
1
vote
3answers
106 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 ...
1
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1answer
35 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
47 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
31 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
29 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 ...
5
votes
2answers
1k 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?
3
votes
1answer
80 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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vote
3answers
106 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 ...
0
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2answers
31 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
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1answer
54 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
39 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
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2answers
104 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 ...
5
votes
1answer
155 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
2answers
81 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
53 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
57 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
265 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
559 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
1answer
33 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.
2
votes
4answers
85 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 ...
6
votes
2answers
167 views

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

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 $-(r_1r_2r_3r_4r_5+r_1r_3r_4r_5r_6+...+r_6r_7r_8r_9r_{10})$ Since ...
-1
votes
1answer
46 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
33 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
46 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
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} = ...
0
votes
2answers
210 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
118 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
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 ...
2
votes
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 ...
0
votes
1answer
28 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
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 $\mathbb{Z}[X]$. ...
7
votes
3answers
137 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
74 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
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1answer
58 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
68 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 ...
11
votes
0answers
279 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 ...