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

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1answer
25 views

Limit sup and inf hint

I have problem in finding the Limsup and liminf for the following sequences. Any hint pls? $(s_n) = [1-r^n]\sin \frac{n\pi}{2}$ and $(s_n) = [(-1)^n + 1]n^2$.
3
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0answers
32 views

Area of a circumcenter triangle equals area of medial triangle

Let $X$, $Y$, $Z$ be the midpoints of sides $BC$, $AC$, $AB$ respectively in triangle $ABC$. Let $O_{A}$, $O_{B}$, and $O_{C}$ be the circumcenters of triangles $AZX$, $BXY$, and $CYZ$ respectively. ...
2
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0answers
53 views

How prove this complex inequality with same as (2014 china CMO) Cauchy-Schwarz inequality

let $r$ is give numbers,let $z_{1},z_{2},\cdots,z_{n}$ such $|z_{i}-1|\le r,i=1,2,\cdots,n,r\in(0,1)$ show that ...
0
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1answer
27 views

Find the volume $z \geq 3x^2+2y^2, \ \ 3x^2+2y^2+5z^2 \le 1$

Find the volume of solid defined by the following inequalities : $$z \geq 3x^2+2y^2, \ \ 3x^2+2y^2+5z^2 \le 1$$ We have an ellipse, which the semi-axis are $\sqrt{\frac{z}{2}}$ and ...
2
votes
8answers
114 views

How to show that $f(x) = 0$ if $\int_a^bf(x)\,\text{d}x=0$ for all $a,b\in\mathbb{R}$?

I found this problem on the web: Let $f(x)$ be a real-valued, continuous function with the property that $$\int_a^bf(x)\,\text{d}x=0$$for all real numbers $a,b$. Prove that $f$ is identically $0$. ...
1
vote
1answer
59 views

2014 Fall OMO #28

Here is a problem from this year’s OMO: Let $S$ be the set of all pairs $(a,b)$ of real numbers satisfying $1+a+a^2+a^3 = b^2(1+3a)$ and $1+2a+3a^2 = b^2 - \frac{5}{b}$. Find $A+B+C$, where $$ A = ...
4
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4answers
526 views

Is it possible to permute an unknown binary sequence so that two particular bits are equal?

A blind mathematician is give a $2015$ bit sequence. The mathematician can take any two bits and switch them (so the bit in position $A$ goes to position $B$ and vice-versa). He knows at what position ...
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2answers
39 views

How to find Bitwise AND of all numbers for a given range?

How can I find Bitwise AND of all numbers for a given range say from A to B, including both? I found a beautiful answer for finding XOR for such range. http://stackoverflow.com/a/10670524/2046703How ...
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0answers
41 views

Where can Gaussian Elimination be used?

I have searched for this and came to know about it that it is traditionally used to solve linear equations, finding determinant, rank of matrix, inverse of matrix. There was a problem on codechef: ...
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0answers
30 views

How prove this the number of ordered $n$-tuples $(\varepsilon_{1},\cdots,\varepsilon_{n})$such this following inequality is $2^{n-100}$

Interesting Question: for any complex numbers $z_{1},z_{2},\cdots,z_{n}$ such $$\begin{cases} |z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2=1\\ |z_{i}|\le\dfrac{1}{10},i=1,2,\cdots,n \end{cases}$$ ...
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0answers
39 views

Motivation for Putnam (soft question)

This question may be too specific and too vague. But I'm curious about this. How highly are the applicants evaluated in PhD admission if they were ranked above the cutoff of honorable-mention in ...
5
votes
1answer
105 views

Set with distinct subset sums

The problem is as follows : Given a set A with distinct positive integer elements, prove that there always exists another set B consisting of positive integers, s.t., The size of B is less than or ...
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6answers
104 views

$2x^2+ 3y^2+4z^2 =1$ find the maximum of $4x+3y+2z$

If $2x^2+ 3y^2+4z^2 =1$ find the maximum of $4x+3y+2z$. This is a question from a regional math olympiad and thus there must exist solutions without application of calculus. I have no idea how to ...
0
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1answer
42 views

Do two altitudes uniquely determine the third

BdMO 2014 Nationals If the lengths of two altitudes drawn from two vertices of a triangle on their opposite sides are $2014$ and $1$ unit, then what will be the length of the altitude drawn from ...
3
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1answer
126 views

When does the Putnam release solutions to this year's exam? Has anyone released their own solutions?

I was just wondering when the Putnam committee releases the solutions to this year's exam or if anyone has posted their own solutions.
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1answer
30 views

Finding the invariant

There are A white, B black, and C red chips on a table. In one step, you may choose two chips of different colors and replace them by a chip of the third color. If just one chip will remain at the ...
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2answers
36 views

No 37 question in knight of pi math tournament Dec 15, 2012

The five digit integer ABCDE, where each letter represents a digit, not necessary distinct, is divided by the numbers $2$,$3$,$4$,$5$, and $6$. The remainders are A, B, C, D, and E respectively. What ...
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2answers
47 views

How many consecutive squares can be subtracted from a number?

Let's say I am given a number N. I want to check how many consecutive squares of integers(starting from 1) can be subtracted from this number. Example- For N=13, I will first subtract 1(=1^2), leaves ...
3
votes
1answer
62 views

Taking a Putnam (General Questions) [duplicate]

I've just discovered an undergrad math competition (William Lowell Putnam Competition) and that my school offers it. The competition looks extraordinarily difficult, but I thought I'd give it a go. ...
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1answer
16 views

Find if there exist some combination of these digits that will be divisible by 8 or not

Let's say I am given some 100 digits and I have to find whether there can be any combination of these digits such that the number formed will be divisible by 8, how can I do that? I know divisibility ...
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1answer
54 views

How can I determine the center of this circumference?

I have the following question: if I have an irregular symmetric polygon, how can I determinate the circumference with the least area that contains this polygon? I believe (in case that the polygon ...
3
votes
3answers
136 views

AHSME 1981 #22 - Number of lines that pass through four distinct points

How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form $(i, j, k)$ where $i$, $j$, and $k$ are positive integers not exceeding four? ...
4
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2answers
57 views

Find minimum of $a+b$ if $13|a+11b $ and $11|a+13b$

Find minimum of $a+b$ if $13|a+11b $ and $11|a+13b$ where $a,b>0$. My attempt : $13|a+11b \implies 13|a+24b$ . Similarly we get $11|a+24b$. Now $\gcd(11,13)=1$, so, $143|a+24b$. Therefore $a+24b ...
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0answers
118 views
+50

How to prove there exists $n_{1}a_{n_{0}}+n_{2}a_{n_{1}}+\cdots+n_{k}a_{n_{k-1}}<3(a_{1}+a_{2}+\cdots+a_{N})$

Let $a_{1},a_{2},\cdots,a_{N}$ be nonnegative reals, not all $0$. Prove that there exists a sequence $$1=n_{0}<n_{1}<\cdots<n_{k}=N+1$$ of integers such that ...
0
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1answer
37 views

How find this Exponential constant $C$,if such this $Ax^C\le N(x)\le Bx^C$

interesting problem Let sequence $$a_{0}=x\in (0,1),a_{n}=a_{n-1}+a^3_{n-1},n=1,2,\cdots$$ and define $$N(x)=\min{\{n|a_{n}>1\}}$$ Assmue that there exsit postive constant $A,B$,and ...
4
votes
1answer
42 views

Modulo Arithmetic of Complex Numbers

Suppose $a,b,c \in \mathbb{C}$ such that $$a+b+c\in \mathbb{Z},$$ $$a^2+b^2+c^2=-3,$$ $$a^3+b^3+c^3=-46,$$ $$a^4+b^4+c^4=-123$$ then find $(a^{10}+b^{10}+c^{10})\pmod{1000}$. I only observed that ...
9
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0answers
164 views

Determining information in minimum trials (combinatorics problem)

A student has to pass a exam, with $k2^{k-1}$ questions to be answered by yes or no, on a subject he knows nothing about. The student is allowed to pass mock exams who have the same questions as the ...
2
votes
1answer
28 views

Determining diagonalizability of a linear transformation defined by a matrix.

Suppose $A\in M_n(\Bbb C)$ satisfies $A^6-A^3+I=O$. Prove that if a linear transformation $T:M_n(\Bbb C)\rightarrow M_n(\Bbb C)$ is given by $T(B)=AB$, then $T$ is diagonalizable. How to prove it? ...
2
votes
0answers
61 views

How prove this polynomials $p_{j}(x)$ and $p_{k}(x)$ are relatively prime (2014,Putnam problem)

Question: Let $P_n(x)=1+2x+3x^2+\cdots+nx^{n-1}.$ Prove that the polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j\ne k.$ this problem it seem ...
3
votes
1answer
51 views

How find prime numbers $p_{i}$ such $p_{1}+p_{2},p_{2}+p_{3},p_{3}+p_{4},\cdots,p_{n}+p_{1}$ is square number

Question: Let $n\ge 5$ be an odd number, show that: there exist (or does not exist) primes $p_{i}\:;\:i=1,2,\cdots,n$ such that $$p_{1}+p_{2},p_{2}+p_{3},p_{3}+p_{4},\cdots,p_{n}+p_{1}$$ all ...
2
votes
2answers
51 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 ...
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1answer
91 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 ...
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0answers
17 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 ...
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1answer
76 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.
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0answers
49 views

Bisectors of opposite angles of a circular quadrilateral meet at the diagonal.

Let ABCD be a circular quadrilateral so that the bisectors of angles ABC and ADC meet at the diagonal AC. Let M be the midpoint of AC. Let q be a line parallel with the side BC so that q passes ...
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2answers
110 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
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3answers
91 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
95 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
50 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
48 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
134 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
35 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
44 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 ...
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2answers
54 views

RMO 1991 problem no.4 [closed]

There are two urns each containing an arbitrary number of balls, both are non-empty to begin with. We are allowed 2 types of operations: (a) Remove an equal number of balls simultaneously from ...
1
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1answer
61 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
27 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
83 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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0answers
18 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 ...
1
vote
3answers
40 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.
0
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1answer
30 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, ...