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

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MOSP $2002$ Combinatorics Problem

I only want a hint(I already have the solution near me, but the book doesn't give a hint (MOSP) Assume that each of the $30$ MOPpers has exactly one favorite chess variant and exactly one ...
2
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0answers
13 views

Use Lagrange Interpolation polynomial to find this $\sum_{cyc}\frac{x^3}{(x^2-y^2)(x^2-z^2)}$

let $x,y,z$ are $t^3-t^2+2t-3=0$ three complex solution, find $$\dfrac{x^3}{(x^2-y^2)(x^2-z^2)}+\dfrac{y^3}{(y^2-x^2)(y^2-z^2)}+\dfrac{z^3}{(z^2-x^2)(z^2-y^2)}$$ How to use interpolation ...
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1answer
44 views

If Sam read 60 pages in 100 min, how many can Sam read in 45 [on hold]

Sam reads 60 pages of his novel in 100 minutes. How many pages of his novel can Sam expect to read in 45 minutes if he reads at the same rate?
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0answers
46 views

Fermat's last theorem generalization

Are there solutions to Fermat's last theorem in transcendental numbers greater than two, for further details please follow the link below: http://www.quora.com/What-is-the-new-Pythagoras-Theorem Let ...
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12 views

Logic problems : references

I'm looking for problems from mathematical contests about logic (similar to the problem PMWC Problem T5).
8
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1answer
44 views

Is it true that $\sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb N^+\text{ such that } a^2-b=k^2 $?

This is a curiosity question: Question Given two positive integers $a$ and $b$ do we have the following equivalence: $$\sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb ...
0
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2answers
58 views

Show that $f(a)$ converges after some point

There is a row of 1000 integers. There is a second row below, which is constructed as follows. Under each number $a$ of the first row, there is a positive integer $f(a)$ such that $f (a)$ equals ...
2
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1answer
66 views

Making change with prime-valued coins

Am I understanding this question correctly and how do I approach these problems? In Numberland, the unit of currency is the El (E). The value of each Numberlandian coin is a prime number of Els. So ...
2
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1answer
39 views

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits.

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits. It is an intuitive re-statement of Baltic Way 2012 (I think there are shortlists in Baltic Way every ...
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1answer
32 views

Grid Problem Proof

I have a 2x2 grid square say, I can fit a shape like this: Such that there is one missing square. I can arrange this in any way so that the missing square can be located anywhere. I can do ...
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4 views

Applicaton Word Problems that apply with the wind

A citation CV Jet travels 460 MPH in still air and flies 525 Miles into the wind 525 Miles with the wind in a total of 2.3 hours. Find the wind speed?
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2answers
32 views

algebra word problem that aplies with the speed of still water [on hold]

the speed of the current in catamount creek is 3 mph. Sean can kayak 4 mi. upstream in the same time that it takes him to kayak 10 mi.downstream. What is the speed of seans kayak in still water?
2
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1answer
26 views

Cute convergence problem. Proving convergence of sequence regarding reciprocals of least common multiple converges.

This is the first problem of the second day of the $2014$ CIIM. Let $\{a_n\}$ be a strictly increasing sequence of positive integers. Prove the sequence ...
5
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1answer
55 views

Rocks and squares, balls and sticks. [closed]

Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, ...
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0answers
139 views
+100

On $a^4 + b^4 = c^4 + d^4 = e^5$.

Let $a, b, c, d, e$ be distinct positive integers such that $a^4 + b^4 = c^4 + d^4 = e^5$. Show that $ac + bd$ is a composite number.
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2answers
51 views

Show that an even integer exists at the end

Start with positive integers: $1, 7, 11, 15, ..., 4n - 1$. In one move you may replace any two integers by their difference. Prove that an even integer will be left after $4n - 2$ steps. I said, ...
1
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1answer
23 views

Solution Invariant Explanation Trick

Suppose not all 4 integers, $a,b,c,d$ are equal. Start with $(a,b,c,d)$ and repeatedly replace $(a,b,c,d)$ by $(a−b,b−c,c−d,d−a)$. Then show that at least one number of the quadruple will become ...
13
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1answer
107 views

Quadrilateral $APBQ$.

Quadrilateral $APBQ$ is incsribed in a circle $\omega$ with $\angle P = \angle Q = 90^{\circ}$ and $AP = AQ < BP$. Let $X$ be a variable point on segment $\overline{PQ}$. Line $AX$ meets $\omega$ ...
7
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1answer
46 views

Sequences of real numbers, arithmetic mean.

Given a sequence of real numbers, a move consists of choosing two terms and replacing each with their arithmetic mean. Show that there exists a sequence of 2015 distinct real numbers such that after ...
12
votes
1answer
43 views

Subset coloring, additive structure.

Let $S = \{1, 2, \dots, n\}$, where $n \ge 1$. Each of the $2^n$ subsets of $S$ is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set $T ...
11
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2answers
141 views

Diophantine equation $x^2 + xy + y^2 = \left({{x+y}\over{3}} + 1\right)^3$.

Solve in integers the equation$$x^2 + xy + y^2 = \left({{x+y}\over3} + 1\right)^3.$$
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2answers
31 views

Show that there is a large value

Suppose not all 4 integers, $a, b, c, d$ are equal. Start with $(a, b, c, d)$ and repeatedly replace $(a, b, c, d)$ by $(a - b, b - c, c - d, d - a)$. Then show that at least one number of the ...
2
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0answers
45 views

Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$

Let $P_1 P_2 \dots P_n$ be a convex polygon in the plane. Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$ Show ...
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3answers
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Number of times $2^k$ appears in factorial

For what $n$ does: $2^n | 19!18!...1!$? I checked how many times $2^1$ appears: It appears in, $2!, 3!, 4!... 19!$ meaning, $2^{18}$ I checked how many times $2^2 = 4$ appears: It appears in, ...
3
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1answer
57 views

Find all odd positive integers $n$ greater than $1$ such that for any coprime divisors …

Find all odd positive integers $n$ greater than $1$ such that for any coprime divisors $a$ and $b$ of $n$, the number $a + b − 1$ is also a divisor of $n$. This was taken from the Russian ...
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3answers
47 views

How to apply Chinese Remainder Theorem for $x$

If: $$x \equiv 0 \pmod{17}$$ and $$x \equiv -1 \pmod{9}$$ Then how is: $$x \equiv 17 \pmod{153}$$ I get that since $\gcd(9, 17) = 153 $ the solution will be $\pmod{153}$ but how do you get the $17 ...
1
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1answer
91 views

A math contest question related to Ramsey numbers

In a group of 17 nations, any two nations are either mutual friends, mutual enemies, or neutral to each other. Show that there is a subgroup of 3 or more nations such that any two nations in the ...
0
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2answers
36 views

Strategy to find the most money to use.

As a reward for a week of good behavior, Tommy was given 7 dollars to spend at the canteen. By the time Tommy got to the canteen, there were only chocolate bars, meat pies and pizza pieces left. The ...
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1answer
20 views

Invariance Principle Question

A circle is divided into six sectors. Then the numbers $1, 0, 1, 0, 0, 0$ are written into the sectors (counter-clockwise say). You may increase two neighboring numbers by $1$. Is it possible to ...
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2answers
69 views

If $\prod\limits_{k=0}^5(5^{2^k}+6^{2^k})=6^x-5^y$, what is the value of $x-y$?

I think this might be a contest math question, so I'm tagging it as such. I don't know how to do something like this by hand (or if it's even possible, though I would presume it is if it's from a ...
1
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1answer
36 views

Complex Number - root

The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < ...
10
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1answer
290 views

How prove this systems-equation has least two postive integers solution

Show that: for any $k\ge 100,(k\in N^{+})$, there exsit $p\in N^{+}$, such $$\begin{cases} a+b+c=k\\ abc=p\\ a>b>c \end{cases}$$ has at least two postive integers solution $(a,b,c)$ ...
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0answers
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BMO1 2007/08 Question 5 Geometry Problem

$5.$ Let $P$ be an internal point of triangle $ABC$. The line through $P$ parallel to $AB$ meets $BC$ at $L$, the line through $P$ parallel to $BC$ meets $CA$ at $M$, and the line through $P$ parallel ...
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2answers
57 views

Given $n$ points, the difference of $2$ of them is $1/n$ close to an integer

From today's ENS Ulm Math D exam Let $x_1,\ldots,x_n$ be real numbers Prove there exists $i\neq j $ and $h\in \mathbb Z$ such that $|x_i-x_j-h|\leq \frac{1}{n}$ I tried contradiction and ...
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1answer
35 views

Real Numbers are Roots $r, s$.

Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$. Using Vieta's Formulas, $r+s+x_1$ $=0$ ...
0
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1answer
101 views

Solve the system $ x \lfloor y \rfloor = 7 $ and $ y \lfloor x \rfloor = 8 $.

Solve the following system for $ x,y \in \mathbb{R} $: \begin{align} x \lfloor y \rfloor & = 7, \\ y \lfloor x \rfloor & = 8. \end{align} It could be reducing to one variable, but it is ...
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2answers
66 views

Analog clock with same hands - sometimes one can't tell time [duplicate]

There is an accurate analog clock, however both hands are the same size and shape. How many moments during a day a person can not conclude current time from the position of the hands? This is from a ...
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2answers
87 views

Inequality with condition $x+y+z=xy+yz+zx$

I'm trying to prove the following inequality: For $x,y,z\in\mathbb{R}$ with $x+y+z=xy+yz+zx$, prove that $$ \frac{x}{x^2+1}+\frac{y}{y^2+1}+\frac{z}{z^2+1}\ge-\frac{1}{2} $$ My approach: After ...
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0answers
18 views

What is the isotomic conjugate version of the six point circle of isogonal conjugates?

As it is well known, the pedal triangles of a pair of isogonal conjugates in a triangle share a circumcircle. This is a nice theorem, but is there an analogous version of it for a pair of isotomic ...
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3answers
155 views

Sum of digits of $11\dots 11^2$ where $11\dots 11$ is a 1992 digit number with all digits $1$ [duplicate]

I read this on a non-math forum where the OP says this is a question for Grade 6 elementary school students. Grade 6 elementary school level is somehow ambiguous but clearly this means no advanced ...
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3answers
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AIME I 2015 #14:Area under a function

(This isn't the exact wording of the problem on the AIME) Find the number of $n,2\le n \le 1000$ such that $$\int_1^n x \lfloor \sqrt x \rfloor dx\in \Bbb Z$$ During the test, I noticed that for ...
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1answer
22 views

End digit of numbers raised to a certain power

In a math competition I came across the following question: What digit does the result of 2^2006 end with? This competition tested how fast you are at solving math problems. So, I was wondering ...
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1answer
22 views

if $f(n+1)-f(n)=P(n)$, exist a polynomial $Q(x)$ such that for all $n \in \mathbb{Z}$ : $Q(n)=f(n)$

Let $f:\mathbb{Z} \to \mathbb{Z}$ such that, exist a polynomial $P(x)$: $$f(n+1)-f(n)=P(n)$$ for all $n \in \mathbb{Z}$ Prove that exist a polynomial $Q(x)$ such that for all $n \in ...
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0answers
35 views

Combination problems

During numerous math contests I have come across questions such as: I have __ shirts, __ shoes and ___ pants... How many combinations of the __ are possible... As well as many other combination ...
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1answer
59 views

What are some good problem solving techniques for Math Olympiad style tests? [duplicate]

I am taking part in a Math Olympiad style test at my school in a few weeks. This test is mainly problem solving based and tests you on topics such as counting techniques, algebra, geometry as well as ...
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1answer
27 views

Find a number that is evenly divisible by all numbers between 1 and 20

I'm solving this for a programming challenge, in fact I already solved it but I'd like to know if there's some kind of rule that could improve such thing? For example if I needed the numbers ...
28
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1answer
428 views

How prove this geometry inequality $R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2$

Zhautykov Olympiad 2015 problem 6 This links discusses the olympiad problem which none of students could solve , meaning it is very hard. Question: The area of a convex pentagon $ABCDE$ is $S$, ...
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Snow White split 3 liters [duplicate]

Snow White split 3 liters of milk into the cup of the Seven Dwarfs. Before the meal, the Dwarfs play a game as follows: Dwarves are first divided all his cup of milk into the cup of the remaining six ...
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4answers
239 views

Writing numbers as a sum of 2s and 3s

Is there a way to count the number of ways a positive integer N, can be written as a sum of twos and threes? Are there any patterns? Re-arranging the twos and threes are distinct..(makes sense right?? ...
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0answers
28 views

What resources are necessary for IMC (International Mathematical Competition among Undergraduate Students)?

I am studying Azerbaijan as a undergraduate student. This year I am going to participate in the IMC, which is organised every year in Bulgaria. But unfortunately there is not a math department in my ...