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

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8
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1answer
37 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
votes
1answer
32 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
votes
1answer
51 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
votes
1answer
22 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 ...
1
vote
1answer
29 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 ...
0
votes
0answers
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?
-1
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2answers
30 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
votes
1answer
25 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 ...
4
votes
1answer
51 views

Rocks and squares, balls and sticks. [on hold]

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, ...
11
votes
0answers
73 views

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.
0
votes
2answers
50 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
vote
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
votes
1answer
96 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
votes
0answers
32 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
42 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
votes
2answers
137 views

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

Solve in integers the equation$$x^2 + xy + y^2 = \left({{x+y}\over3} + 1\right)^3.$$
0
votes
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
votes
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 ...
0
votes
3answers
42 views

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
votes
1answer
55 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 ...
1
vote
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
vote
1answer
89 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
votes
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 ...
2
votes
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 ...
4
votes
2answers
68 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
vote
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
votes
1answer
288 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 $(a,b,c)$ ...
0
votes
0answers
24 views

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 ...
8
votes
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 ...
1
vote
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
votes
1answer
100 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 ...
1
vote
2answers
65 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 ...
2
votes
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 ...
0
votes
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 ...
9
votes
3answers
154 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 ...
1
vote
3answers
74 views

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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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 ...
3
votes
1answer
57 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 ...
0
votes
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 ...
26
votes
1answer
389 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 $S$, ...
-1
votes
0answers
30 views

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 ...
3
votes
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?? ...
-1
votes
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 ...
0
votes
1answer
59 views

How many natural numbers less than $10^{2015}$ have their digits in non-decreasing order?

I am having pretty hard time with combinatorics. Could someone explain me step-by-step how to get to solution? Note: digits are observed from left to right.
0
votes
1answer
27 views

Convex optimization problem: linear equality and inequality constraints

When linear equality constraints can be converted in an inequality constraints for a strongly convex optimization problem? I mean, I got the same solution for both the following problem: 1) $\min_x ...
2
votes
1answer
48 views

BMO1 2007/08 Question 3 Geometry Problem [closed]

Let ABC be a triangle, with an obtuse angle at A. Let Q be a point (other than A, B or C ) on the circumcircle of the triangle, on the same side of chord BC as A, and let P be the other end of the ...
0
votes
2answers
35 views

why the all the coefficient terms of this integral share the least common factor 1/594

why the all the coefficient terms of this integral share the least common factor 1/594? Refer to this: $\int 1/(x^{23}+x^{50}) dx$ There are a lot of weird terms in the answer but they all share the ...
0
votes
1answer
57 views

Sum of Number of non-decreasing sequences [duplicate]

I know that the number of non-decreasing sequences of length $n$ and numbers in the sequence lying in the range $[l,r]$ is given by $$\binom{n+r-l}{n}$$ What is the formula to find the ...