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

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3
votes
1answer
58 views

$\alpha$ exists so that for any points $x_n$ there is a point at average distance $\alpha$ from the $x_n$.

Let $X$ be a connected and compact metric space. Prove a real number $\alpha$ exists so that for every finite set of points $x_1,x_2,\dots, x_n\in X$ (not necessarily distinct) there exists $x\in X$ ...
5
votes
1answer
71 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 ...
2
votes
2answers
89 views

$x^3 +y^2 +z =100z+10y+x$ What is the largest and smallest integer that satisfies this equation.

$x^3 + y^2 +z=zyx$,where $zyx$ denotes the sequence of the digits. $x^3 +y^2 +z =100z+10y+x$,where $x,y,z>0$ The maximum value of x,y,z individually can only be 9. Maximum value: $= 9^3 + 9^2 + ...
-1
votes
2answers
34 views

PAMO G Qualification Exam Question

ABCD is rectangular court with AB = 50m and BC = 30m. Four girls stand at different positions in that court so that the distance between the two girls next to each other is maximised. What is this ...
4
votes
1answer
71 views

Find all odd positive integers $n$, which there exists odd positive integers $x_1,x_2,..,x_n$, such that $x_1^2+x_2^2+\cdots+x_n^2=n^4$

Find all odd positive integers $n$, which there exists odd positive integers $x_1,x_2,..,x_n$, such that $$x_1^2+x_2^2+\cdots+x_n^2=n^4$$ My work so far 1) $n=3$ $$x_1^2+x_2^2+x_3^2=81$$ no ...
0
votes
2answers
54 views

What is the largest of the five missing numbers?

This is Q28 from Australian Maths Competition 2014. A circle is surrounded by 6 other circles,in a hexagonal formation.The leftmost circle is 0,which the rightmost circle is 1000.Each of the five ...
3
votes
2answers
80 views

$5$ numbers add up to 3231.What is the $6$th number?

This is Q27 from Australian Maths 2013. $3$ different non-zero digits are used to form $6$ different $3$-digit numbers.The sum of $5$ of them is $3231$.What is the $6$ th number? What I tried: Let $...
3
votes
2answers
97 views

What is the $2012th$ number in this pattern?

This is question 30 from Australian Maths 2012 $(0,1,2,1,2,3,2,3,4,1,2,3,2,3,4,3,4,5,2,3,4,...)$ What is the $2012th $ number in this list? What I did: I broke up the first few numbers into ...
1
vote
1answer
33 views

Prove that $a^2 pq + b^2 qr + c^2 rp \leq $ given a,b and c are sides of triangle and p+q+r=0

The question is asking to prove that $a^2 pq + b^2 qr + c^2 rp \leq 0 $ given that $a,b$ and $c$ are the sides of a triangle and that $p+q+r=0$. I have tried AM GM as well as countless pages of ...
0
votes
0answers
28 views

What are the characteristics of numbers whose count of its prime divisors is itself prime? [closed]

How does one find the characteristics of numbers whose count of prime divisors is equal to some prime number? Let c denote the number of prime numbers dividing s. Number c should be a prime number. I ...
3
votes
1answer
78 views

An Olympiad question on arithmetic progressions.

I stuck in the following problem that was one of math Olympiad questions. Can anybody give me some hints please? Suppose that $s_1,s_2,s_3,\ldots$ is a strictly increasing sequence of positive ...
2
votes
4answers
80 views

$pqrs \cdot 4 =srqp $,then what is the value of $qrs$?

This is question 26 from Australian Maths Competition 2013. $pqrs $ is a 4-digit number and has the property that $pqrs \cdot 4 = srqp$.If p=2,what's the value if the 3-digit number qrs? Here's what ...
14
votes
1answer
413 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 $$x_{1}+x_{2}+\cdots+x_{n}=\dfrac{...
0
votes
0answers
16 views

Special integer system values couting

Well I saw this question in a competition: A city uses a special system to represent integers. In the system, there are 5 different numerals $A, B, C, D, E$, corresponding to the values $1, 6, 36, ...
-2
votes
2answers
43 views
0
votes
0answers
49 views

Is there a trick to Australian Maths Competition Questions? [closed]

I have been going to Australian Maths Competitions for 2 years now,3rd year if I count this year.Every thing I do the contest,I get stuck at the last few questions,where the majority of the marks are,...
1
vote
1answer
75 views

longest way to rearrange students before returning to original arrangement? [closed]

This is Q24 from the 2012 Intermediate Australian Mathematics Competition: "A teacher has a class of twelve students. She thinks it would be a nice idea if they change desks every day, so she has ...
4
votes
1answer
76 views

Sally and I EACH flip a fair coin.

We then each guess what the other person got: I guess what side Sally's coin landed on, and Sally guesses what side my coin landed on. We win as long as at least one of us is correct. I understand ...
0
votes
2answers
73 views

AMC 2012(Senior) Q28

A quadrilateral with sides $15,15,15$ and $20$ is drawn with each vertex on a circle.Around this circle,a square is drawn,with each side tangent to the circle.What is the area of this square? I know ...
0
votes
1answer
83 views

AMC 2012 Junior Question [closed]

$x^2 +y^2 +z^2 = 100x+10y+z $. Find the smallest number and largest number that fit the equation.The numbers are below 1000 I am just baffled at the question.Is there a way to tackle such questions?
2
votes
3answers
85 views

Q27 from AMC 2012(Senior)

Five consecutive integers $p,q,r,s,t$,each less than $10000$, produce a sum which is a perfect square,while the sum of $q,r,s$ is a perfect cube.What is the value of $ \sqrt{p+q+r+s+t}$ ? What I have ...
2
votes
5answers
313 views

$33^{33}$ is the sum of $33$ consecutive odd numbers. Which one is the largest? (Q25 from AMC 2012)

The number $33^{33}$ can be expressed as the sum of $33$ consecutive odd numbers. The largest of these odd numbers is $\mathrm{A.}\ 33^{32} +32$ $\mathrm{B.}\ 33^{31} +32$ $\mathrm{C.}\...
1
vote
0answers
114 views

Strategies for solving rational Diophantine equations

Are there any strategies for solving Diophantine equations where the solutions can be any rational number, not just an integer, besides substituting $x=p/q$ and $y=r/s$, with $p,q,r,s$ integers with $\...
10
votes
2answers
338 views

Prove that $\gcd(3^n-2,2^n-3)=\gcd(5,2^n-3)$

Prove that $\gcd(3^n-2,2^n-3)=1$ if and only if $\gcd(5,2^n-3)=1$ where $n$ is a natural number. I didn't see an easy way to prove this using the Euclidean algorithm, but it seems true that both gcd'...
33
votes
4answers
1k views

Olympiad Inequality $\sum_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$

$x,y,z >0$, prove $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$ Note: Often Stack Exchange asked to show some work before answering the ...
30
votes
8answers
37k views

Expected Number of Coin Tosses to Get Five Consecutive Heads

A fair coin is tossed repeatedly until 5 consecutive heads occurs. What is the expected number of coin tosses?
11
votes
2answers
410 views

IMC 2015 - Problem 10 - Inequality between polynomials and exponential

This is problem 10 from the International Mathematical Competition for University Students of 2015, from day 2, in Bulgaria. I think it is an interesting problem! Let $n$ be a positive integer, and $...
6
votes
2answers
212 views

Find the $k$ such that $2^{(k-1)n+1}$ does not divide $\frac{(kn)!}{n!}$.

Find all positive integers $k$ such that for any positive integer $n$, $2^{(k-1)n+1}$ does not divide $\frac{(kn)!}{n!}$. From olympiad problem I'm curious So far no one to solve this problem,Maybe ...
4
votes
1answer
122 views

IMO 1988 question No. 6 Possible values of $a$ and $b$, $\displaystyle\frac{a^2+b^2}{ab+1}$

I have a confusion in the question.The question is as follows: $a$ and $b$ are positive integers and $ab+1$ is a factor of $a^2+b^2$. Prove that $\displaystyle\frac{a^2+b^2}{ab+1}$ is a perfect ...
1
vote
4answers
104 views
0
votes
1answer
70 views

Permuting the roots of a cubic polynomial with a quadratic polynomial cyclicaly

The polynomial $Q(x)=x^3-21x+35$ has three distinct real roots $r,s,t$. Find reals $a,b$ so that $P(x)=x^2+ax+b$ satisfies $P(r)=s,P(s)=t,P(t)=r$ or $P(r)=t,P(t)=s,P(s)=r$. I tried using cardano to ...
0
votes
1answer
69 views

Find the Smallest Value

Find the smallest value of $$a + \frac {1}{(a-b)b} $$ where a>b>0 I found this question in AM-GM inequality problems but I am stuck at this
1
vote
1answer
59 views

PRIMES 2016 entrance problem

PROBLEM G4 In a couples therapy session, n couples are to be seated at a round table (in 2n chairs), but no person is allowed to sit next to his/her spouse. How many seat assignments are there? ...
2
votes
3answers
68 views

How do you evaluate this sum of multiplied binomial coefficients: $\sum_{r=2}^9 \binom{r}{2} \binom{12-r}{3} $?

We have to find the value of x+y in: $$\sum_{r=2}^9 \binom{r}{2} \binom{12-r}{3} = \binom{x}{y} $$ My approach: I figured that the required summation is nothing but the coefficient of $x^3$ is the ...
2
votes
2answers
37 views

What are the valid deductions of a congruence equation?

So I was just sitting here, doing math, and I came over this: $9+16a\equiv 12 \pmod 5$ Obviously, through some simple manipulations: $9+16a-15a-9\equiv 12-9 \pmod 5$ $a\equiv 3 $ And that is a ...
1
vote
3answers
97 views

Prove $\forall n \in \mathbb{N}: \int_{0}^{\frac{\pi}{2}} |\frac{\sin(nx)}{x}|dx \geq \frac{2}{\pi}\sum_{k=1}^{n}\frac{1}{k}$

As my further preparation to Putnam competition, I came across such inequality to prove: $$\forall n \in \mathbb{N}: \int_{0}^{\pi} \left|\frac{\sin(nx)}{x}\right|dx \geq \frac{2}{\pi}\sum_{k=1}^{n}\...
0
votes
1answer
23 views

combinatorics: give an upper bound for the cardinality of a set of 100-ary sequences

Let $S$ be a $1990$-element set and let $P$ be a set of $100$-ary sequences $(a_1, a_2, ..., a_{100})$, where $a_i$'s are distinct elements of $S$. An ordered pair $(x,y)$ of elements of $S$ is said ...
1
vote
1answer
57 views

Evaluate $\cos \frac{\pi}{7} \cos \frac{2\pi}{7}\cos \frac{4\pi}{7}$

Evaluate $$\cos \frac{\pi}{7} \cos \frac{2\pi}{7}\cos \frac{4\pi}{7}.$$ The first thing i noticed was that $$\cos \frac{\pi}{7}=\frac{\zeta_{14}+\zeta_{14}^{-1}}{2},$$ where $\zeta_{14}=e^{2\pi i/14}$...
11
votes
1answer
172 views

Prove the inequality $\deg{P(x)}\cdot \deg{Q(x)}\cdot \deg{R(x)}\ge 656$

Let three non-constant polynomials $P(x),Q(x),R(x)\in \mathbb Z[x]$, and suppose that the equation $P(x)Q(x)R(x)=2015$ has $49$ distinct integer roots. Prove that $$\deg{P(x)}\cdot \deg{Q(x)}\cdot ...
1
vote
1answer
94 views

For which $a,b\in \mathbb{N},$ is $\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$ is a rational number.

I found the following problem on a Olympiad question paper: For which $a,b\in \mathbb{N},$ is $$\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$$ a rational number. I am unable to solve it. Any help ...
1
vote
2answers
54 views

Proving $\cos A \cdot \cos 2 A \cdot \cos 4 A \cdots \cos 2^{n-1} A = \frac{\sin 2^n A}{2^n \sin A}$

Just a bit of background on the question: When proving: $$\cos\frac{\pi}{15}\cdot \cos\frac{2\pi}{15} \cdot \cos\frac{3\pi}{15}\cdot \cos\frac{4\pi}{15} \cdot \cos\frac{5\pi}{15} \cdot \cos\frac{6\pi}...
1
vote
3answers
92 views

find the the greatest value of $m$ such that $lcm(1,2,3,..,n)=lcm(m,m+1,..,n).$

I am stuck and unable to proceed. Value of n can be very large. For eg:if $n=6,lcm(1,2,...,6)=60$, so answer will be $4$ in this case. Since $lcm(2,3,4,5,6)=60,lcm(3,4,5,6)=60,lcm(4,5,6)=60$ and $...
0
votes
2answers
108 views

What is the size of the angle $\angle AMC$? [duplicate]

Suppose we have a triangle $\triangle ABC$ where the size of two angles are given: $\angle B=15^\circ$ and $\angle C=30^\circ$. We draw the median $AM$, so now what is the size of angle $\angle AMC$? ...
0
votes
2answers
114 views

Does there exist such a number?

Does there exist a $2n$-digit number $\overline{a_{2n}a_{2n-1}\ldots a_1}$ (for an arbitrary $n$) for which the following equality holds: $$\overline{a_{2n}\ldots a_1}= (\overline{a_n \ldots a_1})^2?...
3
votes
3answers
81 views

Evaluation of $\iint_D \frac {\ln(2 - \sin \xi \cos \eta)\sin \xi} {2 - 2\sin \xi \cos \eta + \sin^2 \xi \cos^2 \eta} \mathrm d\xi \; \mathrm d\eta$

Evaluate the following integral: $$\iint_D \frac {\ln(2 - \sin \xi \cos \eta)\sin \xi} {2 - 2\sin \xi \cos \eta + \sin^2 \xi \cos^2 \eta} \mathrm d\xi \; \mathrm d\eta$$ where $D = [ 0, \pi/2] \...
4
votes
3answers
357 views

The square of a number's last few digits remain the same.

The number $9376$ has a property that the last four digits of $9376^2$ remain the same. How many $4$ digit numbers have this property? Are there values of $n>4$ such that a $n$-digit number has $n$...
21
votes
1answer
343 views

How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$?

How can I prove the following identity? $$\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$$
5
votes
1answer
91 views

USA $2011$ contest inequality problem, proving $\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\ge 3$, under given condition.

If $a^2+b^2+c^2+(a+b+c)^2\le4$, then $$\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\ge 3.$$ My attempt: From the given criteria, one can easily obtain that $$(a+b)^2+(b+c)^2+(c+a)^...
1
vote
0answers
69 views

Polynomial taking irrationals to irrationals

Problem: Find all polynomials from $\mathbb{R}\to \mathbb{R}$ $f$ with integer coefficients taking irrationals to irrationals. My attempt: It is clear that the problem statement is equivalent to ...
5
votes
2answers
88 views

Functional Equation of iterations

Problem: Let $f : \mathbb{Q} \to \mathbb{Q}$ satisfy $$f(f(f(x)))+2f(f(x))+f(x)=4x$$ and $$f^{2009}(x)=x$$ ($f$ iterated $2009$ times). Prove that $f(x)=x$. This is a contest type problem ...