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

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1answer
17 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 ...
3
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2answers
87 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 ...
3
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1answer
73 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
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4answers
70 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 ...
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0answers
13 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, ...
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2answers
39 views

What can be a good programming algorithm to solve the given equation other than the brute force? [on hold]

Find all $x$, $y$ and $z$ for $n=100$; $$x^2 + y^2 + z^2 = n$$ $x,\ y,$ and $z$ should be positive integers.
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0answers
45 views

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

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,...
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0answers
27 views

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

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 ...
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1answer
68 views

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

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 ...
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2answers
70 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 ...
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1answer
81 views

AMC 2012 Junior Question [on hold]

$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?
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3answers
84 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
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5answers
296 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.}\...
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4answers
97 views

Help in proving an inequality

Show that $$a^4 + b^4\ge\frac{1}{8}$$ if $a+b=1.$
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1answer
67 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
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1answer
65 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
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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? ...
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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}\...
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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
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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 $\zeta_{14}=e^{2\pi i/14}$...
10
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2answers
328 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'...
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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
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3answers
91 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
109 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?...
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 ...
3
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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] \...
1
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0answers
68 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 ...
0
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2answers
107 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 $\angle AMC$? ...
5
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1answer
89 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)^...
4
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1answer
113 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 ...
5
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2answers
87 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 ...
3
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1answer
45 views

Prove existence of a triangle with least angle $\leq 30$ degrees

Problem: Let $A$ be a set of $6$ points in a plane such that no $3$ are collinear. Show that there exist 3 points in $A$ which form a triangle having an interior angle not $30$ degrees. I am supposed ...
3
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0answers
98 views

Using Group Theory to Solve this IMO problem

A few weeks ago, I found a fascinating solution to a USAMO combinatorics problem that used group theory. Look at the 2nd solution on this link to view it. I think there might be a way to use group ...
5
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1answer
60 views

Find all whole number solutions of the following equation

While training for a math olympiad(university level) I stumbled upon the following problem. Find all $n, k \in \mathbb{N}$ such that $${ n \choose 0 } + {n \choose 1}+{n \choose 2} + {n \choose 3} = ...
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1answer
71 views

Proving existence of a triangle of area $\leq \frac{1}{8}$

Problem: Given any set $S$ of $9$ points within a unit square, show that there always exist $3$ distinct points in $S$ such that the area of the triangle formed by these $3$ points is less than or ...
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0answers
35 views

Sticky boots and modular arithmetic: Find the formula!

Suppose a trek begins and on this trek the road is paved by squares with labels on them. The warning sign next to the beginning of the first square, labeled $1$, states: Beware that due to natural ...
2
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1answer
87 views

If $a+b+c=0$ what is the value of $\frac{a^2}{2a^2 +bc }+\frac{b^2}{2b^2 +ca }+\frac{c^2}{2c^2 +ab }$

Let $s=\frac{a^2}{2a^2 +bc }+\frac{b^2}{2b^2 +ca }+\frac{c^2}{2c^2 +ab }$. If we use inequality $\frac{x^2}{a}+\frac{y^2}{b} \ge \frac{(x+y)^2}{(a+b)}$ we get $s \ge 0$ as $a+b+c=0$. Again $s \le \...
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2answers
29 views

UIL Math Contest Problem #60 Regional

Given the circle O with perpendicular diameters and a chord, find the area of the circle if EF = 8" and DE = 20" (DF = 12"). (integer) DE is a chord that intersects one of the diameters and shares a ...
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1answer
52 views

Probability of rolling product of 200

Alec rolls a fair $6$-sided die repeatedly until the product of the rolls is at least $100$, at which point he stops rolling. Compute the probability that he stops rolling with a product of $200$. ...
1
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1answer
52 views

Sum of 3 smallest divisors add to 17

How many $3$-digit integers $N$ have the property that the sum of the $3$ smallest divisors of $N$ is equal to $17$? My approach: We know that $1$ is the smallest divisor of any number. We then need ...
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0answers
45 views

Prove an inequation about x,y,z [closed]

Suppose x,y,z are positive real number,prove $$\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}>2\sqrt[3]{x^3+y^3+z^3}$$
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3answers
171 views

Find the missing digits in the expansion of $34!$ [closed]

If $34!=295232799cd96041408476186096435ab000000$ then find the value of $a,b,c$ and $d.$ My Attempt: I can find that $b=0$ because it has seven five integers. Note: calculator is not allowed.
2
votes
1answer
80 views

Prove that Triangle ABC is an equilateral triangle iff $\tan{A}+\tan{B}+\tan{C} = 3^\frac32$.

This question is picked from AM GM HM inequalities, so this is to be proved form that concept only, I think it isn't possible because there is no inequality, but if it is please tell me how.
12
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1answer
232 views

Doing a magic trick with limited memory (from a problem solving course)

I got the following question in a problem solving course: There are four different objects lying on places 1, 2, 3, 4. A magician closes his eyes and someone from the audience comes. He switches ...
2
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0answers
29 views

A congruence of sum of kth powers of first p-1 numbers [duplicate]

Problem: For $k < p-1$ where $p$ is an odd prime and $k$ is a natural number, prove that $$1^k+2^k+\cdots+(p-1)^k \equiv 0 \mod p.$$ My attempt: It's obvious for odd $k$, as we can pair the ...
11
votes
2answers
149 views

Maximum value of the smallest number of operations to obtain configuration from original configuration

Let $n$ be a positive integer. There are $n(n+1)/2$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each ...
2
votes
2answers
49 views

To show that the variables in the system are same in magnitude

I am stuck with this interesting problem, If for non-negative integers $a, b, \text{and} c$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are both integers then ...
1
vote
1answer
35 views

Prove Two Functions are Simultaneously Continuous

Let $f,g,h: \mathbb{R} \rightarrow \mathbb{R}$ so that $f$ is differentiable, $g,h$ monotone and $f'=f+g+h$. Prove that $g$ is continuous in $x_0$ iff $h$ continuous in $x_0$. My attempt ...