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

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2
votes
0answers
22 views

a particular linear combination

Fix $a_1,\ldots,a_n\in\mathbb{N}$. I'd like to know if one can characterize the natural numbers that belong to the set $$\{b_1a_1+\ldots+b_na_n:\,b_j\in\{-1,0,1\}\}.$$ EDIT: Maybe this question doesn'...
0
votes
1answer
55 views

Olympiad Books for Primary Students

I am a teacher of gifted program in primary school and currently I am developing Olympiad Curriculum (topic-wise) for my students. I have those topics that could need some help in terms of questions: ...
4
votes
3answers
106 views

Books for maths olympiad

I want to prepare for the maths olympiad and I was wondering if you can recommend me some books about combinatorics, number theory and geometry at a beginner and intermediate level. I would appreciate ...
0
votes
0answers
22 views

What is the steps for finding a formula [on hold]

I Got a problem then tried to solve it it was mathematical but i failed. Problem statement is given below: Several ages ago Berland was a kingdom. The King of Berland adored math. That's why, when he ...
6
votes
1answer
90 views

If $x+y=10^{200}$ then prove that 50 divides $x$

Let $x$ be a positive integer and $y$ is another integer obtained after rearranging the digits of $x$. If $x+y=10^{200}$ then prove that $x$ is divisible by 50. My attempt Since $y$ is the digit ...
11
votes
3answers
171 views

Decompose $5^{1985}-1$ into factors

Decompose the number $5^{1985}-1$ into a product of three integers, each of which is larger than $5^{100}$. We first notice the factorization $x^5-1 = (x-1)(x^4+x^3+x^2+x+1)$. Now to factorize $x^4+...
4
votes
1answer
75 views

Is there an 'interesting' way to derive this expression?

So I was asked to prove the following term is equal to $2016$: $$ \left( \frac{251}{ \frac{1}{ \sqrt [3] {252} - 5 \sqrt [3] {2} } -10 \sqrt [3] {63} } + \frac {1} { \frac {251} { \sqrt [3] {252} +5 ...
1
vote
1answer
32 views

Proving circumcenter lies on altitude

Problem: In $\triangle ABC$, let $D$ be the intersection of the tangents to the circumcircle at $B$ and $C$, let $B'$ be the reflection of $B$ across $AC$, let $C'$ be the reflection of $C$ across $AB$...
0
votes
1answer
62 views

Discriminant of Cubics and Math Olympiad

Let $a,b,c$ be distinct nonzero real numbers. If the equations $E_1: ax^3+bx+c=0, E_2: bx^3+cx+a=0$ and $E_3: cx^3+ax+b=0$ have a common root, prove that at least one of these equations has three real ...
10
votes
1answer
87 views

finite polynomials satisfy $|f(x)|\le 2^x$

This is a problem from TsingHua University math competition for high school students. Prove there exists only finite number of polynomials $f\in \mathbb{Z}[x]$ such that for any $x\in \mathbb{N}$ ,...
1
vote
1answer
61 views

Maths Puzzle!!! [duplicate]

I am planning on taking an interview in the near future and was practicing on some previously asked aptitude questions. During my prep I came across a problem for which I couldn't find an answer. ...
2
votes
0answers
56 views

olympiad-type inequality

Prove that for any $x_1,\dots,x_n>0$ $$ {\root{n}\of{\prod _{k=1}^{n}\ \sum_{t=1}^{k}\ \frac{1}{t^2\cdot\sqrt[t]{x_1\cdot\ldots\cdot x_t}} }} \ \cdot\ \sum _{k=1}^{n}\frac{\sum_{j=1}^{k}\sum_{i=1}^...
9
votes
0answers
197 views

Prove that $\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$ if $(a+b+c)^2(a^2+b^2+c^2)=27$

Let $a$, $b$ and $c$ be non-negative numbers such that $(a+b+c)^2(a^2+b^2+c^2)=27$. Prove that: $$\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$$ A big problem here around $(a,b,c)=(1.6185...,...
1
vote
1answer
48 views

Prove $\sum_{k=1}^{n} 2^n \text{ mod }k > 2n$ where $n > 1000$

This problem is taken from a Russian textbook of past Olympiads. Its statement looks like this : Given a natural number $n > 1000$ prove that $\sum_{k=1}^{n} 2^n \text{ mod }k > 2n$. ...
4
votes
0answers
369 views

IMO 2016 P3, number theory with the area of a polygon

Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \cdots A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is ...
0
votes
1answer
46 views

Integer solution to the equation below

I wanted to know integer solutions to the equation (1/k1) + (10/k2) + (100/k3) + ..... + (10^18/k19) = 1 (where k1,k2,k3.... are integers) which I believe is ...
1
vote
1answer
59 views

Economics : Game-theory (Nash equilibrium)

This is a homework question, but resources online are exceedingly complicated, so I was hoping there was a fast, efficient way to solving following question. Question: Six students are going on a ...
4
votes
1answer
65 views

Sharing and odd pizza

Here is a classical problem, which every mathematician will have seen at least onece in their life: Anne and Ben are sharing a pizza. The pizza is divided into an even number of pieces of unequal ...
4
votes
2answers
115 views

The problem of congruent areas in a triangle.

A problem was posed in front of me and I couldn't solve it after multiple attempts-- Consider any triangle and 3 concurent cevians are drawn from each of its 3 points . Now the figure formed has 6 ...
2
votes
1answer
39 views

$a$ and $b$ are factors of $6^6$ and $a$ is a factor of $b$

How many pairs of ($a$,$b$) of positive integers are there such that $a$ and $b$ are factors of $6^6$ and $a$ is a factor of $b$? What I tried I know $6^6$ an be broken down into $(2)^6 (3)^6$ If $...
6
votes
3answers
640 views

Finding the roots (contest math)

So the problem is : $x^4-4x^3-x^2-8x+4=0$, find all solutions A tip that I have gotten, is to divide both sides by $x^2$. I've tried so, but I do not manage to see any further. Do anyone know how ...
-3
votes
1answer
68 views

Who becomes king? [closed]

5 earls argue which becomes king and which becomes treasurer. A will be happy only if D or E is treasurer. B will be happy only if C is treasurer. C will be happy only if D is either king or ...
0
votes
3answers
53 views

The fraction of the larger hexagon that is shaded?

This is from Australian Maths 2013. In a regular hexagon,the midpoints of the sides are joined to form he shaded regular hexagon.What fraction of the larger hexagon is shaded? Since the larger ...
9
votes
1answer
135 views

Is $n^7 - 77$ ever a Fibonacci number?

As the question title suggests, is $n^7 - 77$ ever a Fibonacci number, where $n$ is a integer?
0
votes
2answers
57 views

Inductive reasoning question

Can someone help with this inductive reasoning question. What should come next in this series of 5 and what is the reasoning? Can someone also help with the following question.
10
votes
2answers
261 views

Determine all functions $f$ on $\mathbb R$ such that $f(x^2+yf(x))=f(x)f(x+y)$ for all $x,y$

Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that $$f(x^2+yf(x))=f(x)f(x+y). $$ for all $x,y$ real numbers. I think that the only three solutions are: $f(x)=0$, $f(x)=1$ and $f(x)=x$...
2
votes
0answers
49 views

About periodicity of $f(\frac{m}{n})=\frac{3m-1}{2n+1}$ when $\frac{m}{n}$ is reduced form.

Consider a function $f\colon\mathbb{Q}_{>0}\longrightarrow\mathbb{Q}_{>0}$ such that $f(x)=\frac{3m-1}{2n+1}$ where $x=\frac{m}{n}$ and $\frac{m}{n}$ is reduced form. (i.e., $\gcd(n,m)=1$ and $...
4
votes
0answers
56 views

Independence of radicals: First-principles proof of special case

I've known this problem for a long time: Problem. Show that the number $\alpha=\sqrt{1} + \sqrt{2} + \ldots + \sqrt{n}$ is irrational for $n\geq 2$. but I haven't been able to find a solution from ...
5
votes
2answers
90 views

Find a polynomial with integer coefficients which has a global minimum equal to (a)$- \sqrt{2}$, (b)$\sqrt{2}$

Find a polynomial with integer coefficients which has a global minimum equal to (a)$- \sqrt{2}$, (b)$\sqrt{2}$. It it a high-school math contest problem. The answer is given: $$(a) ~~~~~~~P(x)=N(2x^...
4
votes
1answer
61 views

A number which can be factored into a product of $k$ and $k+2$ consecutive natural numbers (each $>1$)

We say that the number $N \in \mathbb{N}$ has the property $P(k)$ if it can be factored into a product of $k$ consecutive natural numbers (not equal to $1$). Find the value of $k$ such that some $...
9
votes
2answers
100 views

Favourite problem books at university level

As background let me start by stating what I perceive to be the point of problem books, or to put the matter in perhaps more acceptable way, how I define problem books. A large majority of textbooks ...
2
votes
2answers
49 views

A little bit confused about the solution for 2016 AIME II #7

Squares $ABCD$ and $EFGH$ have a common center at $\overline{AB} || \overline{EF}$. The area of $ABCD$ is 2016, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so ...
2
votes
2answers
68 views

Chessboard Kings and no check [closed]

What is the largest number of kings which can be placed on a chessboard so that no two of them put each other in check?
3
votes
2answers
107 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
43 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 ...
0
votes
2answers
57 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 ...
5
votes
1answer
77 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 ...
3
votes
2answers
91 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 $...
1
vote
1answer
37 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
votes
2answers
101 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
votes
1answer
85 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
82 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 ...
0
votes
0answers
20 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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votes
2answers
44 views

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

Find all $x$, $y$ and $z$ for $n=100$; $$x^2 + y^2 + z^2 = n$$ $x,\ y,$ and $z$ should be positive integers.
1
vote
1answer
82 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 ...
0
votes
2answers
80 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
90 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
89 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
327 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
4answers
115 views

Help in proving an inequality

Show that $$a^4 + b^4\ge\frac{1}{8}$$ if $a+b=1.$