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

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6
votes
0answers
64 views
2
votes
1answer
32 views

Number of People Required (Arena Survival Question)

Let's say that in an arena slaves are fighting 1v1. Only slaves with the identical number of wins can fight with each other. If a slave has 3 losses, the slave will be kicked out of the arena. If a ...
2
votes
1answer
29 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'...
11
votes
3answers
282 views

Solve $\lim_{x\to +\infty}\frac{x^x}{(\lfloor x \rfloor)^{\lfloor x \rfloor }}$

Determine if the following limits exist $$\lim_{x\to +\infty}\dfrac{x^x}{(\lfloor x \rfloor)^{\lfloor x \rfloor }}$$ note that $\lfloor x \rfloor \leq x < \lfloor x \rfloor + 1 \implies x-1 ...
36
votes
6answers
2k 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 ...
7
votes
1answer
97 views

Theoretical way to prove no positive integer $n$ exists such that $n+3$ and $n^2+3n+3$ are both perfect cubes.

I have to prove that for any positive integer $n$ at least one of $n+3$ and $n^2+3n+3$ is not a perfect cube. Is there a methodical way to solve this problem? I managed to solve it by contradiction, ...
13
votes
2answers
427 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 $...
5
votes
1answer
139 views

Prove that: $ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq j}b_{i}b_{j} \right)$

Let $a_{1}, \cdots, a_{n}, b_{1}, \cdots, b_{n}$ be positive real numbers. Prove that: $$ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq j}...
0
votes
1answer
58 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: ...
0
votes
1answer
30 views

Real polynomials from repunits to repunits ( Putnam 2007 A4) [closed]

Find all polynomials $ f$ with real coefficients such that if $ n$ is a repunit, then so is $ f(n).$ [Note this is a Putnam question, so it is intended to be of easy to middling difficulty as contest ...
4
votes
3answers
108 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 ...
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 $x\in \mathbb{N}$ ,...
9
votes
0answers
201 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...,...
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
176 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 ...
0
votes
1answer
64 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 ...
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 $AB$...
4
votes
0answers
375 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 ...
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 $f(x)=x$...
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. ...
-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 ...
15
votes
3answers
6k views

How to improve mathematics for Programming Contests?

You might close this question or downvote it, but I can't stop myself from asking the experts of mathematics who solve thousands of math problems. I'm a C++/C programmer who wants to improve his ...
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}^...
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 $\sum_{k=1}^{n} 2^n \text{ mod }k > 2n$. ...
1
vote
1answer
60 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 ...
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 ...
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 ...
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
116 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 $...
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.
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
1answer
152 views

An identity satisfying the divisors of a positive integer

I saw a hard competition problem with long and ugly proof in http://solmu.math.helsinki.fi/olympia/valmennus/2013/vt2013_12var.pdf ? The question is from Australian mathematical olympiad 1985. Is ...
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 $...
8
votes
0answers
136 views

Find the least possible value of $n$ such that there exist $P(x), Q(x) \in \mathbb{Z}[x]$

Find the least possible value of $n, n \geq 2015$ such that there exists polynomial $P(x)$ with degree $n$, integer coefficients, the coefficient of the term $x^n$ is positive and polynomial $Q(x)$ ...
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?
5
votes
1answer
87 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 ...
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
45 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 ...
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 ...
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 ...