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

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5
votes
2answers
127 views

given $2f(x) + f(1-x) = x^2$ find $f(-5)$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. A function $f$ has property that $2f(x)+ f(1-x) = x^2$ ...
1
vote
1answer
71 views

(Putnam) Let $f:[1,3] \rightarrow \mathbb{R}$ such that $-1 \leq f(x) \leq 1 $ for all x and

The following is a Putnam math competition problem: Let $f:[1,3] \rightarrow \mathbb{R}$ such that $-1 \leq f(x) \leq 1 $ for all x and $ \int_{1}^{3}f(x)dx = 0 $. What is the max value of ...
3
votes
0answers
27 views

Show that ordered pairs are solutions to an equation if and only if they are consecutive elements of a recursive sequence (contest question)

The following question appeared on the 1998 Canada National Olympiad. I need help proving that the only solutions to the equation are consecutive elements of the recursively-defined sequence. I ...
0
votes
1answer
36 views

Number of ways to invite people to dinner

So I have this maths contest problem which goes like this: Alfred has seven friends (we'll call them A,B,C,D,E,F and G). Each night for a week/7 days he can invite any group of three friends over to ...
3
votes
0answers
82 views

Minimize Value of Function with Constrain

Let $x$ and $y$ be real number with $xy\neq-1$ and $$\frac{x^7y+xy^7}{1+x^5y^5}=4$$ What is the minimum value of $x^2+y^2?$ I've been trying to solve it by Lagrange Multiplier but it's getting ...
1
vote
5answers
174 views

Find $x$ if $\frac {1} {x} + \frac {1} {y+z} = \frac {1} {2}$ [closed]

I found this question from past year's maths competition in my country. I've tried any possible way to find it, but it is just way too hard. Find $x$ if \begin{align}\frac {1} {x} + \frac {1} ...
2
votes
2answers
50 views

What is $k$ so that $\frac {1001\times 1002 \times … \times 2008} {11^k}$ will be an integer?

I found this question from last year's maths competition in my country. I've tried any possible way to find it, but it is just way too hard. What is the largest integer $k$ such that the following ...
4
votes
4answers
86 views

$\frac {1} {ab} + \frac {1} {ac} + \frac {1} {ad} + \frac {1} {bc} + \frac {1} {bd} + \frac {1} {cd}$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. given $$ \frac {1} {a} + \frac {1} {b} + \frac {1} {c} + ...
3
votes
3answers
92 views

$ x^2 + \frac {x^2}{(x-1)^2} = 2010 $

I found this question from last year's maths competition in my country. I've tried any possible way to find it, but it is just way too hard. Given $$ x^2 + \frac {x^2}{(x-1)^2} = 2010,$$ find ...
1
vote
3answers
69 views

Number theory with binary quadratic

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. Given $$ \frac {x^2-y^2+2y-1}{y^2-x^2+2x-1} = 2$$ find ...
3
votes
2answers
62 views

Maximum and minimum of a sum involving floor functions of rational numbers (contest question)

This question originates from the 1996 Canada National Olympiad. Let $r_1, r_2, \dots, r_m$ be a given set of $m$ positive rational numbers such that $\sum\limits^{m}_{k=1}{r_k} = 1 \tag{1}$ ...
2
votes
2answers
72 views

How to solve “ways of seating around a circular table”

Recently I asked a question about seating, here it is again: The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five ...
2
votes
0answers
79 views

A sequence of polynomials [duplicate]

I posted this question a while back, and I think I may not have made myself clear or shown what I got so far. So let me post this question again with more information and clarification. I have a ...
1
vote
2answers
78 views

How many ways to arrange the seating?

The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with ...
1
vote
0answers
76 views

A Summation Challenge

I am trying to understand the solution of problem from its editorial by djdolls' answer,I am not able to understand a particulare step which is as follows: $$S(n)=\sum_0^D (-1)^i \cdot ...
2
votes
1answer
27 views

How find all finite sets $ M$ such that $ |M|\ge 2$ and $ \frac {2a}{3} - b^2\in M$ for all $ a,b\in M$

How find all finite sets of real numbers $ M$ such that $ |M|\ge 2$ and $ \frac {2a}{3} - b^2\in M$ for all $ a,b\in M$?
0
votes
2answers
77 views

Olympiad inequality

I want to prove that for $a, b, c > 0$ we have $\frac{a}{2a + b} + \frac{b}{2b + c} + \frac{c}{2c + a} \leq 1$. My approach: I know that each of the individual terms is lesser than $\frac{1}{2}$ ...
5
votes
1answer
201 views

$\sum_{i=1}^n \frac{x_i}{\sqrt[n]{x_i^n+(n^n-1)\prod _{j=1}^nx_j}} \ge 1$, for all $x_i>0$

Can you help with the following inequality? I found it experimentally. Prove that, for all $x_1,x_2,\ldots,x_n>0$, $$\sum_{i=1}^n\frac{x_i}{\sqrt[n]{x_i^n+(n^n-1)\prod _{j=1}^nx_j}} \ge ...
4
votes
0answers
48 views

Find all integers $a,b,c$ [duplicate]

This question comes from the 2007 IMO shortlist: Find all integers $a,b,c$ such that $ab-c$, $bc-a$ and $ca-b$ are powers of two (of the form $2^k$ where $k \geq0$). What are some methods of ...
1
vote
0answers
66 views

Graph Theory number of handshakes of couples

This is an Olympiad question which I now know the answer to, but I am a bit unsatisfied with it. So maybe someone can shed some light: Question: $5$ couples go to a party. Each person shakes the ...
0
votes
1answer
35 views

Find all real solutions to the following system of equations (involving fixed point iteration)

From the 1996 Canada National Olympiad. I have emphasised the real point of the question. Find all real solutions to the following system of equations. Carefully justify your answer. ...
3
votes
1answer
56 views

Largest sum of compatible triples

A triple $(a,b,c)$ of distinct integers is called compatible if at least one of them, say $b$ has the property that either $n\mid b$ or $b\mid n,$ for each $n\in\{a,c\}.$ Let $X$ be the set of all ...
1
vote
2answers
64 views

How many such polynomial exist?

Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0)=2010$. I got: $$P(x) = ax^2 + bx + c \implies P(0) = c = 2010$$ Let $P(r_1, r_2) ...
1
vote
1answer
54 views

Probability of not making a shoe pair.

Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The ...
3
votes
1answer
66 views

Why doesnt this Combinatoric work two ways?

There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are identical blue flags, and $9$ are identical green flags. Let $N$ be the number of distinguishable arrangements ...
1
vote
0answers
38 views

What can be said about triangle with certain condition?

This question comes from 1988 Irish Mathematical Olympiad, for all those interested. A mathematical moron is given the values $b,c,\alpha$ for a triangle $ABC$ and is required to find $a$. He does ...
5
votes
1answer
105 views

Let $f : [0,1] \to \mathbb{R}$, prove that $2 \int_{0}^{1} f(x)dx \ge f\Big(\frac{1}{n}\Big) + \sum_{k=1}^{n-1}\frac{1}{k} f\Big(\frac{k}{n}\Big)$

Let $f : [0,1] \to \mathbb{R}$ be a differentiable function with a continuous derivative such that $f(x) \ge xf'(x), \forall x \in [0,1]$. Prove that: $$2 \int_{0}^{1} f(x)dx \ge ...
1
vote
3answers
59 views

How many possible guesses?

A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $ 1$ to $ 9999$ inclusive. The contestant wins the prizes by correctly guessing the ...
8
votes
3answers
112 views

Prove that $(\sqrt2 − 1)^n, \forall n \in \mathbb{Z^+}$ can be represented as $\sqrt{m} − \sqrt{m−1}$ for some $m \in \mathbb{Z^+}$ (no induction).

From the 1994 Canada National Olympiad: Prove that $(\sqrt2 − 1)^n, \forall n \in \mathbb{Z^+}$ can be represented as $\sqrt{m} − \sqrt{m−1}$ for some $m \in \mathbb{Z^+}$. I think one ...
0
votes
2answers
75 views

BMO1 2003/04 Question 2 - Geometry Prolem

$ABCD$ is a rectangle, $P$ is the midpoint of $AB$, and $Q$ is the point on $PD$ such that $CQ$ is perpendicular to $PD$. Prove that the triangle $BQC$ is isosceles. Clearly, we need to prove that ...
5
votes
1answer
81 views

How many ways are there to shake hands?

In a group of $9$ people, each person shakes hands with exactly $2$ of the other people from the group. Let $X$ be the number of possible ways to perform these handshakes. Take $2$ handshake ...
6
votes
1answer
63 views

Is it possible to choose $10$ distinct numbers from the set $\{0, 1, 2, . . . , 14\}$ so that various differences are all distinct?

From the 1991 Canada National Olympiad: Can ten distinct numbers $a_1, a_2, b_1, b_2, b_3, c_1, c_2, d_1, d_2, d_3$ be chosen from $\{0, 1, 2, \dotsc, 14\}$ so that the $14$ differences $$ ...
4
votes
1answer
163 views

Cover the grid graph with simple cycles

I have a two dimensional n x m grid graph. And I want to find in how many ways this grid can be covered with simple cycles (it can be a one cycle or it can be many ...
5
votes
1answer
50 views

Consider the 1000-element subsets

Consider all 1000-element subsets of the set $A = \{ 1, 2, 3, ... , 2015 \}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, ...
5
votes
1answer
67 views

Infinite number of ways to write $1=\frac{1}{n}+\frac{1}{a_1}+\cdots+\frac{1}{a_k}$

How can I show that there is an infinite number of ways in which $1$ can be written in the form $$1=\frac{1}{n}+\frac{1}{a_1}+\cdots+\frac{1}{a_k},$$ where $n>1$ is an integer (this number is ...
5
votes
1answer
223 views

IMO 2015 #1: “Balanced” and “Centre-Free” sets of points in the plane [closed]

International Mathematical Olympiad 2015, Problem 1: We say that a finite set $S$ of points in the plane is $\color{\red}{\text{balanced}}$ if, for any two different points $A$ and $B$ in $S$, ...
0
votes
0answers
19 views

Create a recursion here [duplicate]

Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain either exactly two adjacent chairs or no adjacent chairs. I had this question before, but I ...
1
vote
1answer
39 views

Unfairish Probability

Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that it comes up six with probability $\frac{2}{3}$ and each of the other five sides has probability ...
3
votes
1answer
64 views

Ten chairs arranged in a circle

Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain either exactly two adjacent chairs or no adjacent chairs. Let $1$ be chair, and $0$ be an empty ...
0
votes
0answers
72 views

Why doesn't combinatorics work here?

A while ago I asked one-to-one in combinatorics and then using one-to-one I'll repeat my answer here: There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are ...
-3
votes
1answer
85 views

what is the probabilty that sum of two random numbers between A and B is less than third number C [closed]

What is the probabilty that sum of two random numbers uniformly distributed in $[A,B]$ is less than a fixed $C$? I have tried answering this question using graph method to find the area under the ...
6
votes
2answers
89 views

$1,2,…,n(n+1)/2$ placed at random in bottom-heavy nxn triang. array. Prob. that largest num in every row is smaller than largest in any row below?

From the 1990 Canada National Olympiad: $\dfrac{n(n+1)}{2}$ distinct numbers are arranged at random into $n$ rows. The first row has $1$ number, the second has $2$ numbers, the third has ...
0
votes
1answer
73 views

Minimum moves to make all coins have Heads facing up

Given a circular list of coins with Tails facing up. In each move, if we flip coin at position $i$, coins at positions $i-1$ and $i+1$ get flipped as well. That is, consider: $H H H T T$ : if I flip ...
11
votes
2answers
110 views

Subgroups of $S_n$ with exactly one fixed point for each element all have the same fixed point.

Let $G$ be a subgroup of $S_n$ (where $n$ is a positive integer) such that each non identity element $g\in G$ has exactly one fixed point. Prove there is an element of $[n]$ that is fixed by every ...
1
vote
1answer
63 views

Solve $n(n+1) \equiv 0 \pmod{1004}$

Solve: $$n(n+1) \equiv 0 \pmod{1004}$$ For the smallest possible $n > 0$. It's either $n \equiv 0$ or $n \equiv -1 \pmod{1004}$. The correct answer is $251$, I'm not sure how though.
5
votes
3answers
112 views

How many ways to arrange the flags?

There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are identical blue flags, and $9$ are identical green flags. Let $N$ be the number of distinguishable arrangements ...
3
votes
1answer
46 views

One-to-One correspondence in Counting

I have a confusion on the one-to-one correspondence in combinatorics. Take the problem: In how many ways may five people be seated in a row of twenty chairs given that no two people may sit next ...
1
vote
1answer
48 views

Recursive sum of digits of $1989^{1989}$

It's a Big Sum Of Digits From the 1989 Canadian Maths Olympiad: Define the sequence $\{a_n\}, n\ge1$ as follows: $a_n = \begin{cases} 1989^{1989}, & \text{if } n = 1, \\ \text{sum ...
2
votes
0answers
38 views

Finding relations of variables

Suppose that \begin{align*} x&=t+t^{-1}+t^2s+t^{-2}s^{-1}+ts^{-1}+t^{-1}s-6\\ y&=t+t^{-2}+ts+s^{-1}-4\\ z&=t^{-1}+t^2+t^{-1}s^{-1}+s-4 \end{align*} Find a polynomial $P(x, y, z)=0$ ...
3
votes
3answers
102 views

How many of the numbers in $A=\{1!,2!,…,2015!\}$ are square numbers?

Problem How many of the numbers in $A=\{1!,2!,...,2015!\}$ are square numbers? My thoughts I have no idea where to begin. I see no immediate connection between a factorial and a possible square. ...