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

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3
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1answer
60 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
67 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 $P(r_1, r_2) = 0$...
2
votes
1answer
87 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
68 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
47 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
115 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 f\Big(\frac{1}{n}\...
1
vote
3answers
64 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 price ...
9
votes
3answers
122 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
87 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
124 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 patterns (...
6
votes
1answer
87 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
190 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 ...
6
votes
1answer
67 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 $\frac{p}{q}$, where ...
5
votes
1answer
72 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 fixed)...
5
votes
1answer
345 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 tried ...
1
vote
1answer
43 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 $\frac{1}{...
3
votes
1answer
86 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
75 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
259 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
92 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
108 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
147 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
64 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
206 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
99 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
70 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
40 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
120 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. ...
1
vote
1answer
122 views

Math contest question - prove unsolvability of equations.

Prove that the system of equations has no real solutions: $$\begin{cases} y=\sqrt{x+\sqrt{1-x}} \\ x=\sqrt{y-\sqrt{1+y}}\end{cases}. $$ This is a former problem from a national math contest which I´...
3
votes
5answers
106 views

How many $5$ element sets can be made?

Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ is ...
3
votes
3answers
100 views

Double Factorial Sum

Let $ n!!$ to be $ n(n-2)(n-4)\ldots3\cdot1$ for odd $ n$ values and let $ n(n-2)(n-4)\ldots4\cdot2$ for even $ n$ values. Also let $ \displaystyle \sum_{n=1}^{2009} \frac{(2n-1)!!}{(2n)!!}$ be ...
5
votes
2answers
65 views

Floor Function Equation

How many positive integers $ N$ less than $ 1000$ are there such that the equation $ x^{\lfloor x\rfloor} = N$ has a solution for $ x$? (The notation $ \lfloor x\rfloor$ denotes the greatest integer ...
7
votes
2answers
739 views

Olympiad question on Pigeonhole principle

Given a set $M$ of $1985$ distinct positive integers, none of which has a prime divisor greater than $26$, prove that $M$ contains at least one subset of four distinct elements, whose product is the ...
0
votes
2answers
107 views

Find the coefficient of $x^{17}$

Find the coefficient of $x^{17}$ in:$$ (1 + x^5 + x^7)^{20}$$ $x^{17} = x^{5} x^5 x^{7}$ I would say: $$\frac{17!}{5!5!7!} $$ But this isnt the correct answer. I know I need to use combinations, ...
4
votes
2answers
126 views

A positive integer is equal to the sum of digits of a multiple of itself.

Let $n$ be a positive integer, prove there is a positive integer $k$ so that $n$ is equal to the sum of digits of $nk$. I'm not really sure how I should approach this problem, I tried to do a ...
1
vote
1answer
78 views

“At least” type probability question.

Recently, I asked a question: Team A has more Points than team B Though I ultimately got the right answer, it took extreme casework, and long computations. My question is: suppose the question was ...
0
votes
1answer
58 views

Maximum number of teams of three people such that each team is built in one of two ways

A coach picks team members in two ways:   A. The team of three people should consist of one experienced participant and two newbies. Thus, each experienced participant can share the ...
7
votes
1answer
201 views

Elegant applications of advanced techniques to “olympiad” problems

I am interested in applications of somewhat "advanced machinery" (with respect to the usual machinery involved in these cases, which is usually elementary) to olympiad or (high school-level) contest ...
9
votes
1answer
238 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x^b_{1}+x^b_{2}+\cdots+x^b_{k}|\ge 1$

This problem is a 2014 Sydney mathematics competition problem (11 grade). It seems difficult to solve. (I previously posted the n=2 case for which André Nicolas and Dan Robertson proposed solutions) ...
4
votes
2answers
152 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x_{1}+x_{2}+\cdots+x_{k}|\ge 1$

Edit: This problem 1 is a 2014 Sydney mathematics competition problem (8th grade). It seems difficult to solve. Show that: There exist complex numbers $x_{1},x_{2},\cdots,x_{k}(k\ge 2)$ such ...
2
votes
4answers
123 views

Probability that team $A$ has more points than team $B$

Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games ...
2
votes
0answers
546 views

Two circumcircles of triangles defined relative to a fixed acute triangle are tangent to each other (IMO 2015)

I'm posting here the question because I want to see a nice synthetic solution (not using complex numbers or inversive geometry) for the 3rd problem from IMO 2015. The problem is as follows: Let $...
5
votes
4answers
359 views

Find the roots of the summed polynomial

Find the roots of: $$x^7 + x^5 + x^4 + x^3 + x^2 + 1 = 0$$ I got that: $$\frac{1 - x^8}{1-x} - x^6 - x = 0$$ But that doesnt make it any easier.
0
votes
3answers
58 views

discount and percentage question, how to solve this

To attract more visitors, Zoo authority announces $20\%$ discount on every ticket which cost $\$25$. For this reason, sales of tickets increases by $28\%$. Find the $\%$ of increase in the number of ...
16
votes
1answer
415 views

On the inequality $\frac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.$

For all positive integers $n$, $p(n)$ is the number of partitions of $n$ as the sum of positive integers (the partition numbers); e.g. $p(4)=5$ since $4=1+1+1+1=1+1+2=1+3=2+2=4.$ Prove ...
2
votes
0answers
159 views

Expected Power Product of rolling a dice .

A 15 sided dice is rolled 1000 times. Let k1,k2,k3,k4,..k15 denote the number of times 1,2,3...15 appears. How can I compute the following expected value :$$E( (k_1 k_2 k_3 k_4)^5).$$ My attempts:: ...
3
votes
2answers
201 views

IMO 1995 Shortlist problem C5

IMO 1995 Shortlist problem C5 At a meeting of $12k$ people, each person exchanged greetings with exactly $3k+6$ people. For any two people, the number who exchange greetings with both is always ...
0
votes
2answers
41 views

Triangle and Ratio : Find the length of a side.

Let $\theta = \angle CAD, \phi = \angle CDB, \varphi=\angle DBC, \alpha = \angle BCD$ and $\beta=\angle ACD$. Then we have the following system of equations $\theta + \varphi = 90^{\circ},$ $\theta +...
6
votes
0answers
155 views

Balanced, center-free set. [closed]

We say that a finite set $\mathcal{S}$ of points in the plane is balanced if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say ...