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

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2
votes
1answer
47 views
+50

On divisibility of sum of positive integers

Let $ a,b,c$ positive integer such that $ a + b + c \mid a^2 + b^2 + c^2$. Show that $ a + b + c \mid a^n + b^n + c^n$ for infinitely many positive integer $ n$. (problem composed by Laurentiu ...
6
votes
7answers
125 views

How to find $ab+cd$ given that $a^2+b^2=c^2+d^2=1$ and $ac+bd=0$?

It is given that $a^2+b^2=c^2+d^2=1 $ And it is also given that $ac+bd=0$ What then is the value of $ab+cd$ ?
1
vote
0answers
21 views

non-countable subset of $\mathbb 2^{\mathbb Z}$ with finite pairwise intersection. [duplicate]

Does a non countable subset of the power-set of $\mathbb Z$ exist so that the intersection of any two elements is a finite set? If we ask for the sets to be pairwise disjoint then the answer is a ...
3
votes
0answers
17 views

Iterated circumcenters - proving collinearity and establishing distance ratios

Let $P_0, P_1, P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle ...
0
votes
0answers
25 views

Showing $pk+1|p^p-1$ implies that $k$ is even

Suppose $p$ is an odd prime such that $pk+1$ divides $p^p-1$. Prove that it is not possible for $k$ to be odd. Here's my solution: Assume to the contrary that $pk+1$ does divide $p^p-1$ We can ...
2
votes
3answers
36 views

Trouble understanding inequality proved using AM-GM inequality

I am studying this proof from Secrets in Inequalities Vol 1 using the AM-GM inequality to prove this question from the 1998 IMO Shortlist. However, I'm lost on the very first line of the solution. ...
-3
votes
0answers
28 views

What are the possible values of d [on hold]

Let $n$ be a positive integer greater than or equal to $6$, and suppose that $a_1, a_2, \ldots$ an are real numbers such that the sums $a_i + a_j$ for $1\le i < j \le n$, taken in some order, form ...
-3
votes
1answer
69 views

real analysis olympiad question [on hold]

Define $f:$ $\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ and consider $f(1,n) = 2n-1 $ & $f(m+1,n)$ = $2^m$$(2n-1)$ Prove that $f$ is a bijection.
2
votes
3answers
104 views

Probability that the eventually a six on a dice will appear.

Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $ m$ and $ n$ be relatively prime ...
0
votes
2answers
76 views

How many pairs $(m, n)$ exist?

For certain pairs $ (m,n)$ of positive integers with $ m\ge n$ there are exactly $ 50$ distinct positive integers $ k$ such that $ |\log m - \log k| < \log n$. Find the sum of all possible ...
-1
votes
1answer
41 views

Find the number of members of a family [on hold]

One morning, each member of Manjul’s family drank an 8-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank 1/7-th of the ...
3
votes
1answer
52 views

Russian MO 2004 Question involving the AM-GM inequality

I'm reading Secrets in Inequalities by Pham Kim Hung, and I'm having trouble understanding this proof from a problem from the 2004 Russian MO. Let a,b,c be positive real numbers and $a + b +c = 3$. ...
14
votes
3answers
273 views

Prove that there exists infinitely many positive integers $n$ such that $\sin^2{(na)}+\sin^2{(nb)}\le \frac{2\pi^2}{n}$

Can anyone please help me with the following proof: Prove that there exists infinitely many positive integers, $n$, such that $$\sin^2{(na)}+\sin^2{(nb)}\le \dfrac{2\pi^2}{n}\quad a,b\in \Bbb R$$
0
votes
1answer
37 views

Factorial Series Compute

Given $\sum_{n=2}^{\infty}\sum_{j=2}^{\infty}\frac{1}{(j^n)(j!)}=a+be$ where $a$ and $b$ are integers, find $a$ and $b$.
2
votes
2answers
58 views

a matrix of rank $r$ satisfies a polynomial of degree $r+1$.

Let $M$ be an $n\times n$ matrix with coefficients in $\mathbb C$. Suppose $M$ has rank $r$ with $r<n$. Prove there is a polynomial $P(x)$ with degree $r+1$ and coefficients in $\mathbb C$ such ...
1
vote
1answer
33 views

Inequality about unit vectors

$2n$ unit vectors with non-negative $y$-coordinates are given. Prove that if the sum of $x$-coordinates is an odd integer, then the sum of $y$-coordinates is at least $1$. $n=1$ case is easy; I also ...
0
votes
2answers
70 views

$f:\mathbb{N}\rightarrow \mathbb{N}$ is a one-to-one function such that $f(mn)=f(m)f(n).$ Find the lowest possible value of $f(999)$. [duplicate]

$f:\mathbb{N}\rightarrow \mathbb{N}$ is a one-to-one function such that $f(mn)=f(m)f(n).$ Find the lowest possible value of $f(999)$. The answer is given as $24$ but I never get that.
5
votes
2answers
63 views

Trapezoids in a square

Good day As part of a problem I need to show that AB is parallel to CD, with the given info on the image. All the segments marked red are equal, all 1-stripe grey equal etc. I'd like to prove ...
1
vote
1answer
15 views

Calculating some probability of buying different cards

This problem was recently featured in an ICPC contest. The problem stated, without taking the story into account and as well as I recall: Peter likes a fad card game. This game has various cards, and ...
15
votes
1answer
117 views
+50

For all $n$ there exists $x$ such that $\varphi(x)<\varphi(x+1)<\ldots<\varphi(x+n)$

Let $\varphi$ be the Euler's function, i.e. $\varphi(n)$ stands for the number of integers $m \in \{1,\ldots,n\}$ such that $\text{gcd}(m,n)=1$. Let $n\ge 2$ be a positive integer. Show that there ...
2
votes
0answers
44 views

Inequality problem involving log function

Given $|f(x+y)-f(x)-f(y)| \leq x+y$ for all $x > y > 0$, prove that real valued function $f$ satisfies the inequality $|\frac{f(x)}{x} - \frac{f(y)}{y}| \leq M(1+\log_2\frac{x}{y})$ where M is ...
1
vote
1answer
29 views

Find the maximum value of the quotient

Given a real number $x,$ let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{1}, n_{2}, \ldots, ...
-2
votes
1answer
38 views

Been trying to work this out for a while, please help me.

It is known that among any three students in a class, two of them are friends. The total number of students is $25$ prove that their is a student with at least $12$ friends. How do I work this out?
1
vote
2answers
20 views

Variation of geometric, harmonic and arithmetic means in sequence.

A question I got on my test was - Let ${A}_{1}, {G}_{1}$ and ${H}_{1}$, denote the arithmetic, geometric and harmonic means of two distinct positive numbers. For $n\geq 2$, Let ${A}_{n-1}$ and ...
1
vote
2answers
99 views

Determine all positive integers $n$ which have a divisor $d$ with the property that $dn+1$ is a divisor of $d^2 + n^2$

Determine all positive integers $n$ which have a divisor $d$ with the property that $dn+1$ is a divisor of $d^2 + n^2$. So i formed the equation that $$\frac{n}{d} = \frac{d^2 + n^2}{dn + 1}$$ And ...
1
vote
1answer
33 views

Two cevians divide a triangle into 4 parts. Calculate the area of the 4th part, given the other 3.

Good day Here is the question: Connecting $AF$ and setting areas $\triangle ADF = x$ and $\triangle AFE = y$: $\frac {9+x}{12} =\frac y{15}$ $\frac{15+y}{12} =\frac x9$ from the ratios of the ...
7
votes
1answer
61 views

What is maximum a number of to form right-triangles from in n straight lines

I am interested what is maximum a number of to form right-triangles from in $n=100$ straight lines such $n=3$,then maximum number of is $1$,see fig:$\Delta ABC$ is right-triangles. $n=4$ then ...
2
votes
3answers
110 views

Are these lines going to meet in exactly 2002 points?

There is a plane P.100 lines are on P.Is it possible to arrange them in a way such that they intersect in exactly 2002 points given that no three of them are concurrent? Any help is highly ...
1
vote
1answer
33 views

Prove that $\sum\sqrt{\frac{a+b}{c}}\ge2\sum\sqrt{\frac{c}{a+b}}$

Let $a,b,c$ be positive numbers. Then we need to prove $\sqrt{\frac{a+b}{c}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{c+a}{b}}\ge2\left(\sqrt{\frac{c}{a+b}}+\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}\right).$ ...
1
vote
3answers
125 views

Why do we subtract [Combinatorics]

I asked Here This question and I am still confused. I got that, for at least one group together there are: $$3 \cdot 9 \cdot \binom{6}{3, 3}$$ But why do we subtract: $3 \cdot 9 \cdot 4$. Lets ...
8
votes
1answer
147 views

Prove that, if $A, B$ are matrices from $M_4(R)$ so that $AB=BA$

Prove that, if $A, B$ are matrices from $M_4(\Bbb R)$ so that $AB=BA$ and $\det(A^2 −AB + B^2) = 0$ then: $$ \det(A + B) + 3\det(A − B) = 6 (\det(A) + \det(B)) \tag 1 $$ What I tried: Because of ...
0
votes
1answer
21 views

Making all row sums and column sums non-negative by a sequence of moves

Real numbers are written on an $m\times n$ board. At each step, you are allowed to change the sign of every number of a row or of a column. Prove that by a sequence of such steps, you can always ...
12
votes
2answers
137 views

Exist complex $z_{0}$ ,such $|z_{0}|=1$,and $|f(z_{0})|\le|f(z)|,\forall |z|\ge 1$

Let $a\in (0,1), f(z)=z^2-z+a, z\in \mathbb C$. Does there exist a complex number $z_{0}$ such that $|z_{0}|=1$, and $$|f(z_{0})|\le|f(z)|,\forall |z|\ge 1$$ I just have no idea where to even begin ...
6
votes
1answer
63 views

Tzaloa 2015 game problem (piles with $1,2,4 \dots 2^{19}$ coins each)

We have $20$ piles with $1,2,4,8\dots 2^{19}$ coins repectively and two players. In each turn a player must select five piles that have at least one coin and remove exactly one coin from each. Player ...
15
votes
1answer
296 views

About sequences of positive integers

Prove there is no sequence of positive integers $(x_n)_{n \ge 1}$ so that: $$ x_{n+2} = x_{n+1} + x_{x_n} \quad \forall n\ge1 $$ I think the idea is to find two different values for the same index.
4
votes
1answer
140 views

Prove that, if $p \in \mathbb{N}, p>5$, p prime

Prove this: Hypothesis Let $p \in \mathbb{N}, p>5$, p prime so that $p | (2^q + 3^q)$ where $q \in \mathbb{N}$, $q$ prime. Conclusion $p>q$ No idea how to start...
1
vote
2answers
65 views

Ghosts closing and opening doors [duplicate]

There are $1000$ doors $D_1,D_2,D_3,\dots,D_{1000}$ and $1000$ persons $P_1,P_2,\dots,P_{1000}$. Initially all the doors were closed. Person $P_1$ goes and opens all the doors. Then person $P_2$ ...
2
votes
1answer
27 views

Prove the function is periodic

If $f:[0, \infty) \rightarrow \mathbb{R}$ is continuous so that: $$ \int_{0}^{n} f(x)f(n-x)dx = \int_{0}^{n} (f(x))^2dx \quad \forall n\in \mathbb{N}^* \tag1 $$ then f is a periodic function. What I ...
1
vote
1answer
46 views

Find the sign of $a,b,c$ in $ax^2+bx+c$ given the graph and a coordinate on it.

So my first approach was that, we see that there are $2$ roots. And one is negative and one is positive. $a$ would be evidently positive. The positive one's modulus is bigger than the negative ...
2
votes
1answer
35 views

Ordered triples of n-powerful integers

Let’s say that an ordered triple of positive integers (a, b, c) is n-powerful if: $a \le b \le c$, $gcd(a, b, c) = 1$ and $a^n + b^n + c^n$ is divisible by $a + b + c$. For ...
1
vote
1answer
42 views

Maximum number of non-zero entries ,such that no two non-zero entries are on the same row or column.

In an M x N matrix such that all non-zero entries are covered in "a" rows and " b" columns. Then the maximum number of non-zero entries ,such that No two non-zero entries are on the same row or column ...
2
votes
2answers
148 views

2011 AIME Problem 12, probability round table

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate ...
3
votes
2answers
58 views

If $2xy$ divides $x^2+y^2-x$, prove that $x$ is a perfect square [duplicate]

This problem is from ( BMO Exam1991 ). I tried to solve but it was difficult. The problem is: If $ x^{2} + y^{2} - x $ is a multiple of $ 2xy $ where $x$ & $y$ are integers, prove that $x $ ...
2
votes
2answers
78 views

Prove the root is less than $2^n$

A polynomial $f(x)$ of degree $n$ such that coefficient of $x^k$ is $a_k$. Another constructed polynomial $g(x)$ of degree $n$ is present such that the coefficeint of $x^k$ is $\frac{a_k}{2^k-1}$. ...
15
votes
1answer
226 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) ...
-1
votes
4answers
60 views

How many ways can we arrange 7 books, including 2 math books and 1 physics book, with the math books next to each other and left of the physics book?

I have 7 books I want to arrange on a shelf. Two of them are math books, and one is a physics book. How many ways are there for me to arrange the books if I want to put the math books next to each ...
3
votes
1answer
85 views

Determine all functions $f:\mathbb{Q}\to\mathbb{Q}$ satisfying the functional equation $f(2f(x) + f(y)) = 2x + y$

Determine all functions $f$ defined on the set of rational numbers that take rational values for which $$f(2f(x) + f(y)) = 2x + y \tag{1}$$ for each x and y. This question is from the 2008 ...
1
vote
1answer
39 views

Distinguishability in Round Table Combinatorics

I have stumbled upon many questions, and one of the weaknesses is the ability to test if the concept is distinguishable or not. For example this: Nine delegates, three each from three different ...
-1
votes
1answer
33 views

Combinatoric meaning of multinomial coefficients

$$\binom{n}{k}$$ means how many ways there are to choose $k$ objects from $n$ total objects. What is the combinatoric meaning of: $$\binom{n}{k_1, k_2, ... , k_n}$$ ??
0
votes
3answers
124 views

A triangle with vertices on the sides of a square, with one at a midpoint, cannot be equilateral

Let A, B, C, D, L, M, N be distinct points in the plane such that A, B, C, D are the vertices of a square with sides AB, BC, CD, DA and L, M, N lie on the sides AB, CD, BC respectively. If M is the ...