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

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0
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2answers
15 views

$Q=\{ 1,2,…n \}$. $S \subset Q$, let $p(S)$ be the product of elements of $S$, Find the sum of reciprocals $\frac{1}{p(S)}$ for all $S \subset Q$.

Consider the set $Q=\{ 1,2,...n \}$. For each $S \subset Q$, let $p(S)$ be the product of elements of $S$, Find the sum of reciprocals $\frac{1}{p(S)}$ for all $S \subset Q$. I have no idea how to ...
4
votes
1answer
66 views

find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y \in \mathbb{R}^n$

find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying \begin{equation*} f(x+y)+f(x-y)=2f(x)+2f(y)~\forall x,y \in \mathbb{R}^n. \end{equation*} My attempt: I manage to show ...
0
votes
2answers
12 views

let $q$ be the number of pairs of linearly independent vectors from $S$. What is the smallest and the largest possible value of $q$?

Let $S$ be a set of $n$ nonzero vectors in $\mathbb{R}^2$ such that $S$ spans the whole $\mathbb{R}^2$ and let $q$ be the number of pairs of linearly independent vectors from $S$. What is the smallest ...
0
votes
0answers
48 views

combinatorics contest problem

Question: Calvin has a bag containing $50$ red balls, $50$ blue balls, and $30$ yellow balls. Given that after pulling out 65 balls at random (without replacement), he has pulled out $5$ more red ...
2
votes
0answers
20 views

Let $p_1, p_2,…,p_n$ be polynomials of $k$ variables $x_1,…,x_k$ and $p_1^2 + \cdots p_n^2=x_1^2 + \cdots + x_k^2$ Prove that $n \geq k$.

Let $p_1, p_2,...,p_n$ be real polynomials of $k$ variables $x_1,...,x_k$ and assume that $$p_1^2 + \cdots p_n^2=x_1^2 + \cdots + x_k^2$$ Prove that $n \geq k$. Out of so many questions that I ...
0
votes
1answer
34 views

Show that the equation has at least two solutions on the interval $0 \leq x \leq 1$

Let $0 < a < 1$. Show that the equation $$\int_0^x{\left( \sin \left(\frac{\pi \sin\frac{\pi t}{2}}{2} \right)+ \frac{2}{\pi} \sin^{-1} \left( \frac{2}{\pi} \sin^{-1}(t) \right) -2t \right)}dt ...
-1
votes
0answers
26 views

Hollywood press-agent age formula (according to Time magazine 1949) [on hold]

In the Wikipedia article on Age fabrication, it quotes Time magazine from 1949: To find the age of a star, a Hollywood press-agent takes the year of her (or his) birth, subtracts it from itself, ...
1
vote
2answers
40 views

Establishing a trigonometric identity for $n\in\mathbb{N}$

The original problem was showing that this infinite sum converges to $\tan\theta$: $$\sum_{n=1}^\infty \frac{\tan\dfrac{\theta}{2^n}}{\cos\dfrac{\theta}{2^{n-1}}}$$ One hint was given: the series ...
0
votes
2answers
25 views

During a night, each chameleon changes its colour to one of the other four colours with equal probability.

Five chameleons of all different colours meet one evening. During the night, each chameleon changes its colour to one of the other four colours with equal probability. Find the probability that the ...
6
votes
2answers
66 views

Suppose $a_n>0$ and $\sum_{n=1}^{\infty}{a_n}$ diverges. Determine whether $\sum_{n=1}^{\infty}{\frac{a_n}{s_n^2}}$, where $s_n=a_1+a_2+\cdots+a_n$.

Suppose $a_n>0$ and $\sum_{n=1}^{\infty}{a_n}$ diverges. Determine whether $\sum_{n=1}^{\infty}{\frac{a_n}{s_n^2}}$, where $s_n=a_1+a_2+ \cdots + a_n$. My attempt: By testing a few examples, the ...
0
votes
2answers
34 views

Find the number of possible values of $a$

Positive integers $a, b, c$, and $d$ satisfy $a > b > c > d, a + b + c + d = 2010$, and $a^2 − b^2 + c^2 − d^2 = 2010$. Find the number of possible values of $a.$ Obviously, factoring, ...
0
votes
0answers
38 views

Sum of zeros polynomial

There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x)={x}^{3}-a{x}^{2}+bx-65$. For each possible combination of $a$ and $b$, let ...
2
votes
1answer
36 views

Suppose entries of $A$ are either $-1$ or $1$, and $\det(A)=n!$. Find all positive numbers $n$ such that such matrix exists

A matrix $A$ is interesting if entries of $A$ are either $-1$ or $1$, and $\det(A)=n!$. Find all positive numbers $n$ such that there exists an interesting matrix of size $n \times n$. Claim: If ...
0
votes
3answers
34 views

Given a fixed positive integer $k$, find the number of pairs of integers $(x,y) \in \mathbb{Z}^2$ such that $x^2+y^2=5^k$

Given a fixed positive integer $k$, find the number of pairs of integers $(x,y) \in \mathbb{Z}^2$ such that $$x^2+y^2=5^k$$ Attempt: Clearly $x$ and $y$ cannot have the same parity. Assume that ...
15
votes
2answers
129 views

$xf(y)+yf(x)\leq 1$ for all $x,y\in[0,1]$ implies $\int_0^1 f(x) \,dx\leq\frac{\pi}{4}$

I want to show that if $f\colon [0,1]\to\mathbb{R}$ is continuous and $xf(y)+yf(x)\leq 1$ for all $x,y\in[0,1]$ then we have the following inequality: $$\int_0^1 f(x) \, dx\leq\frac{\pi}{4}.$$ The ...
-6
votes
0answers
72 views

IMO 2015 Problem 3 [on hold]

Let $n$ and $k$ be positive integers. Prove that if $n$ is relatively prime with $30$, then there exist integers $a$ and $b$, each relatively prime with $n$, such that $\frac{a^2-b^2+k}{n}$ is an ...
1
vote
1answer
42 views

Assume that the sum of absolute values of all entries of $A$ equals to $1$. What is the maximal possible value of $\det(A)$?

Let $A$ be an $n \times n$ matrix and assume that the sum of absolute values of all its entries equals to $1$. What is the maximal possible value of $\det(A)$? My attempt: We know that $|a_{i,j}| ...
0
votes
4answers
37 views

For every integer vector $\overrightarrow{a}$,there is a integer vector $\overrightarrow{b}$ such that $\overrightarrow{a}\bot\overrightarrow{b}$

In $R^3$,show that for every integer vector $\overrightarrow{a}$,there is a integer vector $\overrightarrow{b}$ such that $\overrightarrow{a}\bot\overrightarrow{b}$ Generally,in $R^n$,for every ...
26
votes
1answer
278 views

Prove that there is a real number $a$ such that $\frac{1}{3} \leq \{ a^n \} \leq \frac{2}{3}$ for all $n=1,2,3,…$

Prove that there is a real number $a$ such that $\frac{1}{3} \leq \{ a^n \} \leq \frac{2}{3}$ for all $n=1,2,3,...$ Here, $\{ x \}$ denotes the fractional part of $x$. My attempt: Clearly $a$ cannot ...
2
votes
1answer
48 views

Prove that all the five sequences converge to the same point $P \in \mathbb{R}^3$.

Let five sequences $A_n, B_n, C_n, D_n, E_n \in \mathbb{R}^3$ be constructed as follows: $A_0, B_0, C_0, D_0$ and $E_0$ are some given points of the space and $A_{n+1}, B_{n+1}, C_{n+1}, D_{n+1}, ...
6
votes
4answers
71 views

Prove that for any given $c_1,c_2,c_3\in \mathbb{Z}$,the equations set has integral solution.

$$ \left\{ \begin{aligned} c_1 & = a_2b_3-b_2a_3 \\ c_2 & = a_3b_1-b_3a_1 \\ c_3 & = a_1b_2-b_1a_2 \end{aligned} \right. $$ $c_1,c_2,c_3\in \mathbb{Z}$ is given,prove that $\exists ...
3
votes
1answer
95 views

Maximum $C$ such that every shape in $\Bbb R^2$ with area $<C$ can be placed to avoid $\Bbb Z^2$

For $C=1$, it has been proved here that every shape in the plane having area less than $1$ can be translated and rotated so that it does not touch any element of $\mathbb Z^2$. (In fact, for $C=1$, ...
2
votes
2answers
28 views

Prove that for any $f_1,f_2,…f_k \in I$, there exists a point $x_0 \in [a,b]$ such that $f_1(x_0)=…=f_k(x_0)=0$.

Let $C[a,b]$ be the ring of real-valued functions continuous on $[a,b]$ and let $I \subset C[a,b]$ be its proper ideal. Prove that for any $f_1,f_2,...f_k \in I$, there exists a point $x_0 \in [a,b]$ ...
5
votes
1answer
62 views

Find all real numbers $c$ satisfying the following condition: For all $n \in \mathbb{N}$, we have $n^c \in \mathbb{N}$.

Find all real numbers $c$ satisfying the following condition: For all $n \in \mathbb{N}$, we have $n^c \in \mathbb{N}$. My attempt: Clearly all $c \in \mathbb{N}$ works while negative integer $c$ ...
1
vote
0answers
39 views

Polynomial With Complex Zeros

There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x) = x^3 - ax^2 + bx - 65$. For each possible combination of $a$ and $b$, let ...
3
votes
1answer
39 views

Polynomials and Commutativity

Let $f(x)=2013x+1$. Suppose $g(x), h(x)$ are polynomials with real coefficients such that $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$. Prove that $g(h(x))=h(g(x))$. I tried to look at the coefficients of ...
2
votes
2answers
71 views

BMO1 2006/07 Question 4 Geometry Problem

$4.$ Two touching circles $S$ and $T$ share a common tangent which meets $S$ at $A$ and $T$ at $B$. Let $AP$ be a diameter of $S$ and let the tangent from $P$ to $T$ touch it at $Q$. Show that $AP = ...
2
votes
2answers
59 views

Suppose $A^2B+BA^2=2ABA$.Prove that there exists a positive integer $k$ such that $(AB-BA)^k=0$.

Let $A, B \in M_n(\mathbb{C})$ be two $n \times n$ matrices such that $$A^2B+BA^2=2ABA$$ Prove that there exists a positive integer $k$ such that $(AB-BA)^k=0$. Here is the source of the problem. ...
4
votes
1answer
78 views

Is it possible to express $x^4-x^3+3x^2-4x+6$ as a product of polynomials of smaller degree with integer coefficients?

Is it possible to express $x^4-x^3+3x^2-4x+6$ as a product of polynomials of smaller degree with integer coefficients? My attempt: By equating the polynomial to $0$, one obtains $x=1\pm i, ...
0
votes
0answers
31 views

How many ways can I pick 3 marbles from this bag? [on hold]

Let's say you have a bag of 300 marbles (100 blue, 100 red, 100 yellow), and you draw 3 marbles from the bag. How many different outcomes are there? (How many different groups of 3 marbles?) Also, ...
3
votes
2answers
83 views

BMO1 2006/07 Question 2 Geometry Problem

$2.$ In the convex quadrilateral $ABCD$, points $M,N$ lie on the side $AB$ such that $AM = MN = NB$, and points $P,Q$ lie on the side $CD$ such that $CP = PQ = QD$. Prove that Area of $AMCP=$ Area of ...
1
vote
2answers
30 views

ARML: Tangent congruent circles forming a right circular cone

Four congruent circles are tangent to each other and tangent to the edges of a sector as shown. If the straight edges are joined to form a right circular cone with the vertex at P, the radius ...
2
votes
1answer
20 views

Denote $y_n=\int_0^1{\frac{f^{n+1}(x)}{g^n(x)}}dx$ for all integer $n \geq 0$. Prove that $(y_n)_{n \geq 1}$ is an increasing and divergent sequence. [duplicate]

Let $f,g:[0,1] \rightarrow (0,\infty)$ be two distinct, continuous functions such that $$\int_0^1 f(x)dx=\int_0^1 g(x)dx$$ Denote $$y_n=\int_0^1{\frac{f^{n+1}(x)}{g^n(x)}}dx$$ for all integer $n \geq ...
0
votes
2answers
53 views

Let $H = \frac{A+A^T}{2}$. Assume that $H$ is positive definite. Prove that $\det(H) \geq \det(A)$.

Let $A$ be an $n \times n$ matrix with real entries and let $H = \frac{A+A^T}{2}$. Assume that $H$ is positive definite. Prove that $\det(H) \geq \det(A)$. This question is obtained from Moscow (I ...
1
vote
3answers
57 views

Determine the number of possible values for $\det(A)$, given that $A$ is an $n \times n$ matrix with real entries such that $A^3 - A^2 -3A +2I=0$.

Determine the number of possible values for $\det(A)$, given that $A$ is an $n \times n$ matrix with real entries such that $A^3 - A^2 -3A +2I=0$. here is the source of the problem. In the last ...
2
votes
2answers
40 views

Find all functions $f(x)$ satisfying $f(x)+f^{\prime}(\pi-x)=1$ for all $x \in \mathbb{R}$.

Find all functions $f(x)$ satisfying $f(x)+f^{\prime}(\pi-x)=1$ for all $x \in \mathbb{R}$. This is a question from Moscow. I have tried $f(x)=x^m$ and it clearly does not work. Clearly $f(x)=1$ ...
13
votes
6answers
203 views

show that $\frac{1}{F_{1}}+\frac{2}{F_{2}}+\cdots+\frac{n}{F_{n}}<13$

Let $F_{n}$ is Fibonacci number,ie.($F_{n}=F_{n-1}+F_{n-2},F_{1}=F_{2}=1$) show that $$\dfrac{1}{F_{1}}+\dfrac{2}{F_{2}}+\cdots+\dfrac{n}{F_{n}}<13$$ if we use Closed-form expression ...
0
votes
1answer
33 views

show that $\sum_{n=1}^{2015}a_{n}\equiv 3\pmod 4$

Assmue that real sequence $\{a_{n}\}$ such $$a_{1}=1,|a_{n+1}|=2|a_{n}|$$ show that $$\sum_{n=1}^{2015}a_{n}\equiv 3\pmod 4$$ I have solve $$|a_{n}|=|a_{1}|\cdot 2^{n-1}=2^{n-1}\Longrightarrow ...
1
vote
2answers
68 views

two objects moving in opposite directions.

I don't need a specific answer for this question, and would rather prefer to know how to solve questions like this one. So far I've tried using the $v=d/t$ formula to form equations, but haven't ...
1
vote
2answers
32 views

A question on perfect square

Prove that if $ab$ is a perfect square and $\gcd(a,b)=1$, then both $a$ and $b$ must be perfect squares. Their Answer: Consider the prime factorization $ab=p_1^{e_1}\cdots p_k^{e_k}$. If $ab$ ...
5
votes
3answers
111 views

Given $a,b,c,d>0$ and $a^2+b^2+c^2+d^2=1$, prove $a+b+c+d\ge a^3+b^3+c^3+d^3+ab+ac+ad+bc+bd+cd$

Given $a,b,c,d>0$ and $a^2+b^2+c^2+d^2=1$, prove $$a+b+c+d\ge a^3+b^3+c^3+d^3+ab+ac+ad+bc+bd+cd$$ The inequality can be written in the condensed form ...
1
vote
1answer
59 views

Show that $f(x)$ cant be written like this

For $n>1$ let $a_1$, $a_2$, $\ldots$, $a_n$ be $n$ distinct integers. Prove that the polynomial $$f(x)=(x-a_1)(x-a_2)\cdots(x-a_n)-1$$ cannot be written as $g(x)h(x)$ where $g$ and $h$ are ...
5
votes
1answer
66 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, ...
1
vote
0answers
37 views

Need more insight on a formula

Following is a part of a programming contest problem. Given $C_{1},m,n,o,x,y,z,c,d,K,J$ are positive integers $ C_{i} = \left\{ \begin{array}{l l} (m*C_{i-1}^2 + n*C_{i-1} + o) \bmod J & ...
2
votes
2answers
41 views

Find the number $abc$

The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$. $$N \equiv abcd ...
2
votes
2answers
55 views

Find numbers $a, b, c$ given that $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=168$

Let $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=168$. Find $a,b,c$ Suppose $a, b, c$ are roots of $P(x)$. $$P(x) = k(x - a)(x - b)(x - c)$$ But then I get $(k = 1)$ $$P(x) = x^3 - 12x^2 + ...
1
vote
2answers
37 views

Roots are the reciprocal of $f(x)$

I don't understand if $f(x)$ has roots, $r_1, r_2$ for example and $g(x)$ has roots $\frac{1}{r_1}, \frac{1}{r_2}$ Then how is $g(x) = x^2f(\frac{1}{x})$ What does $$f(\frac{1}{x})$$ have to do ...
2
votes
0answers
63 views

Cognitive processes involved solving IMO level problems [closed]

I am currently 16 years old and, though I'm obviously not as good as most of the people on this site, I have always been considerably better than most of my classmates in mathematics. This, of course, ...
6
votes
1answer
75 views

Prove this Complicated Inequality

Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \le 4$. Prove that $$\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.$$ ...
0
votes
1answer
57 views

What is wrong with this proof of a number theory competition problem?

Let $a$ and $b$ be positive integers. Suppose $a^n+n| b^n+n$ for any positive integer $n$, prove that $a=b$. My trial: Clearly $b\geq a$, write $b=a+d$, we must show that $d=0$. Now by assumption and ...