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

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8
votes
1answer
50 views

For which $n$ does $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ imply $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}$

I'm having trouble finishing a problem on an old national competition. As the title states, the question says asks: Given $a,b,c \neq 0,a+b=c$ such that ...
3
votes
3answers
86 views

Difficult inverse tangent identity

Prove that: $$\arctan\left(\frac{\sqrt{1 + x} - \sqrt{1-x}}{\sqrt{1 + x} + \sqrt{1-x}} \right) = \frac{\pi}{4} - \frac{1}{2}\arccos(x), -\frac{1}{\sqrt{2}} \le x \le 1$$ I'd multiply the ...
1
vote
1answer
46 views

Suppose $x,y$ are fixed real numbers. Does there always exist a real number $z$ such that $\sin(x+z)$ and $\sin(y+z)$ are rational numbers?

Suppose $x,y$ are fixed real numbers. Does there always exist a real number $z$ such that $\sin(x+z)$ and $\sin(y+z)$ are rational numbers? I know that $\sin(x) \in \mathbb{Q}$ implies that $\sin(x) ...
3
votes
1answer
37 views

Inequality and Trigonometric Substitution [duplicate]

Prove that for all positive real $a,b,c$, we have $$(a^2+2)(b^2+2)(c^2+2) \geq 9(ab+bc+ca).$$ Because of the term $a^2+2$, this motiveates me to substitute $a=\sqrt{2}\tan A, b=\sqrt{2}\tan B, ...
0
votes
2answers
87 views

Suppose $a<b<c<d$ and $p(x)=(x-a)(x-b)(x-c)(x-d)$. Show that $\int_a^b \frac{dx}{\sqrt{|p(x)|}} = \int_c^d \frac{dx}{\sqrt{|p(x)|}}$

Suppose $a<b<c<d$ and $p(x)=(x-a)(x-b)(x-c)(x-d)$. Show that $$\int_a^b \frac{dx}{\sqrt{|p(x)|}} = \int_c^d \frac{dx}{\sqrt{|p(x)|}}.$$ My attempt: I perform linear substitution $u=x-a+c$ ...
0
votes
1answer
33 views

Suppose a continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ is nowhere monotone. Show that there exists a local minimum for each interval.

Suppose a continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ is nowhere monotone. Show that there exists a local minimum for each interval. This question is from Moscow institute. First of ...
0
votes
2answers
28 views

Suppose $A$ is an invertible matrix. Is it true that there always exists a polynomial $p(x)$ such that $A^{-1}=P(A)$?

Suppose $A$ is an $ \times n$ invertible matrix. Is it true that there always exists a polynomial $p(x)$ such that $A^{-1}=P(A)$? The question is from Moscow Institute of Physics and Technology My ...
1
vote
3answers
54 views

A point D in a triangle ABC such that $\angle DAB= \angle DBC= \angle DCA$

I got this question from a student of mine, who is participating in a math olympiad competition: How can we construct a point D in a triangle ABC such that $\angle DAB= \angle DBC= \angle DCA$? I've ...
2
votes
1answer
45 views

If $f$ satisfies certain conditions, then show that $\lim_{x \rightarrow \infty}{\frac{f(x)}{x}}=0$

Suppose $a\in \mathbb{R}$, $a \in (0,1)$ and a function $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying the following property: $(1)$ $\lim_{x \rightarrow \infty}{f(x)}=0$ $(2) \lim_{x \rightarrow ...
6
votes
3answers
52 views

“Rationalizing the denominator” of $1/(a + b\sqrt[3]{2} + c\sqrt[3]{4})$?

If $(a, b, c) \in \mathbb{Q}^3 \setminus \{(0, 0, 0)\}$, so that $a + b\sqrt[3]{2} + c\sqrt[3]{4}$ is a nonzero element of $\mathbb{Q}(\sqrt[3]{2})$, is there a formula for $${1\over{a + b\sqrt[3]{2} ...
1
vote
1answer
65 views

Let $F(r)=\sum_{k=1}^m{|P(rz_k)|^2}$ for $r>0$. Prove that the function $F(r)$ is increasing if $m>n>0$.

Let $P(z)$ be a polynomial of degree $n$ with complex coefficients. Further, let $$z_k=e^{\frac{2 \pi i k}{m}}$$ for some $m$ and $k=1,2,...,m$. In other words, $z_1,\cdots z_m$ are the $m$th roots of ...
4
votes
1answer
101 views

Suppose $AB=BA$ and $A^{1965}=B^{2015}=I$. Prove that $A+B+I $ is invertible.

Supppse $A $ abd $B $ are matrices, $AB=BA $ and $A^{1965}=B^{2015}=I $. Prove that $A+B+I $ is invertible. I want to prove that $(A+B+I)C=I $ I have no idea how to start. Can any one give some hint? ...
0
votes
1answer
61 views

Find $f(2015)$ in function $f$ defined below

Let $\mathbb{S}$ be the set $\mathbb{R}^+ \cup \{0\}$ Let a function $f:\mathbb{S} \rightarrow \mathbb{S} $ be defined as: $$f(x^2+y^2) = y^2f(x)+x^2f(y) +x^4+y^4$$ If done so, then what would be ...
-7
votes
1answer
58 views

Simplify $2^3-3^{\frac{5}{8}}+2^2+3^{\frac{5}{8}}+2^1$ [on hold]

How can I simplify this expression? I really need to know how. $2^3-3^{\frac{5}{8}}+2^2+3^{\frac{5}{8}}+2^1$
0
votes
1answer
48 views

How can I determine the value of $a_1 + \displaystyle\sum_{i = 1}^{2012}\frac{a_{i + 1}^3}{a_i^2 + a_ia_{i + 1} + a_{i + 1}^2}$

For reals $x \ge 3$, let $f(x)$ denote the function $f(x) = \frac{-x + x\sqrt{4x - 3}}{2}$. Now suppose that $a_1, a_2, \ldots, a_{2013}$ is a sequence of real numbers such that $a_1 > 3, a_{2013} ...
3
votes
0answers
39 views

Sum of zeros of $P(x)$

I asked this question here before too, but vaguely, hopefully, this time will be a better attempt: There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero ...
0
votes
1answer
18 views

Find the lower and upper bounds

I'm stuck with this question: $-2 < x < 6$ and $-4<y<-2$ What are the bounds of $x^2-y^2$? I thought that they are $(-2)^2-(-4)^2 = -12$ and $6^2-(-2)^2 = 32$, but apparently they are ...
0
votes
3answers
75 views

A function $y(x)$ satisfies the differential equation $y^{\prime}=4\sqrt{y-x^2}$ It is known that $y(1)=2$. Find $y(3)$.

A function $y(x)$ satisfies the differential equation $$y^{\prime}=4\sqrt{y-x^2}$$ It is known that $y(1)=2$. Find $y(3)$. My attempt: Clearly $y^{\prime}=4$ at $x=1$. That's all(LOL). Any hint to ...
1
vote
3answers
21 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
2answers
183 views
+50

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
15 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 ...
1
vote
2answers
73 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
21 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
38 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) [closed]

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, ...
2
votes
2answers
47 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
27 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
73 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
41 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
38 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
35 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
136 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
75 views

IMO 2015 Problem 3 [closed]

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
43 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 ...
32
votes
1answer
518 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
50 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 ...
15
votes
3answers
219 views
+200

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
29 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
63 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
72 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
62 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
32 views

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

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
84 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
31 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 ...