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

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2
votes
0answers
21 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
30 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 ...
14
votes
2answers
121 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 ...
-5
votes
0answers
65 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
41 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
35 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 ...
24
votes
1answer
250 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
45 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
66 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
88 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
60 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
28 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
38 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
66 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
58 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
77 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
81 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
28 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
52 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
196 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
32 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
66 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
103 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
40 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
74 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 ...
5
votes
1answer
221 views
+50

Increasing sequence of divisors of a quadratic trinomial

This question is from a Russian contest, the 2011 Tuymaada Olympiad. It's the fourth question on day two for the problems at grade level 2. Let $P(n)$ be a quadratic trinomial with integer ...
3
votes
0answers
29 views

Picking K counters out of K buckets containing NK counters, N of each different colour, up to N in each

This is a generalisation of a question that recently came up while solving a TopCoder problem. Suppose we have N blue counters, N red counters, N white counters, and so forth, K colours in total. We ...
3
votes
1answer
46 views

My answer to this combi problem doesn't match the answer in the book (Problem-Solving Strategies)

[Problems 31 and 32 from Arthur Engel's Problem-Solving Strategies.] Let $n$ children be seated in a line. How many ways can they change their places if they may only move by one place at most? ...
0
votes
2answers
32 views

Evaluate powers in fraction

This is abstracted from 2007 British Mathematical Olympiad Question 1.I wish to practice mathematics olympiad question for the upcoming Singapore Mathematics Olympiad Secondary 2 (Grade 8). Find ...
1
vote
2answers
59 views

How to solve 3 variable in 2 equation?

This paper is abstracted from 2007 British Mathematics Olympiad Round 1 Question 2. I am currently practicing grade 8 (Singapore Secondary 2) for the upcoming Singapore Mathematics Olympiad(SMO). ...
1
vote
2answers
114 views

Accurate Formula and One Old-Exam Questions?! [closed]

We get stuck in a problem on old-exam. \begin{equation*} A=\sqrt{x+ \frac {2}{x}} -\sqrt{x- \frac {2}{x}}~\text{and}~x>>1. \end{equation*} For calculating $A$ which of the following option ...
-1
votes
1answer
21 views

How long will it take for one of them or both of them?

One knight can storm a castle in 15 days. He and his partner can do it in 10 days. How long does it take the partner to storm the same castle alone? Pipe A can fill a pool in 5 hours, while pipe B ...
2
votes
1answer
68 views

MOSP $2002$ Combinatorics Problem

I only want a hint(I already have the solution near me, but the book doesn't give a hint (MOSP) Assume that each of the $30$ MOPpers has exactly one favorite chess variant and exactly one ...
2
votes
0answers
17 views

Use Lagrange Interpolation polynomial to find this $\sum_{cyc}\frac{x^3}{(x^2-y^2)(x^2-z^2)}$

let $x,y,z$ are $t^3-t^2+2t-3=0$ three complex solution, find $$\dfrac{x^3}{(x^2-y^2)(x^2-z^2)}+\dfrac{y^3}{(y^2-x^2)(y^2-z^2)}+\dfrac{z^3}{(z^2-x^2)(z^2-y^2)}$$ How to use interpolation ...
0
votes
0answers
14 views

Logic problems : references

I'm looking for problems from mathematical contests about logic (similar to the problem PMWC Problem T5).
8
votes
1answer
57 views

Is it true that $\sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb N^+\text{ such that } a^2-b=k^2 $?

This is a curiosity question: Question Given two positive integers $a$ and $b$ do we have the following equivalence: $$\sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb ...
0
votes
2answers
67 views

Show that $f(a)$ converges after some point

There is a row of 1000 integers. There is a second row below, which is constructed as follows. Under each number $a$ of the first row, there is a positive integer $f(a)$ such that $f (a)$ equals ...