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

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1answer
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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
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4answers
38 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 ...
34
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2answers
689 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
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1answer
51 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
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4answers
72 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 ...
21
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3answers
372 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 $C=1$, ...
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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
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1answer
65 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 $c$ ...
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0answers
40 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
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1answer
43 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
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2answers
79 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
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2answers
63 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
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1answer
80 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, ...
3
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2answers
88 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
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2answers
34 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
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1answer
22 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 ...
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2answers
60 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 ...
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3answers
67 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
45 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 $f(x)=1$ ...
14
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6answers
251 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
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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 ...
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38 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
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3answers
127 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
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1answer
66 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 ...
6
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1answer
71 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, ...
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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 & ...
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2answers
51 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
83 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 + ...
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2answers
38 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
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0answers
94 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, ...
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1answer
80 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 $$\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.$$ ...
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1answer
65 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 ...
6
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1answer
298 views

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 ...
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0answers
39 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
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1answer
52 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? ...
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2answers
40 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 ...
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2answers
223 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). ...
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2answers
118 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 ...
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1answer
35 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 ...
3
votes
3answers
559 views

Finding the number of real roots of an unusual(!) equation [closed]

How many real roots does the below equation have? \begin{equation*} \frac{x^{2000}}{2001}+2\sqrt{3}x^2-2\sqrt{5}x+\sqrt{3}=0 \end{equation*} A) 0 B) 11 C) 12 D) 1 E) None of these I could not come ...
2
votes
1answer
76 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 ...
4
votes
2answers
122 views

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

Let $x,y,z$ be the solutions of the equation $t^3-t^2+2t-3=0$. Find the sum $$\dfrac{x^3}{(x^2-y^2)(x^2-z^2)}+\dfrac{y^3}{(y^2-x^2)(y^2-z^2)}+\dfrac{z^3}{(z^2-x^2)(z^2-y^2)}$$ How can I use the ...
0
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0answers
17 views

Logic problems : references

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

Is it true that $\sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb N^+$ 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
68 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 ...
2
votes
2answers
387 views

Making change with prime-valued coins

Am I understanding this question correctly and how do I approach these problems? In Numberland, the unit of currency is the El (E). The value of each Numberlandian coin is a prime number of Els. So ...
3
votes
1answer
52 views

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits.

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits. It is an intuitive re-statement of Baltic Way 2012 (I think there are shortlists in Baltic Way every ...
1
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1answer
36 views

Grid Problem Proof

I have a 2x2 grid square say, I can fit a shape like this: Such that there is one missing square. I can arrange this in any way so that the missing square can be located anywhere. I can do ...
2
votes
2answers
41 views

Cute convergence problem. Proving convergence of sequence regarding reciprocals of least common multiple converges.

This is the first problem of the second day of the $2014$ CIIM. Let $\{a_n\}$ be a strictly increasing sequence of positive integers. Prove the sequence ...
5
votes
1answer
75 views

Rocks and squares, balls and sticks. [closed]

Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, ...