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

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1answer
66 views

Showing that $(a^2-a+1)(b^2-b+1)(c^2-c+1) \leq 7$

How can I show that $(a^2-a+1)(b^2-b+1)(c^2-c+1) \leq 7$ given that $a+b+c = 3?$ Attempt: Setting $x=2a-1, y=2b-1, z=2c-1,$ we obtain that $[(2a-1)^2+3][(2b-1)^2+3][(2c-1)^2+3] \leq 448,$ $s=x+y+z ...
0
votes
1answer
43 views

Midpoint of a set as Mean? [closed]

Given a set of an odd number of terms: $x = \{a, b, c, ..., \}$ Consisting of $n$ elements. How is the midpoint of the set. A proof and explanation would be helpful: $$\frac{a + b + c + ... }{n} ...
3
votes
1answer
86 views

Problem from Olympiad from book Arthur Engel

Each of the numbers $a_1 ,a_2,\dots,a_n$ is $1$ or $−1$, and we have $$S=a_1a_2a_3a_4+a_2a_3a_4a_5 +\dots+ a_na_1a_2a_3=0$$ Prove that $4 \mid n$. If we replace any a i by −a i , then S does not ...
1
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2answers
60 views

Prove the coefficient of $x^2$ is the sum.

In the expansion: $(1 + ax)(1 + bx)(1+cx) \cdots$ find the general coefficient of $x^2$ and prove the formula. Consider $(1 + ax)(1 + bx)$, the coefficient of $x^2$ is: $ab$. Consider $(1 + ...
2
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3answers
185 views

Coefficient Problem (polynomial expansion)

Let $C$ be the coefficient of $x^2$ in the expansion of the product $(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x).$ Find $|C|.$ Just to begin, $(1-x)(1+2x) = -2x^2 + x + 1$ ...
4
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1answer
68 views

If $\lfloor x^i\rfloor =i,i=1,2,3,\cdots,n$ find the maximum of $n$

Find the maximum $n$ for which there exist a real number $x$ such that $$\lfloor x^i\rfloor =i,\quad i=1,2,3,\ldots,n.$$ $\lfloor x\rfloor =1$,then $1<x<2$, $\lfloor x^2\rfloor =2$ then ...
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3answers
133 views

BMO2 Question 1 2015 Iterative Formula Problem

$1.$ The first term $x_{1}$ of a sequence is $2014$. Each subsequent term of the sequence is defined in terms of the previous term. The iterative formula is $x_{n+1} = \frac{(\sqrt2 + 1)x_{n} − 1} ...
2
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1answer
56 views

Find probability of intersection of shortest arcs on sphere

There are four points $A$, $B$, $C$, $D$ which randomly selected on a sphere. Find probability of intersection of shortest arcs (not circles) $AB$ and $CD$. Shortest arc is a intersection of ...
2
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2answers
59 views

Find sum with binomial coefficients and powers of 2

Find this sum for positive $n$ and $m$: $$S(n, m) = \sum_{i=0}^n \frac{1}{2^{m+i+1}}\binom{m+i}{i} + \sum_{i=0}^m \frac{1}{2^{n+i+1}}\binom{n+i}{i}.$$ Obviosly, $S(n,m)=S(m,n)$. Therefore I've tried ...
3
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2answers
55 views

First contest problem

I downloaded a contest and worked the first problem which is: There exists a digit Y such that, for any digit X, the seven-digit number 1 2 3 X 5 Y 7 is not a multiple of 11. Compute Y. My ...
2
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3answers
55 views

Expansion of Generating Functions

If you roll $10$ dice, how many ways can you get a total sum of top faces of $25$? I understand how to write the generating function of $(x+x^2+ \dots +x^6)^{10}$ and the fact that you need to find ...
4
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1answer
65 views

Prove $(a^2+b^2+c^2+d^2)^2≥(a+b)(b+c)(c+d)(d+a)$

I've been unsuccessfully trying to solve this contest-style problem for a while. Tried different substitutions and the such, but nothing helped. I presume the solution is related to Cauchy-Schwarz? ...
2
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0answers
58 views

Proof Verification for Putnam Problem [Alternate Solution] 1997 A4

I have come across an interesting problem from the Putnam 1997 test, question A4: Problem: Let $G$ be a group with identity $e$ and $\phi: G \to G$ a mapping such that $\phi(g_1)\phi(g_2)\phi(g_3) = ...
2
votes
2answers
48 views

Sine Cosine Sequence?

Two real sequences $\{x\}$ and $\{y\}$ satisfy $$x_{n+2}=x_nx_{n+1}-y_ny_{n+1},$$ $$y_{n+2}=x_ny_{n+1}+y_nx_{n+1}.$$ Given $x_1=y_1=1/\sqrt 2$ and $x_2=y_2=1$, find closed forms of $x_n$ and $y_n$. ...
7
votes
3answers
135 views

Functions proof.

Find all functions $$f: \mathbb{Z} \rightarrow \mathbb{Z}$$ such that $$f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a)$$ for all integers $$a, b, c$$ satisfying $$a+b+c=0$$ I have no idea how to ...
10
votes
1answer
158 views

Show that a matrix has positive determinant

For a natural number $i>0$, let $p_i$ be the $i$th prime number, that is, $p_1=2, p_2=3, p_3=5,...$. Show that for all $n$, the following matrix has positive determinant $$ \begin{pmatrix} ...
2
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1answer
22 views

Denote $Z$ as the set of points in $\mathbb{R}^n$ whose coordinates are $0$ or $1$. Find the maximum, of the number of points in $Z \cap V$.

Denote $Z$ as the set of points in $\mathbb{R}^n$ whose coordinates are $0$ or $1$. Let $k$ be given, $0 \leq k \leq n$. Find the maximum, over all vector subspaces $V \subset \mathbb{R}^n$ of ...
2
votes
1answer
37 views

Placing $n$ points so that their distances lie in $[1,a]$

What is the maximum number of points we can place in the plane so that the distance between any two such points is in the interval $[1,a]$? I had initially conjectured that the maximum could be ...
3
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1answer
84 views

Non decreasing real function satisfying $f(x)=f(x+1)$ and/or $f(x)=f(x-1)$. [closed]

Let $f:\mathbb R\to\mathbb R$ be a non-decreasing function. For all $x\in\mathbb R$ we have $(f(x)-f(x-1))(f(x+1)-f(x))=0$. What can we say about the function? [EDITED]
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3answers
77 views

find the value of $\sqrt {(98 \times 100+2)(100\times102+2)+(100\times2)^2}$

From: $2015$ Singapore Mathematical Olympiad Secondary 2 (Grade 8) Question 21 Round 1 on 3rd June. Find the value of $\sqrt {(98 \times 100+2)(100\times102+2)+(100\times2)^2}$ (No use of ...
3
votes
2answers
69 views

Can the expression be made true after replacing the blanks in this sum with 2 numbers from 0-9 and 2 symbols? [closed]

The expression is $$ 2\_\,\_\,\_\,\_5=2015 $$ you have to replace 2 of the blanks with digits (0-9), and the other 2 with one of the operations $+- \times \div$. Is it possible to make the ...
31
votes
4answers
979 views

Is it true that this function $f(n)=n^{13}$?

Assume strictly monotone increasing function; such that $f:N^{+}\to N^{+}$, $h$ for all $n\in N^{+}$, $$f(f(f(n)))=f(f(n))\cdot f(n)\cdot n^{2015}$$ Prove or disprove:$f(n)=n^{13}$ ...
3
votes
1answer
58 views

Do there exist continuous functions $f,g: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(g(x))=x^2$ and $g(f(x))=x^3$ for all $x \in \mathbb{R}$?

Do there exist continuous functions $f,g: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(g(x))=x^2$ and $g(f(x))=x^3$ for all $x \in \mathbb{R}$? My attempt: Since $x^3$ is a bijection, we have $f$ ...
6
votes
1answer
81 views

A runs 7/4 times as fast as B. If A gives B a start of 84m, how far must the winning post be…?

The problem statement in the book is: $A$ runs $7/4$ times as fast as $B$. If $A$ gives $B$ a start of $84$m, how far must the winning post be so that $A$ and $B$ might reach it at the same time? ...
1
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0answers
91 views

The minute hand of a clock overtakes the hour hand at intervals of 65 minutes of correct time. How much a day does the clock gain?

The question in the textbook is: The minute hand of a clock overtakes the hour hand at intervals of 65 minutes of correct time. How much a day does the clock gain? My method: The correct ...
1
vote
1answer
31 views

If $P(x)$ and $Q(x)$ are both factors of $H(x)$

$P(x)$ and $Q(x)$ are both quadratic polynomials and both are factors of a cubic polynomial $H(x)$ such that: $$H(x) = (x - a)P(x) \space \text{AND} \space H(x) = (x - b)Q(x)$$ For distinct $a,b$ ...
1
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1answer
64 views

Finding all solutions to $x^2+y^2=2010$

I need to find all integer solutions to $x^2+y^2=2010$. we can take $x\leq y$ for commodity. The problem can be tackled through brute force. We need $1005\leq x^2\leq 2010$ and so $32\leq x \leq ...
10
votes
1answer
67 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 ...
4
votes
3answers
110 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 ...
39
votes
0answers
2k views

If the decimal expansion of $a/b$ contains “$7143$” then $b>1250$

I recently stumbled upon this really interesting problem: If we have a fraction $\frac{a}{b}$ where $a,b \in \mathbb{N}$ and we know that the decimal fraction of $\frac{a}{b}$ has the numerical ...
1
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1answer
53 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
43 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, ...
1
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2answers
117 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
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1answer
36 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 ...
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2answers
29 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
58 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
53 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 ...
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3answers
80 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
68 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 ...
3
votes
1answer
116 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
69 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 ...
0
votes
1answer
66 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
45 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
21 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
2answers
87 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
29 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 ...
5
votes
2answers
255 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
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
84 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
23 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 ...