3
votes
1answer
75 views

Solve in $\mathbb{R}$: $(x^2-3x-2)^2-3(x^2-3x-2)-2-x=0$

Solve in $\mathbb{R}$: $(x^2-3x-2)^2-3(x^2-3x-2)-2-x=0$ I'm supposed to solve this equation. It's from a math contest so solving it by hand would be preferable (no quartic formulas). I thought ...
1
vote
4answers
77 views

Solve the following equation: $\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$

Solve the following equation: $$\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$$ I know it's from a Math Olympiad but I don't know which and I couldn't find it on the internet. Expanding everything ...
0
votes
2answers
44 views

Roots Of Monic Cubic

I'm currently preparing for the USA Mathematical Talent Search competition. I've been brushing up my proof-writing skills for several weeks now, but one area that I have not been formally taught about ...
6
votes
1answer
91 views

Finding all such polynomials under a gcd condition

Find all such polynomial $f(x)\in \mathbb{Z}[x]$ such that $$ \forall n\in \mathbb{N} \quad \gcd(f(n),f(2^n))=1$$ This is a problem from the Indian Team Selection Test. Can someone give me a solution ...
4
votes
0answers
33 views

Set of Metapolynomials is closed under multiplication

We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integer $m$ and $n$, it can be represented in the form $$f(x_1,\cdots , x_k ...
0
votes
1answer
57 views

Understanding 2012 AMC 12B #23

Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23$, $-21$, $-17$, and $-15$, and $Q(P(x))$ has zeros at $x=-59$,$-57$,$-51$ and $-49$. What is ...
0
votes
2answers
53 views

Polynomial $P(x)$ with degree $1998$

$P(x)$ is a polynomial of degree 1998 such that $P(k) = \frac{1}{k} $ for all values of $k = 1,2,3,...,1999$. What is the value of $P(2000)$? I did try to substitute as $k = 2000$ but the highest ...
4
votes
1answer
96 views

determine all polynomials $P(x)$ such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial. clearly we have to show $(x+1)P(x-1)-(x-1)P(x)=c$ for all values of $x$ ($c$ is a ...
2
votes
2answers
201 views

Solve exponential-polynomial equation

Solve the equation in $\mathbb{R}$ $$10^{-3}x^{\log_{10}x} + x(\log_{10}^2x - 2\log_{10} x) = x^2 + 3x$$ To be fair I wasn't able to make any progress. I tried using substitution for the ...
5
votes
1answer
89 views

P0lyn0mial questi0n

Suppose $P(x)$ is a polynomial of degree $2012$ and $P(x) = 1/x$ when $x$ takes the integer values $1\cdots2013$ (inclusive). What is the value of $P(2014)$? I get $1/1007$ but I'm not sure if ...
8
votes
1answer
382 views

IMO 1979 problem

The question is $$\text{If }\, p, \ q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad \text{Prove that } 1979\mid p.$$ So my solution went like ...
2
votes
1answer
152 views

Iran Math Olympiad 2012 (perfect power)

Prove that if $t$ is a natural number then there exists a natural number $n > 1$ such that $(n, t) = 1$ and none of the numbers $n + t, n^2 + t, n^3 + t…$ are perfect powers. There is a solution ...
7
votes
3answers
209 views

All roots of the quartic equation $a x^4 + b x^3 + x^2 + x + 1 = 0$ cannot be real

Problem Prove that all roots of $a x^4 + b x^3 + x^2 + x + 1 = 0$ cannot be real. Here $a,b \in \mathbb R$, and $a \neq 0$. Source This is one of the previous year problem of Regional ...
4
votes
3answers
148 views

Find all polynomials $P(x)$ such that $2xP(x)=(x+1)P(x-1)+(x-1)P(x+1)$.

Find all polynomials $P(x)$ such that $2xP(x)=(x+1)P(x-1)+(x-1)P(x+1)$. Well, if $\deg P\le 3$ this is easy since we can deduce $P(0)=P(1)=P(-1)$ by letting $x=0,1,-1$
0
votes
1answer
109 views

Polynomials as sum of squares

Sometimes I have seen some math's competition problem solutions made by completing the expression as sum of squares. What is the intuition/computer program behind these solutions? For example, Prove ...
4
votes
1answer
83 views

Minimum difference of roots of a polynomial and its derivative

Let $P(x) = (x-x_1)(x-x_2)...(x-x_n)$ where all the n roots are real and distinct. Let $y_1,y_2,...,y_{n-1}$ be the roots of $P'$. Show that $\min_{i\neq j}|x_i-x_j|<\min_{i\neq j}|y_i-y_j|$. My ...
3
votes
5answers
402 views

What is the value of $f(0)+f(8)$?

Suppose $f$ is a polynomial of degree $7$ which satisfies $f(1) =2$, $f(2)=5$, $f(3)=10$, $f(4)=17$, $f(5)=26$, $f(6)=37$ and $f(7)=50$. What is the value of $f(0)+f(8)$?
8
votes
1answer
218 views

How to prove there exists a polynomial with degree at most $100\sqrt{nk}$ satisfying this condition

Show that for arbitrary positive integers $n,k$, there exists a polynomial $p(x)$, with degree at most $100\sqrt{nk}$, such that ...
1
vote
2answers
63 views

$P(x)\in\mathbb Z$ iff $Q(x)\in\mathbb Z$

Well I have a problem on polynomial, it said like that: Let $P,Q$ be polynomials with real coefficients (that is $P,Q\in\mathbb R[x]$). We assume that for every $x\in\mathbb R$ then $P(x)\in \mathbb ...
0
votes
1answer
263 views

Given a polynomial $P$ find $Q$ such that $Q(x)-Q(x-1)-Q(x-2)=P(x)$ for all $x$

Let $P\in\mathbb{Z}[x]$ be a given polynomial of degree $d$. I want to find the unique polynomial $Q\in\mathbb{Z}[x]$ of degree $d$ such that $Q(x)-Q(x-1)-Q(x-2)=P(x)$. It is possible to construct the ...
3
votes
1answer
431 views

Throw a die three times, and get maximum number of different sums.

The IBM Ponder This problem for July 2013 throws an 8 sided die 3 times, and can get 120 possible different positive integer sums. If all the faces have positive integer sides, what is the lowest ...
6
votes
4answers
296 views

Polynomials Question: Proving $a=b=c$.

Question: Let $P_1(x)=ax^2-bx-c, P_2(x)=bx^2-cx-a \text{ and } P_3=cx^2-ax-b$ , where $a,b,c$ are non zero reals. There exists a real $\alpha$ such that ...
4
votes
1answer
116 views

Polynomial differential equation

I came across this problem in an old olympiad paper (Putnam?) Find all polynomials $p(x)$ with real coefficients satisfying the differential equation $7\dfrac{d }{dx } [xp(x)]=3p(x)+4p(x+1)$ $\ \ ...
4
votes
1answer
107 views

Lowest degree polynomial vanishing on the integers mod $n$?

This problem comes from D.J. Newman's A Problem Seminar. Problem: What is the lowest degree monic polynomial $p(x)$ such that the value of $p(x)$ is divisible by $100$ whenever $x$ is an integer? ...
13
votes
1answer
174 views

Polynomial $P(a)=b,P(b)=c,P(c)=a$

Let $a,b,c$ be $3$ distinct integers, and let $P$ be a polynomial with integer coefficients.Show that in this case the conditions $$P(a)=b,P(b)=c,P(c)=a$$ cannot be satisfied simultaneously. Any hint ...
-2
votes
1answer
157 views

From a Generating Function find $R(x)$ as an infinite product of Quotients

Let $r(n)$ be the number of partitions of $n$ so that no multiple of $3$ appears as a part. For example, $r(8) = 13$. Let $R(x) =\sum_0^\infty r(n)x^ n $ be the generating function for $r(n)$. Find ...
4
votes
1answer
121 views

Prove that the polynomial $P(x_1,x_2…,x_n)=0$ given a set of conditions.

Let $P(x_1,...,x_n)\in\mathbb{R}[x_1,...,x_n]$ (i.e. $P$ is a polynomial of real coefficients in $x_1,..,x_n$). We are given that $\left(\frac{\partial^2}{\partial ...
1
vote
1answer
435 views

Polynomial of polynomials (from Brilliant.org)

Moderator Note: This is a current question on brilliant.org if f(x) is a polynomial satisfying $$27 f(x^3) -4f(x^2) - x^6 f(3x) + 46 = 0 ,$$ what is $f(10)$? We can get $$f(0)=-2,\quad ...
23
votes
6answers
2k views

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
6
votes
2answers
345 views

Help me solve this olympiad challenge?

Given: $$p(x) = x^4 - 5773x^3 - 46464x^2 - 5773x + 46$$ What is the sum of all arctan of all the roots of $p(x)$?
1
vote
2answers
252 views

Ordered quadruples in a grid

Moderator Note: This question is from a contest which ended on 22 Oct 2012. Consider $(\alpha_1, \alpha_2, \alpha_3, \alpha_4)$ such that the ordered quadruple satisfies the following: ...
1
vote
1answer
293 views

What is shortcut to this contest algebra problem about polynomial?

The polynomial $P(x)=x^4 + ax^3 + bx^2 +cx + d$ has the property that $p(k)=11k$ for $k=1,2,3,4$. Compute $c$. The answer is $-39$.
1
vote
2answers
102 views

Interscholastic Mathematics League Senior B #12

Compute the product of the nonreal roots of the equation $x^4+4x^3+6x^2+1004x+1001=0$. So here is what I have done so far. I got two of the roots to be zero and 4 since ...
1
vote
1answer
78 views

Interscholastic Mathematic League Senior B Division #11

The roots of the equation 3x^3-38x^2+cx-192=0 form a geometric progression. Compute c.
2
votes
2answers
151 views

Four polynomials with single-rooted sums

From a 2005 Russian olympiad. Prove that there do not exist four (pairwise) different quadratic polynomials, with leading coefficient 1, such that the sum of any two of them has a single root. ...
11
votes
2answers
261 views

A polynomial determined by two values

From a St. Petersburg school olympiad, 11th grade. Prove or disprove: a non constant polynomial $P$ with non-negative integer coefficients is uniquely determined by its values $P(2)$ and $P(P(2))$.