This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 3597

2015 Iran Team Selection Test, 1
Tags: algebra , polynomial
Find all polynomials $P,Q\in \Bbb{Q}\left [ x \right ]$ such that $$P(x)^3+Q(x)^3=x^{12}+1.$$

2004 Moldova Team Selection Test, 10
Tags: polynomial , algebra
Determine all polynomials $P(x)$ with real coeffcients such that $(x^3+3x^2+3x+2)P(x-1)=(x^3-3x^2+3x-2)P(x)$.

1975 IMO Shortlist, 10
Tags: algebra , polynomial , functional equation
Determine the polynomials P of two variables so that: [b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$ [b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$ [b]c.)[/b] $P(1,0) =1.$

2024 Canadian Junior Mathematical Olympiad, 5
Tags: polynomial , algebra
Let $N{}$ be the number of positive integers with $10$ digits $\overline{d_9d_8\cdots d_0}$ in base $10$ (where $0\le d_i\le9$ for all $i$ and $d_9>0$) such that the polynomial \[d_9x^9+d_8x^8+\cdots+d_1x+d_0\] is irreducible in $\Bbb Q$. Prove that $N$ is even. (A polynomial is irreducible in $\Bbb Q$ if it cannot be factored into two non-constant polynomials with rational coefficients.)

2011 Harvard-MIT Mathematics Tournament, 1
Tags: polynomial , algebra , inequalities
Let $a,b,c$ be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: $ax^2+bx+c, bx^2+cx+a,$ and $cx^2+ax+b $.

PEN F Problems, 2
Tags: trigonometry , rational number , polynomial , algebra
Find all $x$ and $y$ which are rational multiples of $\pi$ with $0<x<y<\frac{\pi}{2}$ and $\tan x+\tan y =2$.

2019 Vietnam TST, P2
Tags: polynomial , algebra
For each positive integer $n$, show that the polynomial: $$P_n(x)=\sum _{k=0}^n2^k\binom{2n}{2k}x^k(x-1)^{n-k}$$ has $n$ real roots.

2002 USAMTS Problems, 2
Tags: polynomial , algebra
Find four distinct positive integers, $a$, $b$, $c$, and $d$, such that each of the four sums $a+b+c$, $a+b+d$,$a+c+d$, and $b+c+d$ is the square of an integer. Show that infinitely many quadruples $(a,b,c,d)$ with this property can be created.

2015 Indonesia MO Shortlist, A2
Tags: natural number , polynomial , algebra , functional equation
Suppose $a$ real number so that there is a non-constant polynomial $P (x)$ such that $\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}$ for each real number $x$, with $x+\pi \ne 0$ and $P(x+\pi)\ne 0$. Show that $a$ is a natural number.

1985 IMO, 6
Tags: fixed point , sequence , calculus , polynomial , limit
For every real number $x_1$, construct the sequence $x_1,x_2,\ldots$ by setting: \[ x_{n+1}=x_n(x_n+{1\over n}). \] Prove that there exists exactly one value of $x_1$ which gives $0<x_n<x_{n+1}<1$ for all $n$.

1977 AMC 12/AHSME, 28
Tags: polynomial , algebra
Let $g(x)=x^5+x^4+x^3+x^2+x+1$. What is the remainder when the polynomial $g(x^{12})$ is divided by the polynomial $g(x)$? $\textbf{(A) }6\qquad\textbf{(B) }5-x\qquad\textbf{(C) }4-x+x^2\qquad$ $\textbf{(D) }3-x+x^2-x^3\qquad \textbf{(E) }2-x+x^2-x^3+x^4$

2005 China Team Selection Test, 3
Tags: computer solutions , 4-variable inequality , symmetric inequality , polynomial , algebra , inequalities
Let $a,b,c,d >0$ and $abcd=1$. Prove that: \[ \frac{1}{(1+a)^2}+\frac{1}{(1+b)^2}+\frac{1}{(1+c)^2}+\frac{1}{(1+d)^2} \geq 1 \]

1999 Putnam, 3
Tags: modular arithmetic , polynomial , limit , algebra
Let $A=\{(x,y): 0\le x,y < 1\}.$ For $(x,y)\in A,$ let \[S(x,y)=\sum_{\frac12\le\frac mn\le2}x^my^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\to(1,1),(x,y)\in A}(1-xy^2)(1-x^2y)S(x,y).\]

1988 IMO Longlists, 42
Tags: vieta , inequality , root , polynomial , algebra
Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

2002 Moldova Team Selection Test, 1
Tags: polynomial , algebra
Prove that for every positive integer n, there exists a polynomial p(x) with integer coefficients such that p(1), p(2),..., p(n-1), p(n) are distinct powers of 2.

III Soros Olympiad 1996 - 97 (Russia), 10.2
Tags: number theory , polynomial , algebra
It is known that the equation $x^3 + px^2 + q = 0$ where $q$ is non-zero, has three different integer roots, the absolute values of two of which are prime numbers. Find the roots of this equation.

1957 Miklós Schweitzer, 9
Tags: polynomial , algebra
[b]9.[/b] Find all pairs of linear polynomials $f(x)$, $g(x)$ with integer coefficients for which there exist two polynomials $u(x)$, $v(x)$ with integer coefficients such that $f(x)u(x)+g(x)v(x)=1$. [b](A. 8)[/b]

1922 Eotvos Mathematical Competition, 2
Tags: algebra , polynomial
Prove that $$x^4 + 2x^2 + 2x + 2$$ is not the product of two polynomials $x^2 + ax + b$ and $x^2 + cx + d$ in which $a$, $b$, $c$, $d$ are integers.

1987 AMC 12/AHSME, 1
Tags: algebra , polynomial
$(1+x^2)(1-x^3)$ equals $ \text{(A)}\ 1 - x^5\qquad\text{(B)}\ 1 - x^6\qquad\text{(C)}\ 1+ x^2 -x^3\qquad \\ \text{(D)}\ 1+x^2-x^3-x^5\qquad \text{(E)}\ 1+x^2-x^3-x^6 $

1988 IMO Shortlist, 7
Tags: divisibility , modular arithmetic , number theory , polynomial , algebra
Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$

2001 Moldova Team Selection Test, 7
Tags: polynomial
Let $(P_n(X))_{n\in\mathbb{N}}$ be a sequence of polynomials defined as: $P_1(X)=X-1, P_2(X)=X^2-X-1, P_n(X)=XP_{n-1}(X)-P_{n-2}(X), \forall n>2$. For every nonnegative integer $n{}$ find all roots of the polynomial $P_n(X)$.

1996 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 7
Tags: polynomial , algebra
If 1,2, and 3 are solutions to the equation $ x^4 \plus{} ax^2 \plus{} bx \plus{} c \equal{} 0,$ then $ a\plus{}c$ equals A. -12 B. 24 C. 35 D. -61 E. -63

2018 Baltic Way, 5
Tags: polynomial , algebra
A polynomial $f(x)$ with real coefficients is called [i]generating[/i], if for each polynomial $\varphi(x)$ with real coefficients there exists a positive integer $k$ and polynomials $g_1(x),\dotsc,g_k(x)$ with real coefficients such that \[\varphi(x)=f(g_1(x))+\dotsc+f(g_k(x)).\] Find all generating polynomials.

2019 LIMIT Category A, Problem 1
Tags: algebra , polynomial
Let $p(x)$ be a polynomial of degree $4$ with leading coefficient $1$. Suppose $p(1)=1$, $p(2)=2$, $p(3)=3$ and $p(4)=4$. Then $p(5)=$? $\textbf{(A)}~5$ $\textbf{(B)}~\frac{25}6$ $\textbf{(C)}~29$ $\textbf{(D)}~35$

2015 JBMO Shortlist, A2
Tags: algebra , polynomial
If $x^3-3\sqrt3 x^2 +9x - 3\sqrt3 -64=0$ find the value of $x^6-8x^5+13x^4-5x^3+49x^2-137x+2015$ .