Found problems: 3597
2024-25 IOQM India, 24
Consider the set $F$ of all polynomials whose coefficients are in the set of $\{0,1\}$. Let $q(x) = x^3 + x +1$. The number of polynomials $p(x)$ in $F$ of degree $14$ such that the product $p(x)q(x)$ is also in $F$ is:
1984 Miklós Schweitzer, 7
[b]7.[/b] Let $V$ be a finite-dimensional subspace of $C[0,1]$ such that every nonzero $f\in V$ attains positive value at some point. Prove that there exists a polynomial $P$ that is strictly positive on $[0,1]$ and orthogonal to $V$, that is, for every $f \in V$,
$\int_{0}^{1} f(x)P(x)dx =0$
([b]F.39[/b])
[A. Pinkus, V. Totik]
2012 India Regional Mathematical Olympiad, 6
Let $a$ and $b$ be real numbers such that $a \ne 0$. Prove that not all the roots of $ax^4 + bx^3 + x^2 + x + 1 = 0$ can be real.
2008 AIME Problems, 11
Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. Examples of such sequences are $ AA$, $ B$, and $ AABAA$, while $ BBAB$ is not such a sequence. How many such sequences have length 14?
1955 AMC 12/AHSME, 19
Two numbers whose sum is $ 6$ and the absolute value of whose difference is $ 8$ are roots of the equation:
$ \textbf{(A)}\ x^2\minus{}6x\plus{}7\equal{}0 \qquad
\textbf{(B)}\ x^2\minus{}6x\minus{}7\equal{}0 \qquad
\textbf{(C)}\ x^2\plus{}6x\minus{}8\equal{}0 \\
\textbf{(D)}\ x^2\minus{}6x\plus{}8\equal{}0 \qquad
\textbf{(E)}\ x^2\plus{}6x\minus{}7\equal{}0$
1987 IMO Longlists, 78
Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$,
\[[r^m] \equiv -1 \pmod k .\]
[i]Remark.[/i] An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients.
[i]Proposed by Yugoslavia.[/i]
2001 Stanford Mathematics Tournament, 14
Find the prime factorization of $\textstyle\sum_{1\le i < j \le 100}ij$.
2017 All-Russian Olympiad, 5
$P(x)$ is polynomial with degree $n\geq 2$ and nonnegative coefficients. $a,b,c$ - sides for some triangle. Prove, that $\sqrt[n]{P(a)},\sqrt[n]{P(b)},\sqrt[n]{P(c)}$ are sides for some triangle too.
2001 IMC, 1
Let $r, s \geq 1$ be integers and $a_{0}, a_{1}, . . . , a_{r-1}, b_{0}, b_{1}, . . . , b_{s-1} $ be real non-negative numbers such that $(a_0+a_1x+a_2x^2+. . .+a_{r-1}x^{r-1}+x^r)(b_0+b_1x+b_2x^2+. . .+b_{s-1}x^{s-1}+x^s) =1 +x+x^2+. . .+x^{r+s-1}+x^{r+s}$.
Prove that each $a_i$ and each $b_j$ equals either $0$ or $1$.
2009 Today's Calculation Of Integral, 428
Let $ f(x)$ be a polynomial and $ C$ be a real number.
Find the $ f(x)$ and $ C$ such that $ \int_0^x f(y)dy\plus{}\int_0^1 (x\plus{}y)^2f(y)dy\equal{}x^2\plus{}C$.
1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 10
Let $ p(x) \equal{} x^6 \plus{} ax^5 \plus{} bx^4 \plus{} cx^3 \plus{} dx^2 \plus{} ex \plus{} f$ be a polynomial such that $ p(1) \equal{} 1, p(2) \equal{} 2, p(3) \equal{} 3, p(4) \equal{} 4, p(5) \equal{} 5,$ and $ p(6) \equal{} 6.$ What is $ p(7)$?
A. 0
B. 7
C. 14
D. 49
E. 727
2022 Belarusian National Olympiad, 9.8
Does there exist a polynomial $p(x)$ with integer coefficients for which
$$p(\sqrt{2})=\sqrt{2}$$
$$p(2\sqrt{2})=2\sqrt{2}+2$$
2008 China Team Selection Test, 2
Let $ x,y,z$ be positive real numbers, show that $ \frac {xy}{z} \plus{} \frac {yz}{x} \plus{} \frac {zx}{y} > 2\sqrt [3]{x^3 \plus{} y^3 \plus{} z^3}.$
1987 IMO Longlists, 35
Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ?
[i]Proposed by Hungary.[/i]
[hide="Remark"]I'm not sure I'm posting this in a right Forum.[/hide]
2006 China Team Selection Test, 3
Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is:
\[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\]
Find the least possible value of $n$.
2003 District Olympiad, 3
Let $\displaystyle \mathcal K$ be a finite field such that the polynomial $\displaystyle X^2-5$ is irreducible over $\displaystyle \mathcal K$. Prove that:
(a) $1+1 \neq 0$;
(b) for all $\displaystyle a \in \mathcal K$, the polynomial $\displaystyle X^5+a$ is reducible over $\displaystyle \mathcal K$.
[i]Marian Andronache[/i]
[Edit $1^\circ$] I wanted to post it in "Superior Algebra - Groups, Fields, Rings, Ideals", but I accidentally put it here :blush: Can any mod move it? I'd be very grateful.
[Edit $2^\circ$] OK, thanks.
2007 Today's Calculation Of Integral, 246
An eighth degree polynomial funtion $ y \equal{} ax^8 \plus{} bx^7 \plus{} cx^6 \plus{} dx^5 \plus{} ex^4 \plus{} fx^3 \plus{} gx^2\plus{}hx\plus{}i\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma ,\ \delta \ (\alpha < \beta < \gamma <\delta).$
Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma ,\ \delta .$
1956 Poland - Second Round, 1
For what value of $ m $ is the polynomial $ x^3 + y^3 + z^3 + mxyz $ divisible by $ x + y + z $?
2010 Indonesia TST, 1
Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number?
[i]Nanang Susyanto, Jogjakarta[/i]
2018 SG Originals, Q5
Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c,$ $$P(a,b,c)=0 \Leftrightarrow a=b=c.$$
Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $m,n,$ $$m^r\mid P(n,n+m,n+2m).$$
[i]Proposed by Ma Zhao Yu
PEN M Problems, 27
Let $ p \ge 3$ be a prime number. The sequence $ \{a_{n}\}_{n \ge 0}$ is defined by $ a_{n}=n$ for all $ 0 \le n \le p-1$, and $ a_{n}=a_{n-1}+a_{n-p}$ for all $ n \ge p$. Compute $ a_{p^{3}}\; \pmod{p}$.
2014 IberoAmerican, 2
Find all polynomials $P(x)$ with real coefficients such that $P(2014) = 1$ and, for some integer $c$:
$xP(x-c) = (x - 2014)P(x)$
2012 USA TSTST, 6
Positive real numbers $x, y, z$ satisfy $xyz+xy+yz+zx = x+y+z+1$. Prove that \[ \frac{1}{3} \left( \sqrt{\frac{1+x^2}{1+x}} + \sqrt{\frac{1+y^2}{1+y}} + \sqrt{\frac{1+z^2}{1+z}} \right) \le \left( \frac{x+y+z}{3} \right)^{5/8} . \]
Russian TST 2014, P2
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
2004 Polish MO Finals, 2
Let $ P$ be a polynomial with integer coefficients such that there are two distinct integers at which $ P$ takes coprime values. Show that there exists an infinite set of integers, such that the values $ P$ takes at them are pairwise coprime.