Found problems: 3597
2001 Moldova National Olympiad, Problem 4
Let $P(x)=x^n+a_1x^{n-1}+\ldots+a_n$ ($n\ge2$) be a polynomial with integer coefficients having $n$ real roots $b_1,\ldots,b_n$. Prove that for $x_0\ge\max\{b_1,\ldots,b_n\}$,
$$P(x_0+1)\left(\frac1{x_0-b_1}+\ldots+\frac1{x_0-b_n}\right)\ge2n^2.$$
2015 Postal Coaching, Problem 2
Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial
\[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n.
\]
2008 AIME Problems, 9
Ten identical crates each of dimensions $ 3$ ft $ \times$ $ 4$ ft $ \times$ $ 6$ ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $ \frac{m}{n}$ be the probability that the stack of crates is exactly $ 41$ ft tall, where $ m$ and $ n$ are relatively prime positive integers. Find $ m$.
2010 Tuymaada Olympiad, 4
Prove that for any positive real number $\alpha$, the number $\lfloor\alpha n^2\rfloor$ is even for infinitely many positive integers $n$.
2017 AIME Problems, 8
Find the number of positive integers $n$ less than $2017$ such that
\[ 1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+\frac{n^4}{4!}+\frac{n^5}{5!}+\frac{n^6}{6!} \]
is an integer.
2015 AIME Problems, 10
Let $f(x)$ be a third-degree polynomial with real coefficients satisfying
\[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\] Find $|f(0)|$.
2009 Germany Team Selection Test, 3
Find all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that $ \forall x,y,z \in \mathbb{R}$ we have: If
\[ x^3 \plus{} f(y) \cdot x \plus{} f(z) \equal{} 0,\]
then
\[ f(x)^3 \plus{} y \cdot f(x) \plus{} z \equal{} 0.\]
2013 ELMO Problems, 5
For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$?
[i]Proposed by Andre Arslan[/i]
2000 AIME Problems, 13
The equation $2000x^6+100x^5+10x^3+x-2=0$ has exactly two real roots, one of which is $\frac{m+\sqrt{n}}r,$ where $m, n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r>0.$ Find $m+n+r.$
2017 Estonia Team Selection Test, 11
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
PEN A Problems, 4
If $a, b, c$ are positive integers such that \[0 < a^{2}+b^{2}-abc \le c,\] show that $a^{2}+b^{2}-abc$ is a perfect square.
PEN G Problems, 7
Show that $ \pi$ is irrational.
2002 Korea - Final Round, 1
For $n \ge 3$, let $S=a_1+a_2+\cdots+a_n$ and $T=b_1b_2\cdots b_n$ for positive real numbers $a_1,a_2,\ldots,a_n, b_1,b_2 ,\ldots,b_n$, where the numbers $b_i$ are pairwise distinct.
(a) Find the number of distinct real zeroes of the polynomial
\[f(x)=(x-b_1)(x-b_2)\cdots(x-b_n)\sum_{j=1}^n \frac{a_j}{x-b_j}\]
(b) Prove the inequality
\[\frac1{n-1}\sum_{j=1}^n \left(1-\frac{a_j}{S}\right)b_j > \left(\frac{T}{S}\sum_{j=1}^{n} \frac{a_j}{b_j}\right)^{\frac1{n-1}}\]
2014 Harvard-MIT Mathematics Tournament, 12
Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]2+\sqrt[3]4)=0$. (A polynomial is called $\textit{monic}$ if its leading coefficient is $1$.)
2010 Contests, 2
Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.
1954 AMC 12/AHSME, 41
The sum of all the roots of $ 4x^3\minus{}8x^2\minus{}63x\minus{}9\equal{}0$ is:
$ \textbf{(A)}\ 8 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \minus{}8 \qquad
\textbf{(D)}\ \minus{}2 \qquad
\textbf{(E)}\ 0$
1995 All-Russian Olympiad, 8
Let $P(x)$ and $Q(x)$ be monic polynomials. Prove that the sum of the squares of the coeficients of the polynomial $P(x)Q(x)$ is not smaller than the sum of the squares of the free coefficients of $P(x)$ and $Q(x)$.
[i]A. Galochkin, O. Ljashko[/i]
2006 Romania National Olympiad, 1
Let $\displaystyle \mathcal K$ be a finite field. Prove that the following statements are equivalent:
(a) $\displaystyle 1+1=0$;
(b) for all $\displaystyle f \in \mathcal K \left[ X \right]$ with $\displaystyle \textrm{deg} \, f \geq 1$, $\displaystyle f \left( X^2 \right)$ is reducible.
2024 Tuymaada Olympiad, 7
Given are two polynomial $f$ and $g$ of degree $100$ with real coefficients. For each positive integer $n$ there is an integer $k$ such that
\[\frac{f(k)}{g(k)}=\frac{n+1}{n}.\]
Prove that $f$ and $g$ have a common non-constant factor.
2008 Balkan MO Shortlist, N2
Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.
2023 Indonesia MO, 8
Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. For example, $S(1, 2, 3) = \{ -1, -2, -1/2 \}$ because the equation $x^2+3x+2$ has roots $-1$ and $-2$, the equation $2x^2+3x+1=0$ has roots $-1$ and $-1/2$, and for all the other permutations of $(1, 2, 3)$, the quadratic equations formed don't have any rational roots.
Determine the maximum number of elements in $S(a, b, c)$.
2004 Czech-Polish-Slovak Match, 1
Show that real numbers, $p, q, r$ satisfy the condition $p^4(q-r)^2 + 2p^2(q+r) + 1 = p^4$ if and only if the quadratic equations $x^2 + px + q = 0$ and $y^2 - py + r = 0$ have real roots (not necessarily distinct) which can be labeled by $x_1,x_2$ and $y_1,y_2$, respectively, in such a way that $x_1y_1 - x_2y_2 = 1$.
2002 India IMO Training Camp, 12
Let $a,b$ be integers with $0<a<b$. A set $\{x,y,z\}$ of non-negative integers is [i]olympic[/i] if $x<y<z$ and if $\{z-y,y-x\}=\{a,b\}$. Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets.
2018 CCA Math Bonanza, TB2
Define a sequence of polynomials $P_0\left(x\right)=x$ and $P_k\left(x\right)=P_{k-1}\left(x\right)^2-\left(-1\right)^kk$ for each $k\geq1$. Also define $Q_0\left(x\right)=x$ and $Q_k\left(x\right)=Q_{k-1}\left(x\right)^2+\left(-1\right)^kk$ for each $k\geq1$. Compute the product of the distinct real roots of \[P_1\left(x\right)Q_1\left(x\right)P_2\left(x\right)Q_2\left(x\right)\cdots P_{2018}\left(x\right)Q_{2018}\left(x\right).\]
[i]2018 CCA Math Bonanza Tiebreaker Round #2[/i]
2010 Contests, 1
The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations
\[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.