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$.

Let $a_1$, $a_2$, $\cdots$, $a_n$ be positive real numbers, and let $S_k$ be the sum of the products of $a_1$, $a_2$, $\cdots$, $a_n$ taken $k$ at a time. Show that \[ S_k S_{n-k} \geq {n \choose k}^2 a_1 a_2 \cdots a_n \] for $k = 1$, $2$, $\cdots$, $n - 1$.

Suppose that $ f(x)\in\mathbb Z[x]$ be an irreducible polynomial. It is known that $ f$ has a root of norm larger than $ \frac32$. Prove that if $ \alpha$ is a root of $ f$ then $ f(\alpha^3\plus{}1)\neq0$.

Knowing that the system \[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\] has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.

Consider the polynomial $P(x)=x^3+x^2-x+2$. Determine all real numbers $r$ for which there exists a complex number $z$ not in the reals such that $P(z)=r$.

Determine all polynomials $P$ with real coeffients such that $1 + P(x) = \frac12 (P(x -1) + P(x + 1))$ for all real $x$.

Let $P$ be a polynomial with each root is real and each coefficient is either $1$ or $-1$. The degree of $P$ can be at most ? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds. Prove that $\sum_{j=1}^n |a_j| \leq 3$.

For which even natural numbers $d{}$ does there exists a constant $\lambda>0$ such that any reduced polynomial $f(x)$ of degree $d{}$ with integer coefficients that does not have real roots satisfies the inequality $f(x) > \lambda$ for all real numbers?

Positive rational number $a$ and $b$ satisfy the equality \[a^3 + 4a^2b = 4a^2 + b^4.\] Prove that the number $\sqrt{a}-1$ is a square of a rational number.

Let $a_0=0<a_1<a_2<...<a_n$ be real numbers . Supppose $p(t)$ is a real valued polynomial of degree $n$ such that $\int_{a_j}^{a_{j+1}} p(t)\,dt = 0\ \ \forall \ 0\le j\le n-1$ Show that , for $0\le j\le n-1$ , the polynomial $p(t)$ has exactly one root in the interval $ (a_j,a_{j+1})$

For every complex number $z$ different from 0 and 1 we define the following function \[ f(z) := \sum \frac 1 { \log^4 z } \] where the sum is over all branches of the complex logarithm. a) Prove that there are two polynomials $P$ and $Q$ such that $f(z) = \displaystyle \frac {P(z)}{Q(z)} $ for all $z\in\mathbb{C}-\{0,1\}$. b) Prove that for all $z\in \mathbb{C}-\{0,1\}$ we have \[ f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}. \]

Let $p_1(x)=p(x)=4x^3-3x$ and $p_{n+1}(x)=p(p_n(x))$ for each positive integer $n$. Also, let $A(n)$ be the set of all the real roots of the equation $p_n(x)=x$. Prove that $A(n)\subseteq A(2n)$ and that the product of the elements of $A(n)$ is the average of the elements of $A(2n)$.

[b]p1.[/b] The English alphabet consists of $21$ consonants and $5$ vowels. (We count $y$ as a consonant.) (a) Suppose that all the letters are listed in an arbitrary order. Prove that there must be $4$ consecutive consonants. (b) Give a list to show that there need not be $5$ consecutive consonants. (c) Suppose that all the letters are arranged in a circle. Prove that there must be $5$ consecutive consonants. [b]p2.[/b] From the set $\{1,2,3,... , n\}$, $k$ distinct integers are selected at random and arranged in numerical order (lowest to highest). Let $P(i, r, k, n)$ denote the probability that integer $i$ is in position $r$. For example, observe that $P(1, 2, k, n) = 0$. (a) Compute $P(2, 1,6,10)$. (b) Find a general formula for $P(i, r, k, n)$. [b]p3.[/b] (a) Write down a fourth degree polynomial $P(x)$ such that $P(1) = P(-1)$ but $P(2) \ne P(-2)$ (b) Write down a fifth degree polynomial $Q(x)$ such that $Q(1) = Q(-1)$ and $Q(2) = Q(-2)$ but $Q(3) \ne Q(-3)$. (c) Prove that, if a sixth degree polynomial $R(x)$ satisfies $R(1) = R(-1)$, $R(2) = R(-2)$, and $R(3) = R(-3)$, then $R(x) = R(-x)$ for all $x$. [b]p4.[/b] Given five distinct real numbers, one can compute the sums of any two, any three, any four, and all five numbers and then count the number $N$ of distinct values among these sums. (a) Give an example of five numbers yielding the smallest possible value of $N$. What is this value? (b) Give an example of five numbers yielding the largest possible value of $N$. What is this value? (c) Prove that the values of $N$ you obtained in (a) and (b) are the smallest and largest possible ones. [b]p5.[/b] Let $A_1A_2A_3$ be a triangle which is not a right triangle. Prove that there exist circles $C_1$, $C_2$, and $C_3$ such that $C_2$ is tangent to $C_3$ at $A_1$, $C_3$ is tangent to $C_1$ at $A_2$, and $C_1$ is tangent to $C_2$ at $A_3$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A cubic polynomial $f$ with a positive leading coefficient has three different positive zeros. Show that $f'(a)+ f'(b)+ f'(c) > 0$.