Find the smallest real number $A$ such that, for every quadratic polynomial $f(x)$ satisfying $ | f(x)| \le 1$ for $0 \le x \le 1$, it holds that $f' (0) \le A$.

Note that if the product of any two distinct members of {1,16,27} is increased by 9, the result is the perfect square of an integer. Find the unique positive integer $n$ for which $n+9,16n+9,27n+9$ are also perfect squares.

Solve the equation $$\left(x^2+3x-4\right)^3+\left(2x^2-5x+3\right)^3=\left(3x^2-2x-1\right)^3.$$

Let $a_1, a_2, \dots, a_{2020}$ and $b_1, b_2, \dots, b_{2020}$ be real numbers(not necessarily distinct). Suppose that the set of positive integers $n$ for which the following equation: $|a_1|x-b_1|+a_2|x-b_2|+\dots+a_{2020}|x-b_{2020}||=n$ (1) has exactly two real solutions, is a finite set. Prove that the set of positive integers $n$ for which the equation (1) has at least one real solution, is also a finite set.

Let $ P(x) \equal{} x^3 \plus{} mx \plus{} n$ be an integer polynomial satisfying that if $ P(x) \minus{} P(y)$ is divisible by 107, then $ x \minus{} y$ is divisible by 107 as well, where $ x$ and $ y$ are integers. Prove that 107 divides $ m$.

It is known that $n$ is a positive integer, and the real number $x$ satisfies $$|1-|2-...|(n-1)-|n-x||...||=x.$$ Find the value of $x$.

Given that $0.3010<\log 2<0.3011$ and $0.4771<\log 3<0.4772$. Find the leftmost digit of $12^{37}$

$A$ can run around a circular track in $40$ seconds. $B$, running in the opposite direction, meets $A$ every $15$ seconds. What is $B$'s time to run around the track, expressed in seconds? $ \textbf{(A)}\ 12\frac12 \qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 27\frac12\qquad\textbf{(E)}\ 55 $

Define a sequence {$a_n$}$^{\infty}_{n=1}$ by $a_1 = 4, a_2 = a_3 = (a^2 - 2)^2$ and $a_n = a_{n-1}.a_{n-2} - 2(a_{n-1} + a_{n-2}) - a_{n-3} + 8, n \ge 4$, where $a > 2$ is a natural number. Prove that for all $n$ the number $2 + \sqrt{a_n}$ is a perfect square.

Let $n$ be a positive integer and let $(1+iT)^n=f(T)+ig(T)$ where $i$ is the square root of $-1$, and $f$ and $g$ are polynomials with real coefficients. Show that for any real number $k$ the equation $f(T)+kg(T)=0$ has only real roots.

Find the greatest $a$ for which there is $b$ such that the system $$\begin{cases} y=x^4+a \\ x=\dfrac{1}{y^4}+b \end{cases}$$ has exactly two solutions.

Let $a$, $b$ and $c$ positive integers. Three sequences are defined as follows: [list] [*] $a_1=a$, $b_1=b$, $c_1=c$[/*] [*] $a_{n+1}=\lfloor{\sqrt{a_nb_n}}\rfloor$, $\:b_{n+1}=\lfloor{\sqrt{b_nc_n}}\rfloor$, $\:c_{n+1}=\lfloor{\sqrt{c_na_n}}\rfloor$ for $n \ge 1$[/*] [/list] [list = a] [*]Prove that for any $a$, $b$, $c$, there exists a positive integer $N$ such that $a_N=b_N=c_N$.[/*] [*]Find the smallest $N$ such that $a_N=b_N=c_N$ for some choice of $a$, $b$, $c$ such that $a \ge 2$ y $b+c=2a-1$.[/*] [/list]

Find all non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying the equality $P(Q(x))=P(x)Q(x)-P(x)$. [i](I. Voronovich)[/i]

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

A $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ where $h_i\left(x_1,x_2, \cdots , x_n\right)$ are $n$ variable polynomials with real coefficients is called [i]good[/i] if the following condition holds: For any $n$ functions $f_1,f_2, \cdots ,f_n : \mathbb R \to \mathbb R$ if for all $1 \le i \le n+1$, $P_i(x)=h_i \left(f_1(x),f_2(x), \cdots, f_n(x) \right)$ is a polynomial with variable $x$, then $f_1(x),f_2(x), \cdots, f_n(x)$ are polynomials. $a)$ Prove that for all positive integers $n$, there exists a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that the degree of all $h_i$ is more than $1$. $b)$ Prove that there doesn't exist any integer $n>1$ that for which there is a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that all $h_i$ are symmetric polynomials. [i]Proposed by Alireza Shavali[/i]

Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$, and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions: (1) For all $n\in \mathbb{Z}$, $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$ (2) For all $n\ge M$, $f(n)\ge 0$ (3) For all $n>m>M$, $n-m|f(n)-f(m)$ Show that $a$ is a perfect $r$-th power.

When rolling a certain unfair six-sided die with faces numbered $1, 2, 3, 4, 5$, and $6$, the probability of obtaining face $F$ is greater than $\frac{1}{6}$, the probability of obtaining the face opposite is less than $\frac{1}{6}$, the probability of obtaining any one of the other four faces is $\frac{1}{6}$, and the sum of the numbers on opposite faces is $7$. When two such dice are rolled, the probability of obtaining a sum of $7$ is $\frac{47}{288}$. Given that the probability of obtaining face $F$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Determine the value of $$\prod^{\infty}_{n=1} 3^{n/3^n}= \sqrt[3]{3} \sqrt[3^2]{3^2} \sqrt[3^3]{3^3} ...$$

Find the smallest integer $k\ge3$ with the property that it is possible to choose two of the number $1,2,...,k$ in such a way that their product is equal to the sum of the remaining $k-2$ numbers.

Compute the number of solutions to $1+\cos(\theta)+\cos(2\theta)+\ldots+\cos(2024\theta) = \tfrac{1}{2}$ for $\theta \in [0,2\pi].$

Let $a \ne 0$ be a real number. Find all functions $f : R_{>0}\to R_{>0}$ with $$f(f(x) + y) = ax + \frac{1}{f\left(\frac{1}{y}\right)}$$ for all $x, y \in R_{>0}$. [i](Proposed by Walther Janous)[/i]

The equation $x^3 + ax^2 + bx + c = 0$ has three (not necessarily distinct) real roots $t, u, v$. For which $a, b, c$ do the numbers $t^3, u^3, v^3$ satisfy the equation $x^3 + a^3x^2 + b^3x + c^3 = 0$?

Construct the $ \triangle ABC$, given $ h_a$, $ h_b$ (the altitudes from $ A$ and $ B$) and $ m_a$, the median from the vertex $ A$.

Let $$a_k = 0.\overbrace{0 \ldots 0}^{k - 1 \: 0's} 1 \overbrace{0 \ldots 0}^{k - 1 \: 0's} 1$$ The value of $\sum_{k = 1}^\infty a_k$ can be expressed as a rational number $\frac{p}{q}$ in simplest form. Find $p + q$.

Let $z_1$, $z_2$, and $z_3$ be the complex roots of the equation $(2z -3\overline{z})^3 = 54i+54$. Compute the area of the triangle formed by $z_1$, $z_2$, and $z_3$ when plotted in the complex plane.