Find a factor of $2^{33}-2^{19}-2^{17}-1$ that lies between $1000$ and $5000$.

Let $a, b > 0$ be reals such that \[ a^3=a+1\\ b^6=b+3a \] Show that $a>b$

Let $Q(x) = a_{2023}x^{2023}+a_{2022}x^{2022}+\dots+a_{1}x+a_{0} \in \mathbb{Z}[x]$ be a polynomial with integer coefficients. For an odd prime number $p$ we define the polynomial $Q_{p}(x) = a_{2023}^{p-2}x^{2023}+a_{2022}^{p-2}x^{2022}+\dots+a_{1}^{p-2}x+a_{0}^{p-2}.$ Assume that there exist infinitely primes $p$ such that $$\frac{Q_{p}(x)-Q(x)}{p}$$ is an integer for all $x \in \mathbb{Z}$. Determine the largest possible value of $Q(2023)$ over all such polynomials $Q$. [i]Authored by Nikola Velov[/i]

Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that $$f(xy)+f(x+y)=f(x)f(y)+f(x)+f(y)$$ for all $x,y\in\mathbb{Q}$.

Given real numbers $a,c,d$ show that there exists at most one function $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfies: \[f(ax+c)+d\le x\le f(x+d)+c\quad\text{for any}\ x\in\mathbb{R}\]

A sequence $(a_k)_{k=1}^{\infty}$ has the property that there is a natural number $n$ such that $a_1 + a_2 +...+ a_n = 0$ and $a_{n+k} = a_k$ for all $k$. Prove that there exists a natural number $N$ such that $$\sum_{i=N}^{N+k} a_i \ge 0 \,\, \,\, for \,\,\,\, k = 0,1,2...$$

Carl chooses a [i]functional expression[/i]* $E$ which is a finite nonempty string formed from a set $x_1, x_2, \dots$ of variables and applications of a function $f$, together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation $E = 0$, and lets $S$ denote the set of functions $f \colon \mathbb R \to \mathbb R$ such that the equation holds for any choices of real numbers $x_1, x_2, \dots$. (For example, if Carl chooses the functional equation $$ f(2f(x_1)+x_2) - 2f(x_1)-x_2 = 0, $$ then $S$ consists of one function, the identity function. (a) Let $X$ denote the set of functions with domain $\mathbb R$ and image exactly $\mathbb Z$. Show that Carl can choose his functional equation such that $S$ is nonempty but $S \subseteq X$. (b) Can Carl choose his functional equation such that $|S|=1$ and $S \subseteq X$? *These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer $i$, the variable $x_i$ is a functional expression, and (iii) if $V$ and $W$ are functional expressions, then so are $f(V)$, $V+W$, $V-W$, and $V \cdot W$. [i]Proposed by Carl Schildkraut[/i]

For positive integer $k$, define integer sequence $\{ b_n \}, \{ c_n \} $ as follows: \[ b_1 = c_1 = 1 \] \[ b_{2n} = kb_{2n-1} + (k-1)c_{2n-1}, c_{2n} = b_{2n-1} + c_{2n-1} \] \[ b_{2n+1} = b_{2n} + (k-1)c_{2n}, c_{2n+1} = b_{2n} + kc_{2n} \] Let $a_k = b_{2014} $. Find the value of \[ \sum_{k=1}^{100} { (a_k - \sqrt{{a_k}^2-1} )^{ \frac{1}{2014}} }\]

Let $f(x)$ be a polynomial of nonzero degree. Can it happen that for any real number $a$, an even number of real numbers satisfy the equation $f(x) = a$?

Factor $a^{10} + a^5 + 1$ into nonconstant polynomials with integer coefficients

Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

Determine all ordered triples $(a, b, c)$ of real numbers such that whenever a function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$|f(x) - f(y)| \le a(x - y)^2 + b(x - y) + c$$ for all real numbers $x$ and $y$, then $f$ must be a constant function.

The positive integers from $ 1$ to $n$ inclusive are written on a whiteboard. After one number is erased, the average (arithmetic mean) of the remaining $n - 1$ numbers is $22$. Knowing that $n$ is odd, determine $n$ and the number that was erased. Explain your reasoning.

Determine all continuous functions $f: \mathbb R \to \mathbb R$ such that \[f(x + y)f(x - y) = (f(x)f(y))^2, \quad \forall(x, y) \in\mathbb{R}^2.\]

A sequence $(a_n)$ of real numbers is defined by $a_0=1$, $a_1=2015$ and for all $n\geq1$, we have $$a_{n+1}=\frac{n-1}{n+1}a_n-\frac{n-2}{n^2+n}a_{n-1}.$$ Calculate the value of $\frac{a_1}{a_2}-\frac{a_2}{a_3}+\frac{a_3}{a_4}-\frac{a_4}{a_5}+\ldots+\frac{a_{2013}}{a_{2014}}-\frac{a_{2014}}{a_{2015}}$.

Numbers $a, b, c$ are such that $3a + 4b = 3c$ and $4a - 3b = 4c$. Show that $a^2 + b^2 = c^2$.

Find all polynomials $f\in Z[X],$ such that for each odd prime $p$ $$f(p)|(p-3)!+\frac{p+1}{2}.$$

Find all functions $f : R \to R$ that satisfy $f((x-y)^2)= f(x)^2 -2x f(y)+y^2$ for all $x,y \in R$

Find all monotonically increasing or monotonically decreasing functions $f: \mathbb{R}_+\to\mathbb{R}_+$ which satisfy the equation $f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1$ for any two numbers $x$ and $y$ from $\mathbb{R}_+$. Hereby, $\mathbb{R}_+$ is the set of all positive real numbers. [i]Note.[/i] A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically increasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\geq f\left(y\right)$. A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically decreasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\leq f\left(y\right)$.

If $x,y,z$ are real numbers satisfying relations \[x+y+z = 1 \quad \textrm{and} \quad \arctan x + \arctan y + \arctan z = \frac{\pi}{4},\] prove that $x^{2n+1} + y^{2n+1} + z^{2n+1} = 1$ holds for all positive integers $n$.

It is well known that the mathematical constant $e$ can be written in the form $e = \tfrac{1}{0!}+\tfrac{1}{1!}+\tfrac{1}{2!}+\cdots$. With this in mind, determine the value of \[\sum_{j=3}^\infty\dfrac{j}{\lfloor\frac j2\rfloor!}.\] Express your answer in terms of $e$.

If $x, y, z, w$ are nonnegative real numbers satisfying \[\left\{ \begin{array}{l}y = x - 2003 \\ z = 2y - 2003 \\ w = 3z - 2003 \\ \end{array} \right. \] find the smallest possible value of $x$ and the values of $y, z, w$ corresponding to it.

Let $ S \equal{} \{0,1,2,\ldots,1999\}$ and $ T \equal{} \{0,1,2,\ldots \}.$ Find all functions $ f: T \mapsto S$ such that [b](i)[/b] $ f(s) \equal{} s \quad \forall s \in S.$ [b](ii)[/b] $ f(m\plus{}n) \equal{} f(f(m)\plus{}f(n)) \quad \forall m,n \in T.$

p1. Nurbaya's rectangular courtyard will be covered by a number of paving blocks in the form of a regular hexagon or its pieces like the picture below. The length of the side of the hexagon is $ 12$ cm. [img]https://cdn.artofproblemsolving.com/attachments/6/1/281345c8ee5b1e80167cc21ad39b825c1d8f7b.png[/img] Installation of other paving blocks or pieces thereof so that all fully covered page surface. To cover the entire surface The courtyard of the house required $603$ paving blocks. How many paving blocks must be cut into models $A, B, C$, and $D$ for the purposes of closing. If $17$ pieces of model $A$ paving blocks are needed, how many the length and width of Nurbaya's yard? Count how much how many pieces of each model $B, C$, and $D$ paving blocks are used. p2. Given the square $PQRS$. If one side lies on the line $y = 2x - 17$ and its two vertices lie on the parabola $y = x^2$, find the maximum area of possible squares $PQRS$ . p3. In the triangular pyramid $T.ABC$, the points $E, F, G$, and $H$ lie at , respectively $AB$, $AC$, $TC$, and $TB$ so that $EA : EB = FA : FC = HB : HT = GC : GT = 2:1$. Determine the ratio of the volumes of the two halves of the divided triangular pyramid by the plane $EFGH$. p4. We know that $x$ is a non-negative integer and $y$ is an integer. Define all pair $(x, y)$ that satisfy $1 + 2^x + 2^{2x + 1} = y^2$. p5. The coach of the Indonesian basketball national team will select the players for become a member of the core team. The coach will judge five players $A, B, C, D$ and $E$ in one simulation (or trial) match with total time $80$ minute match. At any time there is only one in five players that is playing. There is no limit to the number of substitutions during the match. Total playing time for each player $A, B$, and $C$ are multiples of $5$ minutes, while the total playing time of each players $D$ and $E$ are multiples of $7$ minutes. How many ways each player on the field based on total playing time?

Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression