What is the value of \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?\] $ \textbf{(A)}\ -1\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{5}{3}\qquad\textbf{(D)}\ 2013\qquad\textbf{(E)}\ 2^{4024} $

Let $ n \geq 2$ be a natural number and $ p(x)$ a real polynomial of degree at most $ n$ for which \[ \max _{ \minus{}1 \leq x \leq 1} |p(x)| \leq 1, \; p(\minus{}1)\equal{}p(1)\equal{}0 \ .\] Prove that then \[ |p'(x)| \leq \frac{n \cos \frac{\pi}{2n}}{\sqrt{1\minus{}x^2 \cos^2 \frac{\pi}{2n}}} \;\;\;\;\; \left( \minus{}\frac{1}{\cos \frac{\pi}{2n}} < x < \frac{1}{\cos \frac{\pi}{2n}} \\\\\ \right).\] [i]J. Szabados[/i]

An ordered quadruple $(y_1,y_2,y_3,y_4)$ is $\textbf{quadratic}$ if there exist real numbers $a$, $b$, and $c$ such that \[y_n=an^2+bn+c\] for $n=1,2,3,4$. Prove that if $16$ numbers are placed in a $4\times 4$ grid such that all four rows are quadratic and the first three columns are also quadratic then the fourth column must also be quadratic. [i](We say that a row is quadratic if its entries, in order, are quadratic. We say the same for a column.)[/i] [asy] size(100); defaultpen(linewidth(0.8)); for(int i=0;i<=4;i=i+1) draw((i,0)--(i,4)); for(int i=0;i<=4;i=i+1) draw((0,i)--(4,i)); [/asy]

Compute the number of ordered pairs of integers $(a, b),$ with $2 \le a, b \le 2021,$ that satisfy the equation \[a^{\log_b \left(a^{-4}\right)} = b^{\log_a \left(ba^{-3}\right)}.\]

Find all functions $f: R \rightarrow R$ such that \[f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy\] for all $x,y \in R$

Find all polynomials $P(x)$ with integer coefficients such that for all positive integers $n$, we have that $P(n)$ is not zero and $\frac{P(\overline{nn})}{P(n)}$ is an integer, where $\overline{nn}$ is the integer obtained upon concatenating $n$ with itself.

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

Find all possible values of $$\frac{(a+b-c)^2}{(a-c)(b-c)}+\frac{(b+c-a)^2}{(b-a)(c-a)}+\frac{(c+a-b)^2}{(c-b)(a-b)}$$

Find the minimum value of $$K(x,y)=16\frac{x^3}{y}+\frac{y^3}{x}-\sqrt{xy}$$ where $x,y$ are the real allowed values

Let $X \subset \mathbb{R}^2$ be a set satisfying the following properties: (i) if $(x_1,y_1)$ and $(x_2,y_2)$ are any two distinct elements in $X$, then \[\text{ either, }\ \ x_1>x_2 \text{ and } y_1>y_2\\ \text{ or, } \ \ x_1<x_2 \text{ and } y_1<y_2\] (ii) there are two elements $(a_1,b_1)$ and $(a_2,b_2)$ in $X$ such that for any $(x,y) \in X$, \[a_1\le x \le a_2 \text{ and } b_1\le y \le b_2\] (iii) if $(x_1,y_1)$ and $(x_2,y_2)$ are two elements of $X$, then for all $\lambda \in [0,1]$, \[\left(\lambda x_1+(1-\lambda)x_2, \lambda y_1 + (1-\lambda)y_2\right) \in X\] Show that if $(x,y) \in X$, then for some $\lambda \in [0,1]$, \[x=\lambda a_1 +(1-\lambda)a_2, y=\lambda b_1 +(1-\lambda)b_2\]

If $n\ge2$ is an integer, prove the equality $$\lfloor\log_2n\rfloor+\lfloor\log_3n\rfloor+\ldots+\lfloor\log_nn\rfloor=\left\lfloor\sqrt n\right\rfloor+\left\lfloor\sqrt[3]n\right\rfloor+\ldots+\left\lfloor\sqrt[n]n\right\rfloor.$$

Suppose $a$ and $b$ are real numbers such that $0 < a < b < 1$. Let $$x= \frac{1}{\sqrt{b}} - \frac{1}{\sqrt{b+a}},\hspace{1cm} y= \frac{1}{b-a} - \frac{1}{b}\hspace{0.5cm}\textrm{and}\hspace{0.5cm} z= \frac{1}{\sqrt{b-a}} - \frac{1}{\sqrt{b}}.$$ Show that $x$, $y$, $z$ are always ordered from smallest to largest in the same way, regardless of the choice of $a$ and $b$. Find this order among $x$, $y$, $z$.

A quadratic equation $ ax^2\minus{}2ax\plus{}b\equal{}0$ has two real solutions. What is the average of the solutions? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \frac{b}{a} \qquad \textbf{(D)}\ \frac{2b}{a} \qquad \textbf{(E)}\ \sqrt{2b\minus{}a}$

Show that $ \rho (f)$ changes continously over $ f$. It means for every bijection $ f: S^1\rightarrow S^1$ and $ \epsilon > 0$ there is $ \delta > 0$ such that if $ g: S^1\rightarrow S^1$ is a bijection such that $ \parallel{}f \minus{} g\parallel{} < \delta$ then $ |\rho(f) \minus{} \rho(g)| < \epsilon$. Note that $ \rho(f)$ is the rotatation number of $ f$ and $ \parallel{}f \minus{} g\parallel{} \equal{} \sup\{|f(x) \minus{} g(x)| | x\in S^1\}$.

Suppose $x$ and $y$ are real numbers such that \[x^2+xy+y^2=2\qquad\text{and}\qquad x^2-y^2=\sqrt5.\] The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac mn$ where $m$ and $n$ are positive realtively prime integers, what is $100m+n$? [i] Proposed by David Altizio [/i]

Let $a\in\mathbb{R}-\{0\}$. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that $f(a+x) = f(x) - x$ for all $x\in\mathbb{R}$. [i]Dan Schwartz[/i]

Let $A=\sqrt{7+2\sqrt{10}} - \sqrt{7-2\sqrt{10}}.$ We can express $A$ as $a\sqrt{b},$ where $a,b$ are integers and $b$ is square-free. Compute $a+b.$

Find the largest $k$ for which there exists a permutation $(a_1, a_2, \ldots, a_{2022})$ of integers from $1$ to $2022$ such that for at least $k$ distinct $i$ with $1 \le i \le 2022$ the number $\frac{a_1 + a_2 + \ldots + a_i}{1 + 2 + \ldots + i}$ is an integer larger than $1$. [i](Proposed by Oleksii Masalitin)[/i]

Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

Let be three elements $ a,b,c $ of a nontrivial, noncommutative ring, that satisfy $ ab=1-c, $ and such that there exists an element $ d $ from the ring such that $ a+cd $ is a unit. Prove that there exists an element $ e $ from the ring such that $ b+ec $ is a unit. [i]Andrei Nedelcu[/i] and [i] Lucian Ladunca [/i]

Prove that: the polynomial$$(x(x+1)(x+2)(x+3))^{2^{2023}}+1$$is irreducible in $\mathbb{Q}[x].$

All non-zero coefficients of the polynomial $f(x)$ equal $1$, while the sum of the coefficients is $20$. Is it possible that thirteen coefficients of $f^2(x)$ equal $9$? [i](S. Ivanov, K. Kokhas)[/i]

IMONST = [b]I[/b]nternational [b]M[/b]athematical [b]O[/b]lympiad [b]N[/b]ational [b]S[/b]election [b]T[/b]est Malaysia 2020 Round 1 Juniors Time: 2.5 hours [hide=Rules] $\bullet$ For each problem you have to submit the answer only. The answer to each problem is a non-negative integer. $\bullet$ No mark is deducted for a wrong answer. $\bullet$ The maximum number of points is (1 + 2 + 3 + 4) x 5 = 50 points.[/hide] [b]Part A[/b] (1 point each) p1. The number $N$ is the smallest positive integer with the sum of its digits equal to $2020$. What is the first (leftmost) digit of $N$? p2. At a food stall, the price of $16$ banana fritters is $k$ RM , and the price of $k$ banana fritters is $ 1$ RM . What is the price of one banana fritter, in sen? Note: $1$ RM is equal to $100$ sen. p3. Given a trapezium $ABCD$ with $AD \parallel$ to $BC$, and $\angle A = \angle B = 90^o$. It is known that the area of the trapezium is 3 times the area of $\vartriangle ABD$. Find $$\frac{area \,\, of \,\, \vartriangle ABC}{area \,\, of \,\, \vartriangle ABD}.$$ p4. Each $\vartriangle$ symbol in the expression below can be substituted either with $+$ or $-$: $$\vartriangle 1 \vartriangle 2 \vartriangle 3 \vartriangle 4.$$ How many possible values are there for the resulting arithmetic expression? Note: One possible value is $-2$, which equals $-1 - 2 - 3 + 4$. p5. How many $3$-digit numbers have its sum of digits equal to $4$? [b]Part B[/b] (2 points each) p6. Find the value of $$+1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 +... - 2020$$ where the sign alternates between $+$ and $-$ after every three numbers. p7. If Natalie cuts a round pizza with $4$ straight cuts, what is the maximum number of pieces that she can get? Note: Assume that all the cuts are vertical (perpendicular to the surface of the pizza). She cannot move the pizza pieces until she finishes cutting. p8. Given a square with area $ A$. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area $ B$ lies inside the circle, such that all its vertices lie on the circle. Find the value of $A/B$. p9. This sequence lists the perfect squares in increasing order: $$0, 1, 4, 9, 16, ... ,a, 10^8, b, ...$$ Determine the value of $b - a$. p10. Determine the last digit of $5^5 + 6^6 + 7^7 + 8^8 + 9^9$. [b]Part C[/b] (3 points each) p11. Find the sum of all integers between $-\sqrt{1442}$ and $\sqrt{2020}$. p12. Three brothers own a painting company called Tiga Abdul Enterprise. They are hired to paint a building. Wahab says, "I can paint this building in $3$ months if I work alone". Wahib says, "I can paint this building in $2$ months if I work alone". Wahub says, "I can paint this building in $k$ months if I work alone". If they work together, they can finish painting the building in $1$ month only. What is $k$? p13. Given a rectangle $ABCD$ with a point $P$ inside it. It is known that $PA = 17$, $PB = 15$, and $PC = 6$. What is the length of $PD$? p14. What is the smallest positive multiple of $225$ that can be written using digits $0$ and $ 1$ only? p15. Given positive integers $a, b$, and $c$ with $a + b + c = 20$. Determine the number of possible integer values for $\frac{a + b}{c}$. [b]Part D[/b] (4 points each) p16. If we divide $2020$ by a prime $p$, the remainder is $6$. Determine the largest possible value of $p$. p17. A football is made by sewing together some black and white leather patches. The black patches are regular pentagons of the same size. The white patches are regular hexagons of the same size. Each pentagon is bordered by $5$ hexagons. Each hexagons is bordered by $3$ pentagons and $3$ hexagons. We need $12$ pentagons to make one football. How many hexagons are needed to make one football? p18. Given a right-angled triangle with perimeter $18$. The sum of the squares of the three side lengths is $128$. What is the area of the triangle? p19. A perfect square ends with the same two digits. How many possible values of this digit are there? p20. Find the sum of all integers $n$ that fulfill the equation $2^n(6 - n) = 8n$.

Test whether equation $$\frac{1}{x - a} + \frac{1}{x - b} + \frac{1}{x - c} = 0,$$ where $ a $, $ b $, $ c $ denote the given real numbers, has real roots.

Find all $f:\mathbb{R}\to\mathbb{R}$ functions such that $$f(x+y)^3=(x+2y)f(x^2)+f(f(y))(x^2+3xy+y^2)$$ for all real numbers $x,y$