Let $ p(x) $ be a quadratic polynomial for which : \[ |p(x)| \leq 1 \qquad \forall x \in \{-1,0,1\} \] Prove that: \[ \ |p(x)|\leq\frac{5}{4} \qquad \forall x \in [-1,1]\]

Given a positive integer $n \ge 3$. Find the smallest real number $k$ such that for any positive real number except $a_1, a_2,..,a_n$, $$\sum_{i=1}^{n-1}\frac{a_i}{ s-a_i}+\frac{ka_n}{s-a_n} \ge \frac{n-1}{n-2}$$ where, $s=a_1+a_2+..+a_n$

Triangle $ ABC$ has a right angle at $ B$, $ AB \equal{} 1$, and $ BC \equal{} 2$. The bisector of $ \angle BAC$ meets $ \overline{BC}$ at $ D$. What is $ BD$? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair A=(0,1), B=(0,0), C=(2,0); pair D=extension(A,bisectorpoint(B,A,C),B,C); pair[] ds={A,B,C,D}; dot(ds); draw(A--B--C--A--D); label("$1$",midpoint(A--B),W); label("$B$",B,SW); label("$D$",D,S); label("$C$",C,SE); label("$A$",A,NW); draw(rightanglemark(C,B,A,2));[/asy]$ \textbf{(A)}\ \frac {\sqrt3 \minus{} 1}{2} \qquad \textbf{(B)}\ \frac {\sqrt5 \minus{} 1}{2} \qquad \textbf{(C)}\ \frac {\sqrt5 \plus{} 1}{2} \qquad \textbf{(D)}\ \frac {\sqrt6 \plus{} \sqrt2}{2}$ $ \textbf{(E)}\ 2\sqrt3 \minus{} 1$

For positive real numbers $a$ and $b,$ let $f(a,b)$ denote the real number $x$ such that area of the (non-degenerate) triangle with side lengths $a,b,$ and $x$ is maximized. Find \[ \sum_{n=2}^{100}f\left(\sqrt{\tbinom{n}{2}},\sqrt{\tbinom{n+1}{2}}\right). \]

Let $f$ be a function that associates a positive integer $(x, y)$ with each pair of positive integers $f(x, y)$. Suppose that $f(x, y) \le xy$ for all positive integers $x$, $y$. Show that there are $2023$ different pairs $(x_1, y_1)$,$...$, $ (x_{2023}, y_{2023}$) such that $$f(x_1, y_1) = f(x_2, y_2) = ....= f(x_{2023}, y_{2023}).$$

Let $n$ be a positive integer. Determine all positive real numbers $x$ satisfying $nx^2 +\frac{2^2}{x + 1}+\frac{3^2}{x + 2}+...+\frac{(n + 1)^2}{x + n}= nx + \frac{n(n + 3)}{2}$

Consider a real poylnomial $p(x)=a_nx^n+...+a_1x+a_0$. (a) If $\deg(p(x))>2$ prove that $\deg(p(x)) = 2 + deg(p(x+1)+p(x-1)-2p(x))$. (b) Let $p(x)$ a polynomial for which there are real constants $r,s$ so that for all real $x$ we have \[ p(x+1)+p(x-1)-rp(x)-s=0 \]Prove $\deg(p(x))\le 2$. (c) Show, in (b) that $s=0$ implies $a_2=0$.

In a planet $Papella$ year has $400$ days with days coundting from $1-400$ . A holiday would be that day which is divisible by $6$ . The new gonverment decide to reform a new calendar and split in $10$ months with $40$ day each month , and they decide that day of month which is divisible by $6$ to be holiday . Show that after reform the number of holidays after one year decreased less then $ 10 $ percent .

Show that there is no continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for every real number $x$ \[f(x-f(x)) = \dfrac x2.\]

Let $ n, k$ be positive integers and suppose that the polynomial $ x^{2k}\minus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$. Prove that $ x^{2k}\plus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$.

A sequence of integers $a_0,\ a_1,\dots a_n \dots $ is defined by the following rules: $a_0=0,\ a_1=1,\ a_{n+1} > a_n$ for each $n\in \mathbb{N}$, and $a_{n+1}$ is the minimum number such that no three numbers among $a_0,\ a_1,\dots a_{n+1}$ form an arithmetical progression. Prove that $a_{2^n}=3^n$ for each $n \in \mathbb{N}.$

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

Given integer $a_1\geq 2$. For integer $n\geq 2$, define $a_n$ to be the smallest positive integer which is not coprime to $a_{n-1}$ and not equal to $a_1,a_2,\cdots, a_{n-1}$. Prove that every positive integer except 1 appears in this sequence $\{a_n\}$.

Consider 2 non-constant polynomials $P(x),Q(x)$, with nonnegative coefficients. The coefficients of $P(x)$ is not larger than $2021$ and $Q(x)$ has at least one coefficient larger than $2021$. Assume that $P(2022)=Q(2022)$ and $P(x),Q(x)$ has a root $\frac p q \ne 0 (p,q\in \mathbb Z,(p,q)=1)$. Prove that $|p|+n|q|\le Q(n)-P(n)$ for all $n=1,2,...,2021$

Determine all functions $f : R \to R$ satisfying $f(f(x) + 2y)= 6x + f(f(y) -x)$ for all real numbers $x,y$

Let $a, b, n \in \mathbb{N}$, with $a, b \geq 2.$ Also, let $I_{1}(n)=\int_{0}^{1} \left \lfloor{a^n x} \right \rfloor dx $ and $I_{2} (n) = \int_{0}^{1} \left \lfloor{b^n x} \right \rfloor dx.$ Find $\lim_{n \to \infty} \dfrac{I_1(n)}{I_{2}(n)}.$

Using a nozzle to paint each square in a $1 \times n$ stripe, when the nozzle is aiming at the $i$-th square, the square is painted black, and simultaneously, its left and right neighboring square (if exists) each has an independent probability of $\tfrac{1}{2}$ to be painted black. In the optimal strategy (i.e. achieving least possible number of painting), the expectation of number of painting to paint all the squares black, is $T(n)$. Find the explicit formula of $T(n)$.

Find all functions $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ (where $\mathbb{Z}^+$ is the set of positive integers) such that $f(n!) = f(n)!$ for all positive integers $n$ and such that $m-n$ divides $f(m) - f(n)$ for all distinct positive integers $m, n$.

Let $N{}$ be the number of positive integers with $10$ digits $\overline{d_9d_8\cdots d_0}$ in base $10$ (where $0\le d_i\le9$ for all $i$ and $d_9>0$) such that the polynomial \[d_9x^9+d_8x^8+\cdots+d_1x+d_0\] is irreducible in $\Bbb Q$. Prove that $N$ is even. (A polynomial is irreducible in $\Bbb Q$ if it cannot be factored into two non-constant polynomials with rational coefficients.)

Consider two geometric sequences $16$, $a_1$, $a_2$, $ . . .$ and $56$, $b_1$, $b_2$, $. . . $ with the same common nonzero ratio. Given that $a_{2023} = b_{2020}$, compute $b_6 - a_6$.

Welcome to the [b]USAYNO[/b], where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer $n$ problems and get them [b]all[/b] correct, you will receive $\max(0, (n-1)(n-2))$ points. If any of them are wrong (or you leave them all blank), you will receive $0$ points. Your answer should be a six-character string containing 'Y' (for yes), 'N' (for no), or 'B' (for blank). For instance if you think 1, 2, and 6 are 'yes' and 3 and 4 are 'no', you should answer YYNNBY (and receive $12$ points if all five answers are correct, 0 points if any are wrong). (a) $a,b,c,d,A,B,C,$ and $D$ are positive real numbers such that $\frac{a}{b} > \frac{A}{B}$ and $\frac{c}{d} > \frac{C}{D}$. Is it necessarily true that $\frac{a+c}{b+d} > \frac{A+C}{B+D}$? (b) Do there exist irrational numbers $\alpha$ and $\beta$ such that the sequence $\lfloor\alpha\rfloor+\lfloor\beta\rfloor, \lfloor2\alpha\rfloor+\lfloor2\beta\rfloor, \lfloor3\alpha\rfloor+\lfloor3\beta\rfloor, \dots$ is arithmetic? (c) For any set of primes $\mathbb{P}$, let $S_\mathbb{P}$ denote the set of integers whose prime divisors all lie in $\mathbb{P}$. For instance $S_{\{2,3\}}=\{2^a3^b \; | \; a,b\ge 0\}=\{1,2,3,4,6,8,9,12,\dots\}$. Does there exist a finite set of primes $\mathbb{P}$ and integer polynomials $P$ and $Q$ such that $\gcd(P(x), Q(y))\in S_\mathbb{P}$ for all $x,y$? (d) A function $f$ is called [b]P-recursive[/b] if there exists a positive integer $m$ and real polynomials $p_0(n), p_1(n), \dots, p_m(n)$[color = red], not all zero,[/color] satisfying \[p_m(n)f(n+m)=p_{m-1}(n)f(n+m-1)+\dots+p_0(n)f(n)\] for all $n$. Does there exist a P-recursive function $f$ satisfying $\lim_{n\to\infty} \frac{f(n)}{n^{\sqrt{2}}}=1$? (e) Does there exist a [b]nonpolynomial[/b] function $f: \mathbb{Z}\to\mathbb{Z}$ such that $a-b$ divides $f(a)-f(b)$ for all integers $a\neq b$? (f) Do there exist periodic functions $f, g:\mathbb{R}\to\mathbb{R}$ such that $f(x)+g(x)=x$ for all $x$? [color = red]A clarification was issued for problem 33(d) during the test. I have included it above.[/color]

Show that there exists a positive number t such that for all positive numbers $a,b,c,d$ with $abcd = 1$, $$\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+d}> t.$$ and find the largest $t$ with this property.

Compute the sum of the squares of the first $100$ terms of an arithmetic progression, given that their sum is $-1$ and that the sum of those among them having an even index is $1$.

Define the succession $ a_{n}$, $ n>0$ as $ n\plus{}m$, where $ m$ is the largest integer such that $ 2^{2^{m}}\leq n2^{n}$. Find all numbers that are not in the succession.

The function $f(n)$ satisfies $f(0)=0$, $f(n)=n-f \left( f(n-1) \right)$, $n=1,2,3 \cdots$. Find all polynomials $g(x)$ with real coefficient such that \[ f(n)= [ g(n) ], \qquad n=0,1,2 \cdots \] Where $[ g(n) ]$ denote the greatest integer that does not exceed $g(n)$.