Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that $$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$ for all positive integers $n$. Show that $a_{2022}\leq 1$.

Given a polynomial with integer coefficients, which has at least one integer root. The greatest common divisor of all its integer roots equals $1$. Prove that if the leading coefficient of the polynomial equals $1$ then the greatest common divisor of the other coefficients also equals $1$.

If a polynomial $P$ with integer coefficients has three distinct integer zeroes , then show that $P(n)\neq 1$ for any integer $n$.

Evaluate $$2023 \cdot \frac{2023^6 +27}{(2023^2 +3)(2024^3 -1)}-2023^2.$$ [i]Proposed by Evin Liang[/i]

Let $p$ and $q$ be distinct odd primes. Show that $$\bigg\lceil \dfrac{p^q+q^p-pq+1}{pq} \bigg\rceil$$ is even.

For one root of $ ax^2 \plus{} bx \plus{} c \equal{} 0$ to be double the other, the coefficients $ a,\,b,\,c$ must be related as follows: $ \textbf{(A)}\ 4b^2 \equal{} 9c\qquad \textbf{(B)}\ 2b^2 \equal{} 9ac\qquad \textbf{(C)}\ 2b^2 \equal{} 9a\qquad \\ \textbf{(D)}\ b^2 \minus{} 8ac \equal{} 0\qquad \textbf{(E)}\ 9b^2 \equal{} 2ac$

[b]p13.[/b] Emma has the five letters: $A, B, C, D, E$. How many ways can she rearrange the letters into words? Note that the order of words matter, ie $ABC DE$ and $DE ABC$ are different. [b]p14.[/b] Seven students are doing a holiday gift exchange. Each student writes their name on a slip of paper and places it into a hat. Then, each student draws a name from the hat to determine who they will buy a gift for. What is the probability that no student draws himself/herself? [b]p15.[/b] We model a fidget spinner as shown below (include diagram) with a series of arcs on circles of radii $1$. What is the area swept out by the fidget spinner as it’s turned $60^o$ ? [img]https://cdn.artofproblemsolving.com/attachments/9/8/db27ffce2af68d27eee5903c9f09a36c2a6edf.png[/img] [b]p16.[/b] Let $a,b,c$ be the sides of a triangle such that $gcd(a, b) = 3528$, $gcd(b, c) = 1008$, $gcd(a, c) = 504$. Find the value of $a * b * c$. Write your answer as a prime factorization. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose that $ n \geq 2$ and $ x_1, x_2, \ldots, x_n$ are real numbers between 0 and 1 (inclusive). Prove that for some index $ i$ between $ 1$ and $ n \minus{} 1$ the inequality \[ x_i (1 \minus{} x_{i\plus{}1}) \geq \frac{1}{4} x_1 (1 \minus{} x_{n})\]

If a function $f$ defined on all real numbers has at least two centers of symmetry, show that this function can be written as sum of a linear function and a periodic function. [For every real number $x$, if there is a real number $a$ such that $f(a-x) + f(a+x) =2f(a)$, the point $(a,f(a))$ is called a center of symmetry of the function $f$.]

For real numbers $x$ and $y$, suppose that $|x| - |y| = 20$ and $|x| + |y| = 23$. Compute the sum of all possible distinct values of $|x - y|$.

[b]p1.[/b] Twenty five horses participate in a competition. The competition consists of seven runs, five horse compete in each run. Each horse shows the same result in any run it takes part. No two horses will give the same result. After each run you can decide what horses participate in the next run. Could you determine the three fastest horses? (You don’t have stopwatch. You can only remember the order of the horses.) [b]p2.[/b] Prove that the equation $x^6-143x^5-917x^4+51x^3+77x^2+291x+1575=0$ does not have solutions in integer numbers. [b]p3.[/b] Show how we can cut the figure shown in the picture into two parts for us to be able to assemble a square out of these two parts. Show how we can assemble a square. [img]https://cdn.artofproblemsolving.com/attachments/7/b/b0b1bb2a5a99195688638425cf10fe4f7b065b.png[/img] [b]p4.[/b] The city of Vyatka in Russia produces local drink, called “Vyatka Cola”. «Vyatka Cola» is sold in $1$, $3/4$, and $1/2$-gallon bottles. Ivan and John bought $4$ gallons of “Vyatka Cola”. Can we say for sure, that they can split the Cola evenly between them without opening the bottles? [b]p5.[/b] Positive numbers a, b and c satisfy the condition $a + bc = (a + b)(a + c)$. Prove that $b + ac = (b + a)(b + c)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the set $S$ of $100$ numbers: $1; \frac{1}{2}; \frac{1}{3}; ... ; \frac{1}{100}$. Any two numbers, $a$ and $b$, are eliminated in $S$, and the number $a+b+ab$ is added. Now, there are $99$ numbers on $S$. After doing this operation $99$ times, there's only $1$ number on $S$. What values can this number take?

Given a positive integer $n$, find the greatest integer $p$ with the property that for any function $f : \mathbb P(X) \to C$, where $X$ and $C$ are sets of cardinality $n$ and $p$, respectively, there exist two distinct sets $A,B \in \mathbb P(X)$ such that $f(A) = f(B) = f(A \cup B)$. ($\mathbb P(X)$ is the family of all subsets of $X$.)

Find all values of $a$ for which the equation $x^3 - x + a = 0$ has three different integer solutions.

Let the polynomial $P(x)=x^3+19x^2+94x+a$ where $a\in\mathbb{N}$. If $p$ a prime number, prove that no more than three numbers of the numbers $P(0), P(1),\ldots, P(p-1)$ are divisible by $p$.

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^+\to\mathbb{R}^+$ that satisfy \[ \Big(1+yf(x)\Big)\Big(1-yf(x+y)\Big)=1\] for all $x,y\in\mathbb{R}^+$.

Prove that the number $\left(\sqrt2+\sqrt3+\sqrt5\right)^{1997}$ can be represented as $$A\sqrt2+B\sqrt3+C\sqrt5+D\sqrt{30}$$ where $A$, $B$, $C$, $D$ are integers. Find with approximation to $10^{-10}$ the ratio $\frac{D}{A}$

Let $\alpha_1, \alpha_2, \dots, \alpha_n$, and $\beta_1, \beta_2, \ldots, \beta_n$, where $n \geq 4$, be 2 sets of real numbers such that \[\sum_{i=1}^{n} \alpha_i^2 < 1 \qquad \text{and} \qquad \sum_{i=1}^{n} \beta_i^2 < 1.\] Define \begin{align*} A^2 &= 1 - \sum_{i=1}^{n} \alpha_i^2,\\ B^2 &= 1 - \sum_{i=1}^{n} \beta_i^2,\\ W &= \frac{1}{2} (1 - \sum_{i=1}^{n} \alpha_i \beta_i)^2. \end{align*} Find all real numbers $\lambda$ such that the polynomial \[x^n + \lambda (x^{n-1} + \cdots + x^3 + Wx^2 + ABx + 1) = 0,\] only has real roots.

In the equation $ \frac {x(x \minus{} 1) \minus{} (m \plus{} 1)}{(x \minus{} 1)(m \minus{} 1)} \equal{} \frac {x}{m}$ the roots are equal when $ \textbf{(A)}\ m \equal{} 1 \qquad\textbf{(B)}\ m \equal{} \frac {1}{2} \qquad\textbf{(C)}\ m \equal{} 0 \qquad\textbf{(D)}\ m \equal{} \minus{} 1 \qquad\textbf{(E)}\ m \equal{} \minus{} \frac {1}{2}$

Let $d$ be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than $d$ degrees. What is the minimum possible value for $d$?

Given a positive integer $n>1$. Denote $T$ a set that contains all ordered sets $(x;y;z)$ such that $x,y,z$ are all distinct positive integers and $1\leq x,y,z\leq 2n$. Also, a set $A$ containing ordered sets $(u;v)$ is called [i]"connected"[/i] with $T$ if for every $(x;y;z)\in T$ then $\{(x;y),(x;z),(y;z)\} \cap A \neq \varnothing$. a) Find the number of elements of set $T$. b) Prove that there exists a set "connected" with $T$ that has exactly $2n(n-1)$ elements. c) Prove that every set "connected" with $T$ has at least $2n(n-1)$ elements.

Find all triples (p,a,m); p is a prime number, $a,m\in \mathbb{N}$, which satisfy: $a\leq 5p^2$ and $(p-1)!+a=p^m$.

Determine each real root of \[x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0\] correct to four decimal places.

Let $\{x\}$ denote the fractional part of $x$, $x - [x]$. The sequences $x_1, x_2, x_3, ...$ and $y_1, y_2, y_3, ...$ are such that $\lim \{x_n\} = \lim \{y_n\} = 0$. Is it true that $\lim \{x_n + y_n\} = 0$? $\lim \{x_n - y_n\} = 0$?

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.