Find the least positive integer $k$ such that when $\frac{k}{2023}$ is written in simplest form, the sum of the numerator and denominator is divisible by $7$. [i]Proposed byMuztaba Syed[/i]

Let $a > 0$ be a real number and let $f : R \to R$ be a function satisfying $$f(x_1) + f(x_2) \ge a f(x_1 + x_2), \forall x_1 ,x_2 \in R.$$ Prove that $$f(x_1) + f(x_2) +(x_3) \ge \frac{3a^2}{a+2} f(x_1+ x_2 + x_3), \forall x_1 ,x_2,x_3 \in R$$.

The student wrote on the board three natural numbers that are consecutive members of one arithmetic progression. Then he erased the commas separating the numbers, resulting in a seven-digit number. What is the largest number that could result?

[b]p1.[/b] How many real solutions does the following system of equations have? Justify your answer. $$x + y = 3$$ $$3xy -z^2 = 9$$ [b]p2.[/b] After the first year the bank account of Mr. Money decreased by $25\%$, during the second year it increased by $20\%$, during the third year it decreased by $10\%$, and during the fourth year it increased by $20\%$. Does the account of Mr. Money increase or decrease during these four years and how much? [b]p3.[/b] Two circles are internally tangent. A line passing through the center of the larger circle intersects it at the points $A$ and $D$. The same line intersects the smaller circle at the points $B$ and $C$. Given that $|AB| : |BC| : |CD| = 3 : 7 : 2$, find the ratio of the radiuses of the circles. [b]p4.[/b] Find all integer solutions of the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{19}$ [b]p5.[/b] Is it possible to arrange the numbers $1, 2, . . . , 12$ along the circle so that the absolute value of the difference between any two numbers standing next to each other would be either $3$, or $4$, or $5$? Prove your answer. [b]p6.[/b] Nine rectangles of the area $1$ sq. mile are located inside the large rectangle of the area $5$ sq. miles. Prove that at least two of the rectangles (internal rectangles of area $1$ sq. mile) overlap with an overlapping area greater than or equal to $\frac19$ sq. mile PS. You should use hide for answers.

Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be such that $$f(x+f(y))^2\geq f(x)\left(f(x+f(y))+f(y)\right)$$ for all $x,y\in\mathbb{R}^+$. Show that $f$ is [i]unbounded[/i], i.e. for each $M\in\mathbb{R}^+$, there exists $x\in\mathbb{R}^+$ such that $f(x)>M$.

Find all positive real solutions to the equation $x+\left\lfloor\frac x3\right\rfloor=\left\lfloor\frac{2x}3\right\rfloor+\left\lfloor\frac{3x}5\right\rfloor$

Let $n$ be a given positive integer. Alice and Bob play a game. In the beginning, Alice determines an integer polynomial $P(x)$ with degree no more than $n$. Bob doesn’t know $P(x)$, and his goal is to determine whether there exists an integer $k$ such that no integer roots of $P(x) = k$ exist. In each round, Bob can choose a constant $c$. Alice will tell Bob an integer $k$, representing the number of integer $t$ such that $P(t) = c$. Bob needs to pay one dollar for each round. Find the minimum cost such that Bob can guarantee to reach his goal. [i]Proposed by ltf0501[/i]

Let $a_n$ be the number of unordered sets of three distinct bijections $f, g, h : \{1, 2, ..., n\} \to \{1, 2, ..., n\}$ such that the composition of any two of the bijections equals the third. What is the largest value in the sequence $a_1, a_2, ...$ which is less than $2021$?

For a sequence of integers $a_1,a_2,a_3,...$ with $0<a_1<a_2<a_3<...$ applies: $$a_n=4a_{n-1}-a_{n-2} \,\,\, for \,\,\, n > 2$$ It is further given that $a_4 = 194$. Calculate $a_5$.

Let $ f $ be a polynom of degree $ 3 $ and having rational coefficients. Prove that, if there exist two distinct nonzero rational numbers $ a,b $ and two roots $ x,y $ of $ f $ such that $ ax+by $ is rational, then all roots of $ f $ are rational.

[b]p1.[/b] How many lines of symmetry does a square have? [b]p2.[/b] Compute$ 1/2 + 1/6 + 1/12 + 1/4$. [b]p3.[/b] What is the maximum possible area of a rectangle with integer side lengths and perimeter $8$? [b]p4.[/b] Given that $1$ printer weighs $400000$ pennies, and $80$ pennies weighs $2$ books, what is the weight of a printer expressed in books? [b]p5.[/b] Given that two sides of a triangle are $28$ and $3$ and all three sides are integers, what is the sum of the possible lengths of the remaining side? [b]p6.[/b] What is half the sum of all positive integers between $1$ and $15$, inclusive, that have an even number of positive divisors? [b]p7.[/b] Austin the Snowman has a very big brain. His head has radius $3$, and the volume of his torso is one third of his head, and the volume of his legs combined is one third of his torso. If Austin's total volume is $a\pi$ where $a$ is an integer, what is $a$? [b]p8.[/b] Neethine the Kiwi says that she is the eye of the tiger, a fighter, and that everyone is gonna hear her roar. She is standing at point $(3, 3)$. Neeton the Cat is standing at $(11,18)$, the farthest he can stand from Neethine such that he can still hear her roar. Let the total area of the region that Neeton can stand in where he can hear Neethine's roar be $a\pi$ where $a$ is an integer. What is $a$? [b]p9.[/b] Consider $2018$ identical kiwis. These are to be divided between $5$ people, such that the first person gets $a_1$ kiwis, the second gets $a_2$ kiwis, and so forth, with $a_1 \le a_2 \le a_3 \le a_4 \le a_5$. How many tuples $(a_1, a_2, a_3, a_4, a_5)$ can be chosen such that they form an arithmetic sequence? [b]p10.[/b] On the standard $12$ hour clock, each number from $1$ to $12$ is replaced by the sum of its divisors. On this new clock, what is the number of degrees in the measure of the non-reflex angle between the hands of the clock at the time when the hour hand is between $7$ and $6$ while the minute hand is pointing at $15$? [b]p11.[/b] In equiangular hexagon $ABCDEF$, $AB = 7$, $BC = 3$, $CD = 8$, and $DE = 5$. The area of the hexagon is in the form $\frac{a\sqrt{b}}{c}$ with $b$ square free and $a$ and $c$ relatively prime. Find $a+b+c$ where $a, b,$ and $c$ are integers. [b]p12.[/b] Let $\frac{p}{q} = \frac15 + \frac{2}{5^2} + \frac{3}{5^3} + ...$ . Find $p + q$, where $p$ and $q$ are relatively prime positive integers. [b]p13.[/b] Two circles $F$ and $G$ with radius $10$ and $4$ respectively are externally tangent. A square $ABMC$ is inscribed in circle $F$ and equilateral triangle $MOP$ is inscribed in circle $G$ (they share vertex $M$). If the area of pentagon $ABOPC$ is equal to $a + b\sqrt{c}$, where $a$, $b$, $c$ are integers $c$ is square free, then find $a + b + c$. [b]p14.[/b] Consider the polynomial $P(x) = x^3 + 3x^2 + ax + 8$. Find the sum of all integer $a$ such that the sum of the squares of the roots of $P(x)$ divides the sum of the coecients of $P(x)$. [b]p15.[/b] Nithin and Antonio play a number game. At the beginning of the game, Nithin picks a prime $p$ that is less than $100$. Antonio then tries to find an integer $n$ such that $n^6 + 2n^5 + 2n^4 + n^3 + (n^2 + n + 1)^2$ is a multiple of $p$. If Antonio can find such a number n, then he wins, otherwise, he loses. Nithin doesn't know what he is doing, and he always picks his prime randomly while Antonio always plays optimally. The probability of Antonio winning is $a/b$ where $a$ and $b$ are relatively prime positive integers. Find$a + b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n\ge 3$ be a positive integer. Prove that a polynomial of the form \[x^n+a_{n-3}x^{n-3}+a_{n-4}x^{n-4}+\ldots +a_1x+a_0,\] where at least one of the real coefficients $a_0,a_1,\ldots ,a_{n-3}$ is nonzero, cannot have all real roots.

[b]p1.[/b] An $18$oz glass of apple juice is $6\%$ sugar and a $6$oz glass of orange juice is $12\%$ sugar. The two glasses are poured together to create a cocktail. What percent of the cocktail is sugar? [b]p2.[/b] Find the number of positive numbers that can be expressed as the difference of two integers between $-2$ and $2012$ inclusive. [b]p3.[/b] An annulus is defined as the region between two concentric circles. Suppose that the inner circle of an annulus has radius $2$ and the outer circle has radius $5$. Find the probability that a randomly chosen point in the annulus is at most $3$ units from the center. [b]p4.[/b] Ben and Jerry are walking together inside a train tunnel when they hear a train approaching. They decide to run in opposite directions, with Ben heading towards the train and Jerry heading away from the train. As soon as Ben finishes his $1200$ meter dash to the outside, the front of the train enters the tunnel. Coincidentally, Jerry also barely survives, with the front of the train exiting the tunnel as soon as he does. Given that Ben and Jerry both run at $1/9$ of the train’s speed, how long is the tunnel in meters? [b]p5.[/b] Let $ABC$ be an isosceles triangle with $AB = AC = 9$ and $\angle B = \angle C = 75^o$. Let $DEF$ be another triangle congruent to $ABC$. The two triangles are placed together (without overlapping) to form a quadrilateral, which is cut along one of its diagonals into two triangles. Given that the two resulting triangles are incongruent, find the area of the larger one. [b]p6.[/b] There is an infinitely long row of boxes, with a Ditto in one of them. Every minute, each existing Ditto clones itself, and the clone moves to the box to the right of the original box, while the original Ditto does not move. Eventually, one of the boxes contains over $100$ Dittos. How many Dittos are in that box when this first happens? [b]p7.[/b] Evaluate $$26 + 36 + 998 + 26 \cdot 36 + 26 \cdot 998 + 36 \cdot 998 + 26 \cdot 36 \cdot 998.$$ [b]p8. [/b]There are $15$ students in a school. Every two students are either friends or not friends. Among every group of three students, either all three are friends with each other, or exactly one pair of them are friends. Determine the minimum possible number of friendships at the school. [b]p9.[/b] Let $f(x) = \sqrt{2x + 1 + 2\sqrt{x^2 + x}}$. Determine the value of $$\frac{1}{f(1)}+\frac{1}{f(1)}+\frac{1}{f(3)}+...+\frac{1}{f(24)}.$$ [b]p10.[/b] In square $ABCD$, points $E$ and $F$ lie on segments $AD$ and $CD$, respectively. Given that $\angle EBF = 45^o$, $DE = 12$, and $DF = 35$, compute $AB$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.

Prove that (a) for each $n \ge 1$ $$\sum_{k=0}^n C_{n}^{k} \left(\frac{k}{n}-\frac{1}{2} \right)^2 \frac{1}{2^n}=\frac{1}{4n}$$ (b) for every n \ge m \ge 2 $$\sum_{\ell=0}^n \sum_{k_1+...+k_n=\ell,k_i=0,...,m} \frac{\ell!}{k_1!...k_n!} \frac{1}{(m+1)^n} \left(\frac{\ell}{n}-\frac{m}{2} \right)^2= \left(\frac{m^3-3m^2}{12(m+1)}+\frac{m}{2}-\frac{m}{3(m+1)}\right)n$$

Let $ A$ represent the set of all functions $ f : \mathbb{N} \rightarrow \mathbb{N}$ such that for all $ k \in \overline{1, 2007}$, $ f^{[k]} \neq \mathrm{Id}_{\mathbb{N}}$ and $ f^{[2008]} \equiv \mathrm{Id}_{\mathbb{N}}$. a) Prove that $ A$ is non-empty. b) Find, with proof, whether $ A$ is infinite. c) Prove that all the elements of $ A$ are bijective functions. (Denote by $ \mathbb{N}$ the set of the nonnegative integers, and by $ f^{[k]}$, the composition of $ f$ with itself $ k$ times.)

Compute the sum $\sum^{200}_{n=1}\frac{1}{n(n+1)(n+2)}$ .

If $p$ and $q$ are positive numbers with $p+q = 1$, knowing that any real numbers $x,y$ satisfy $(x-y)^2 \ge 0$, show that $\frac{x+y}{2} \ge \sqrt{xy}$, $\frac{x^2+y^2}{2} \ge \big(\frac{x+y}{2}\big)^2$, $\big(p+\frac{1}{p}\big)^2+\big(q+\frac{1}{q}\big)^2 \ge \frac{25}{2}$

A polynomial of degree four with leading coefficient 1 and integer coefficients has two zeros, both of which are integers. Which of the following can also be a zero of the polynomial? $ \textbf{(A)} \ \frac {1 \plus{} i \sqrt {11}}{2} \qquad \textbf{(B)} \ \frac {1 \plus{} i}{2} \qquad \textbf{(C)} \ \frac {1}{2} \plus{} i \qquad \textbf{(D)} \ 1 \plus{} \frac {i}{2} \qquad \textbf{(E)} \ \frac {1 \plus{} i \sqrt {13}}{2}$

Let $f(x)$ be a periodic function of period $T > 0$ defined over $\mathbb R$. Its first derivative is continuous on $\mathbb R$. Prove that there exist $x, y \in [0, T )$ such that $x \neq y$ and \[f(x)f'(y)=f'(x)f(y).\]

1. A finite sequence of integers $a_0,a_1,...,a_n$ is called quadratic if $|a_k -a_{k-1}| = k^2$ for $n\geq k\geq1$. (a) Prove that for any two integers $b$ and $c$, there exist a natural number $n$ and a quadratic sequence with $a_0 = b$ and $a_n =c$. (b) Find the smallest natural number $n$ for which there exists a quadratic sequence with $a_0 = 0$ and $a_n = 1997$

Solve the equation $x^2+ 2 = 4\sqrt{x^3+1}$

Find the sum of all products $a_1a_2...a_{50}$ , where $a_1,a_2,...,a_{50}$ are distinct positive integers, less than or equal to $101$, and such that no two of them add up to $101$.

Let $Q(x)$ be a quadratic trinomial. Given that the function $P(x)=x^{2}Q(x)$ is increasing in the interval $(0,\infty )$, prove that: \[P(x) + P(y) + P(z) > 0\] for all real numbers $x,y,z$ such that $x+y+z>0$ and $xyz>0$.

Prove that if the real numbers $a,b,c$ lie in the interval $[0,1]$, then \[\frac{a}{bc+1}+\frac{b}{ac+1}+\frac{c}{ab+1}\le 2\]