Determine the smallest positive real $K$ such that the inequality \[ K + \frac{a + b + c}{3} \ge (K + 1) \sqrt{\frac{a^2 + b^2 + c^2}{3}} \]holds for any real numbers $0 \le a,b,c \le 1$. [i]Proposed by Fajar Yuliawan, Indonesia[/i]

\[\int_0^2 e^x(x^4+8x^3+18x^2+16x+5)\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

[u]Part 1 [/u] [b]p1.[/b] Calculate $$(4!-5!+2^5 +2^6) \cdot \frac{12!}{7!}+(1-3)(4!-2^4).$$ [b]p2.[/b] The expression $\sqrt{9!+10!+11!}$ can be expressed as $a\sqrt{b}$ for positive integers $a$ and $b$, where $b$ is squarefree. Find $a$. [b]p3.[/b] For real numbers $a$ and $b$, $f(x) = ax^{10}-bx^4+6x +10$ for all real $x$. Given that $f(42) = 11$, find $f (-42)$. [u]Part 2[/u] [b]p4.[/b] How many positive integers less than or equal to $2023$ are divisible by $20$, $23$, or both? [b]p5.[/b] Larry the ant crawls along the surface of a cylinder with height $48$ and base radius $\frac{14}{\pi}$ . He starts at point $A$ and crawls to point $B$, traveling the shortest distance possible. What is the maximum this distance could be? [b]p6.[/b] For a given positive integer $n$, Ben knows that $\lfloor 20x \rfloor = n$, where $x$ is real. With that information, Ben determines that there are $3$ distinct possible values for $\lfloor 23x \rfloor$. Find the least possible value of $n$. [u]Part 3 [/u] [b]p7.[/b] Let $ABC$ be a triangle with area $1$. Points $D$, $E$, and $F$ lie in the interior of $\vartriangle ABC$ in such a way that $D$ is the midpoint of $AE$, $E$ is the midpoint of $BF$, and $F$ is the midpoint of $CD$. Compute the area of $DEF$. [b]p8.[/b] Edwin and Amelia decide to settle an argument by running a race against each other. The starting line is at a given vertex of a regular octahedron and the finish line is at the opposite vertex. Edwin has the ability to run straight through the octahedron, while Amelia must stay on the surface of the octahedron. Given that they tie, what is the ratio of Edwin’s speed to Amelia’s speed? [b]p9.[/b] Jxu is rolling a fair three-sided die with faces labeled $0$, $1$, and $2$. He keeps going until he rolls a $1$, immediately followed by a $2$. What is the expected number of rolls Jxu makes? [u]Part 4 [/u] [b]p10.[/b] For real numbers $x$ and $y$, $x +x y = 10$ and $y +x y = 6$. Find the sum of all possible values of $\frac{x}{y}$. [b]p11.[/b] Derek is thinking of an odd two-digit integer $n$. He tells Aidan that $n$ is a perfect power and the product of the digits of $n$ is also a perfect power. Find the sum of all possible values of $n$. [b]p12.[/b] Let a three-digit positive integer $N = \overline{abc}$ (in base $10$) be stretchable with respect to $m$ if $N$ is divisible by $m$, and when $N$‘s middle digit is duplicated an arbitrary number of times, it‘s still divisible by $m$. How many three-digit positive integers are stretchable with respect to $11$? (For example, $432$ is stretchable with respect to $6$ because $433...32$ is divisible by $6$ for any positive integer number of $3$s.) [u]Part 5 [/u] [b]p13.[/b] How many trailing zeroes are in the base-$2023$ expansion of $2023!$ ? [b]p14.[/b] The three-digit positive integer $k = \overline{abc}$ (in base $10$, with a nonzero) satisfies $\overline{abc} = c^{2ab-1}$. Find the sum of all possible $k$. [b]p15.[/b] For any positive integer $k$, let $a_k$ be defined as the greatest nonnegative real number such that in an infinite grid of unit squares, no circle with radius less than or equal to $a_k$ can partially cover at least $k$ distinct unit squares. (A circle partially covers a unit square only if their intersection has positive area.) Find the sumof all positive integers $n \le 12$ such that $a_n \ne a_{n+1}$. PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3267915p30057005]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let f be a function defined in the set $\{0, 1, 2,...\}$ of non-negative integers, satisfying $f(2x) = 2f(x), f(4x+1) = 4f(x) + 3$, and $f(4x-1) = 2f(2x - 1) -1$. Show that $f $ is an injection, i.e. if $f(x) = f(y)$, then $x = y$.

Find all triples of integers \( (x, y, z) \) such that \[ x - yz = 11 \] \[ xz + y = 13 \]

[b]p1.[/b] It takes $12$ months for Santa Claus to pack gifts. It would take $20$ months for his apprentice to do the job. If they work together, how long will it take for them to pack the gifts? [b]p2.[/b] All passengers on a bus sit in pairs. Exactly $2/5$ of all men sit with women, exactly $2/3$ of all women sit with men. What part of passengers are men? [b]p3.[/b] There are $100$ colored balls in a box. Every $10$-tuple of balls contains at least two balls of the same color. Show that there are at least $12$ balls of the same color in the box. [b]p4.[/b] There are $81$ wheels in storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that one can determine which wheel is incorrectly marked with four measurements. [b]p5.[/b] Remove from the figure below the specified number of matches so that there are exactly $5$ squares of equal size left: (a) $8$ matches (b) $4$ matches [img]https://cdn.artofproblemsolving.com/attachments/4/b/0c5a65f2d9b72fbea50df12e328c024a0c7884.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We consider the $n$ vertices of a regular polygon with $n$ sides. There is a set of triangles with vertices at these $n$ points with the property that for each triangle in the set, the sides of at least one are not the side of any other triangle in the set. What is the largest amount of triangles that can have the set? [hide=original wording]Consideramos los n vértices de un polígono regular de n lados. Se tiene un conjunto de triángulos con vértices en estos n puntos con la propiedad que para cada triángulo del conjunto, al menos uno de sus lados no es lado de ningún otro triángulo del conjunto. ¿Cuál es la mayor cantidad de triángulos que puede tener el conjunto?[/hide]

Determine all polynomials $P(x)$ with real coefficients and which satisfy the following properties: i) $P(0) = 1$ ii) for any real numbers $x$ and $y,$ \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\]

Prove that for arbitrary real numbers $a$ and $b$ the following inequality is true $$a^2 +ab+b^2 \geq 3(a+b-1).$$

Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move: $(3,2),(2,1,2),(4),(4,1),(2,2),$ or $(1,1,2).$ [asy] unitsize(4mm); real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33}; for(real i:boxes){ draw(box((i,0),(i+1,3))); } draw((8,1.5)--(12,1.5),Arrow()); defaultpen(fontsize(20pt)); label(",",(20,0)); label(",",(29,0)); label(",...",(35.5,0)); [/asy] Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth? $\textbf{(A) }(6,1,1) \qquad \textbf{(B) }(6,2,1) \qquad \textbf{(C) }(6,2,2)\qquad \textbf{(D) }(6,3,1) \qquad \textbf{(E) }(6,3,2)$

The five solutions to the equation $$(z-1)(z^2+2z+4)(z^2+4z+6)=0$$ may be written in the form $x_k+y_ki$ for $1\le k\le 5,$ where $x_k$ and $y_k$ are real. Let $\mathcal E$ be the unique ellipse that passes through the points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),$ and $(x_5,y_5)$. The eccentricity of $\mathcal E$ can be written in the form $\sqrt{\frac mn}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$? (Recall that the [i]eccentricity[/i] of an ellipse $\mathcal E$ is the ratio $\frac ca$, where $2a$ is the length of the major axis of $E$ and $2c$ is the is the distence between its two foci.) $\textbf{(A) }7 \qquad \textbf{(B) }9 \qquad \textbf{(C) }11 \qquad \textbf{(D) }13\qquad \textbf{(E) }15$ Proposed by [b]djmathman[/b]

A regular tetrahedron has side $L$. What is the smallest $x$ such that the tetrahedron can be passed through a loop of twine of length $x$?

Let $a, b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that $$\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 $$ When equality holds?

Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.

The sequence ${a_{n}}$ $n\in \mathbb{N}$ is given in a recursive way with $a_{1}=1$, $a_{n}=\prod_{i=1}^{n-1} a_{i}+1$, for all $n\geq 2$. Determine the least number $M$, such that $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$ for all $m\in \mathbb{N}$

If, in the expression $ x^2 \minus{} 3$, $ x$ increases or decreases by a positive amount of $ a$, the expression changes by an amount: $ \textbf{(A)}\ {\pm 2ax \plus{} a^2}\qquad \textbf{(B)}\ {2ax \pm a^2}\qquad \textbf{(C)}\ {\pm a^2 \minus{} 3} \qquad \textbf{(D)}\ {(x \plus{} a)^2 \minus{} 3}\qquad\\ \textbf{(E)}\ {(x \minus{} a)^2 \minus{} 3}$

On a $2\times 40$ chessboard colored black and white in the standard alternating pattern, $20$ rooks are placed randomly on the black squares. The expected number of white squares with only rooks as neighbors can be expressed as $a/b$, where $a$ and $b$ are coprime positive integers. What is $a + b$? (Two squares are said to be neighbors if they share an edge.)

Determine the greatest integer $N$ such that $N$ is a divisor of $n^{13}-n$ for all integers $n$.

Does there exist two infinite sets $A,B$ such that every number can be written uniquely as a sum of an element of $A$ and an element of $B$?

Find all pairs $(a, b)$ of real numbers such that the roots of polynomials $6x^2 -24x -4a$ and $x^3 + ax^2 + bx - 8$ are all non-negative real numbers.

In the triangle $ABC$ the angle bisectors $AL$ and $BT$ are drawn, which intersect at the point $I$, and their extensions intersect the circle circumscribed around the triangle $ABC$ at the points $E$ and $D$ respectively. The segment $DE$ intersects the sides $AC$ and $BC$ at the points $F$ and $K$, respectively. Prove that: a) quadrilateral $IKCF$ is rhombus; b) the side of this rhombus is $\sqrt {DF \cdot EK}$. (Rozhkova Maria)

Given an odd prime $p$, find all functions $f:Z \rightarrow Z$ satisfying the following two conditions: (i) $f(m)=f(n)$ for all $m,n \in Z$ such that $m\equiv n\pmod p$; (ii) $f(mn)=f(m)f(n)$ for all $m,n \in Z$.

Let $n$ be a positive integer. Prove that there exists a finite sequence $S$ consisting of only zeros and ones, satisfying the following property: for any positive integer $d \geq 2$, when $S$ is interpreted in base $d$, the resulting number is non-zero and divisible by $n$. [i]Remark: The sequence $S=s_ks_{k-1} \cdots s_1s_0$ interpreted in base $d$ is the number $\sum_{i=0}^{k}s_id^i$[/i]

How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 1$? (A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.

For positive integers $n,d$, define $n \% d$ to be the unique value of the positive integer $r < d$ such that $n = qd + r$, for some positive integer $q$. What is the smallest value of $n$ not divisible by $5,7,11,13$ for which $n^2 \% 5 < n^2 \% 7 < n^2 \% 11 < n^2 \% 13$?