It is known about positive numbers $a, b, c$ that it is possible to form a triangle from segments of length $a^{2024}, b^{2024}, c^{2024}$. Prove that it is possible to reduce one of the numbers $a, b, c$ by $2024$ times and obtain the numbers $a', b', c'$ so that segments with lengths $a', b', c'$ can also be formed into a triangle. [i]L. Shatunov[/i]

Consider an integer $n \ge 4 $ and a sequence of real numbers $x_1, x_2, x_3,..., x_n$. An operation consists in eliminating all numbers not having the rank of the form $4k + 3$, thus leaving only the numbers $x_3. x_7. x_{11}, ...$(for example, the sequence $4,5,9,3,6, 6,1, 8$ produces the sequence $9,1$). Upon the sequence $1, 2, 3, ..., 1024 $ the operation is performed successively for $5$ times. Show that at the end only one number remains and find this number.

What is the largest $n$ such that $n! + 1$ is a square?

Through an internal point $O$ of $\Delta ABC$ one draws 3 lines, parallel to each of the sides, intersecting in the points shown on the picture. [img]https://cdn.artofproblemsolving.com/attachments/e/3/03d4d1bb61eda8b4a72ff84466d700de47c147.png[/img] Find the value of $\frac{|AF|}{|AB|}+\frac{|BE|}{|BC|}+\frac{|CN|}{|CA|}$.

Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions: (1) for $i=0,1,\cdots,2n-1$, we have $a_i+a_{i+1}\geq 0$; (2) for $j=0,1,\cdots,n-1$, we have $a_{2j+1}\leq 0$; (2) for any integer $p,q$, $0\leq p\leq q\leq n$, we have $\sum_{k=2p}^{2q}b_k>0$. Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$, and determine when the equality holds.

Exactly $2^{1012}$ of the subsets of $\{1, 2, \ldots, 2023\}$ are colored red. Is it always true that there exist three distinct red sets $A$, $B$, and $C$ such that every element of $A$ belongs to at least one of $B$ or $C$?

For which of the following values of real number $t$, the equation $x^4-tx+\dfrac 1t = 0$ has no root on the interval $[1,2]$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ \text{None of the preceding} $

In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members. [i]A. Skopenkov[/i]

Given that $x$ is a real number, compute the minimum possible value of $(x-20)^2 + (x-18)^2$.

Patchouli is taking an exam with $k > 1$ parts, numbered Part $1, 2, \dots, k$. It is known that for $i = 1, 2, \dots, k$, Part $i$ contains $i$ multiple choice questions, each of which has $(i+1)$ answer choices. It is known that if she guesses randomly on every single question, the probability that she gets exactly one question correct is equal to $2018$ times the probability that she gets no questions correct. Compute the number of questions that are on the exam. [i]Proposed by Yannick Yao[/i]

$m,n,p$ are positive integers which do not simultaneously equal to zero. $3$D Cartesian space is divided into unit cubes by planes each perpendicular to one of $3$ axes and cutting corresponding axis at integer coordinates. Each unit cube is filled with an integer from $1$ to $60$. A filling of integers is called [i]Dien Bien[/i] if, for each rectangular box of size $\{2m+1,2n+1,2p+1\}$, the number in the unit cube which has common centre with the rectangular box is the average of the $8$ numbers of the $8$ unit cubes at the $8$ corners of that rectangular box. How many [i]Dien Bien[/i] fillings are there? Two fillings are the same if one filling can be transformed to the other filling via a translation. [hide]translation from [url=http://artofproblemsolving.com/community/c6h592875p3515526]here[/url][/hide]

For any positive integer $n$, let $f(n)$ denote the sum of the positive integers $k \le n$ such that $k$ and $n$ are relatively prime. Let $S$ be the sum of $\frac{1}{f(m)}$ over all positive integers $m$ that are divisible by at least one of $2$, $3$, or $5$, and whose prime factors are only $2$, $3$, or $5$. Then $S = \frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $p+q$.

A factory packs its products in cubic boxes. In one store, they put $512$ of these cubic boxes together to make a large $8\times 8 \times 8$ cube. When the temperature goes higher than a limit in the store, it is necessary to separate the $512$ set of boxes using horizontal and vertical plates so that each box has at least one face which is not touching other boxes. What is the least number of plates needed for this purpose?

Let $A = \{1,...,n\}$ with $n \textgreater 5$. Prove that one can find $B$ a finite set of positive integers such that $A$ is a subset of $B$ and $\displaystyle\sum_{x \in B} x^2 = \displaystyle\prod_{x \in B} x$

Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other

[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].