For positive integers $i$ and $N$, let $k_{N,i}$ be the $i$th smallest positive integer such that the polynomial $\frac{x^2}{2023} + \frac{N_x}{7} - k_{N,i}$ has integer roots. Compute the minimum positive integer $N$ satisfying the condition $\frac{k_{N,2023}}{k_{N,1000}}< 3$. Submit your answer as a positive integer $E$. If the correct answer is $A$, your score for this question will be $\max \left( 0, 25 \min \left( \frac{A}{E} , \frac{E}{A}\right)^{\frac32}\right)$, rounded to the nearest integer.

Let $ABC$ be a triangle with $AC = 28$, $BC = 33$, and $\angle ABC = 2\angle ACB$. Compute the length of side $AB$. [i]2016 CCA Math Bonanza #10[/i]

$ABC$ is a triangle with a circumscribed circle $k$, center $I$ of its inscribed circle $\omega$, and center $I_a$ of its excircle $\omega _a$, opposite to $A$. $\omega$ and $\omega _a$ are tangent to $BC$ in points $P$ and $Q$, respectively, and $S$ is the middle point of the arc $\widehat{BC}$ that doesn’t contain $A$. Consider a circle that is tangent to $BC$ in point $P$ and to $k$ in point $R$. Let $RI$ intersect $k$ for a second time in point $L$. Prove that, $LI_a$ and $SQ$ intersect in a point that lies on $k$.

The diagram below shows a triangle divided into sections by three horizontal lines which divide the altitude of the triangle into four equal parts, and three lines connecting the top vertex with points that divide the opposite side into four equal parts. If the shaded region has area $100$, find the area of the entire triangle. [asy] import graph; size(5cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; filldraw((-1,2.5)--(-1,1.75)--(0.5,1.75)--(0,2.5)--cycle,grey); draw((-1,4)--(-2,1)); draw((-1,4)--(2,1)); draw((-2,1)--(2,1)); draw((-1,4)--(-1,1)); draw((-1,4)--(-0.5,2.5)); draw((-0.25,1.75)--(0,1)); draw((-1,2.5)--(-1,1.75)); draw((-1,1.75)--(0.5,1.75)); draw((0.5,1.75)--(0,2.5)); draw((0,2.5)--(-1,2.5)); draw((-1.25,3.25)--(-0.25,3.25)); draw((-1.5,2.5)--(0.5,2.5)); draw((1.25,1.75)--(-1.75,1.75)); draw((-1,4)--(0,2.5)); draw((0.47,1.79)--(1,1)); dot((-1,1),dotstyle); dot((0,1),dotstyle); dot((1,1),dotstyle); dot((-1.25,3.25),dotstyle); dot((-1.5,2.5),dotstyle); dot((-1.75,1.75),dotstyle); dot((1.25,1.75),dotstyle); dot((0.5,2.5),dotstyle); dot((-0.25,3.25),dotstyle); [/asy]

Let $a, b, c$ be positive real numbers. Prove that $$\frac{1}{a+b+\frac{1}{abc}+1}+\frac{1}{b+c+\frac{1}{abc}+1}+\frac{1}{c+a+\frac{1}{abc}+1}\le \frac{a + b + c}{a+b+c+1}$$

Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma < \pi$, $\alpha+\beta > \gamma$,$ \beta+\gamma > \alpha$, $\gamma + \alpha > \beta.$ Prove that with the segments of lengths $\sin \alpha, \sin \beta, \sin \gamma $ we can construct a triangle and that its area is not greater than \[A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).\] [i]Proposed by Soviet Union[/i]

Let $f:[0, \infty) \to (0, \infty)$ be an increasing function and $g:[0, \infty) \to \mathbb{R}$ be a two times differentiable function such that $g''$ is continuous and $g''(x)+f(x)g(x) = 0, \: \forall x \geq 0.$ $\textbf{a)}$ Provide an example of such functions, with $g \neq 0.$ $\textbf{b)}$ Prove that $g$ is bounded.

Sofia summed all the page numbers from a book starting at $1$ and getting $2013$. Pablo saw how she did this and realized Sofia skipped a page. How many pages does the book have, and what page did Sofia skip?

A four-digit number has the following properties: (a) It is a perfect square; (b) Its first two digits are equal (c) Its last two digits are equal. Find all such four-digit numbers.

The [i]binary sudoku[/i] is a puzzle in which a table should be filled with digits $0$ and $1$ such that in each row and column, the number of 0s is equal to the number of $1$s. Furthermore, there cannot exist three adjacent cells in a row or in a column such that they have the same digit written in them. Solving the given binary sudoku, what is the sum of the numbers in the two diagonals? [img]https://cdn.artofproblemsolving.com/attachments/a/8/be7de94ce02a90b3cabf1b9795b94ec7ec677f.png[/img]

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a continuous function such that for any $x\in \mathbb{R}$, the limit $\lim_{h\to 0} \left|\frac{f(x+h)-f(x)}{h}\right|$ exists and it is finite. Prove that in any real point, $f$ is differentiable or it has finite one-side derivates, of the same modul, but different signs.

The pieces in a game are squares of side $1$ with their sides colored with $4$ colors: blue, red, yellow and green, so that each piece has one side of each color. There are pieces in every possible color arrangement, and the game has a million pieces of each kind. With the pieces, rectangular puzzles are assembled, without gaps or overlaps, so that two pieces that share a side have that side of the same color. Determine if with this procedure you can make a rectangle of $99\times 2007$ with one side of each color. And $100\times 2008$? And $99\times 2008$?

Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.

(a) For non negetive $a,b,c, r$ prove that \[a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r (c-a)(c-b) \geq 0 \] (b) Find an inequality for non negative $a,b,c$ with $a^4+b^4+c^4 + abc(a+b+c)$ on the greater side. (c) Prove that if $abc = 1$ for non negative $a,b,c$, $a^4+b^4+c^4+a^3+b^3+c^3+a+b+c \geq \frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}+3$

Let $n$ points be given inside a rectangle $R$ such that no two of them lie on a line parallel to one of the sides of $R$. The rectangle $R$ is to be dissected into smaller rectangles with sides parallel to the sides of $R$ in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect $R$ into at least $n + 1$ smaller rectangles. [i]Proposed by Serbia[/i]

The sides of the triangle $ABC$ satisfy the relation $c^2 = a^2 + b^2$. Show that angle $C$ is right.

[b]p22.[/b] Consider the series $\{A_n\}^{\infty}_{n=0}$, where $A_0 = 1$ and for every $n > 0$, $$A_n = A_{\left[ \frac{n}{2023}\right]} + A_{\left[ \frac{n}{2023^2}\right]}+A_{\left[ \frac{n}{2023^3}\right]},$$ where $[x]$ denotes the largest integer value smaller than or equal to $x$. Find the $(2023^{3^2}+20)$-th element of the series. [b]p23.[/b] The side lengths of triangle $\vartriangle ABC$ are $5$, $7$ and $8$. Construct equilateral triangles $\vartriangle A_1BC$, $\vartriangle B_1CA$, and $\vartriangle C_1AB$ such that $A_1$,$B_1$,$C_1$ lie outside of $\vartriangle ABC$. Let $A_2$,$B_2$, and $C_2$ be the centers of $\vartriangle A_1BC$, $\vartriangle B_1CA$, and $\vartriangle C_1AB$, respectively. What is the area of $\vartriangle A_2B_2C_2$? [b]p24. [/b]There are $20$ people participating in a random tag game around an $20$-gon. Whenever two people end up at the same vertex, if one of them is a tagger then the other also becomes a tagger. A round consists of everyone moving to a random vertex on the $20$-gon (no matter where they were at the beginning). If there are currently $10$ taggers, let $E$ be the expected number of untagged people at the end of the next round. If $E$ can be written as $\frac{a}{b}$ for $a, b$ relatively prime positive integers, compute $a + b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ n$ be a positive integer and let $ a_1,a_2,a_3,\ldots,a_k$ $ ( k\ge 2)$ be distinct integers in the set $ { 1,2,\ldots,n}$ such that $ n$ divides $ a_i(a_{i + 1} - 1)$ for $ i = 1,2,\ldots,k - 1$. Prove that $ n$ does not divide $ a_k(a_1 - 1).$ [i]Proposed by Ross Atkins, Australia [/i]

Let $ABCD$ be an isosceles trapezoid with bases $AD> BC$. Let $X$ be the intersection of the bisectors of $\angle BAC$ and $BC$. Let $E$ be the intersection of$ DB$ with the parallel to the bisector of $\angle CBD$ through $X$ and let $F$ be the intersection of $DC$ with the parallel to the bisector of $\angle DCB$ through $X$. Show that quadrilateral $AEFD$ is cyclic.

Jamal wants to store $ 30$ computer files on floppy disks, each of which has a capacity of $ 1.44$ megabytes (MB). Three of his files require $ 0.8$ MB of memory each, $ 12$ more require $ 0.7$ MB each, and the remaining $ 15$ require $ 0.4$ MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 16$

Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

Prove that there are only [b]finitely[/b] positive integer $a$ such that $a-2006=\sum\limits_{i=1}^{2006} 2^ia_i$ with $\{a_i\}$ as divisors (not necessary distinct) of $n$.

Let $B$ and $C$ be two fixed points in the plane. For each point $A$ of the plane, outside of the line $BC$, let $G$ be the barycenter of the triangle $ABC$. Determine the locus of points $A$ such that $\angle BAC + \angle BGC = 180^{\circ}$. Note: The locus is the set of all points of the plane that satisfies the property.

$PA_1A_2...A_n$ is a pyramid. The base $A_1A_2...A_n$ is a regular n-gon. The apex $P$ is placed so that the lines $PA_i$ all make an angle $60^{\cdot}$ with the plane of the base. For which $n$ is it possible to find $B_i$ on $PA_i$ for $i = 2, 3, ... , n$ such that $A_1B_2 + B_2B_3 + B_3B_4 + ... + B_{n-1}B_n + B_nA_1 < 2A_1P$?

An hourglass is formed from two identical cones. Initially, the upper cone is filled with sand and the lower one is empty. The sand flows at a constant rate from the upper to the lower cone. It takes exactly one hour to empty the upper cone. How long does it take for the depth of sand in the lower cone to be half the depth of sand in the upper cone? (Assume that the sand stays level in both cones at all times.)