Let $S = \frac{1}{a_1}+\frac{2}{a_2}+ ... +\frac{100}{a_{100}}$ where $a_1, a_2,..., a_{100}$ are positive integers. What are all the possible integer values that $S$ can take ?

Six consecutive positive integers are written on slips of paper. The slips are then handed out to Ethan, Jacob, and Karthik, such that each of them receives two slips. The product of Ethan's numbers is $20,$ and the product of Jacob's numbers is $24.$ Compute the product of Karthik's numbers.

For $a>0$, let $f(a)=\lim_{t\rightarrow +0} \int_{t}^{1} |ax+x\ln x|\ dx.$ Let $a$ vary in the range $0 <a< +\infty$, find the minimum value of $f(a)$.

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [b]Z15.[/b] Let $AOB$ be a quarter circle with center $O$ and radius $4$. Let $\omega_1$ and $\omega_2$ be semicircles inside $AOB$ with diameters $OA$ and $OB$, respectively. Find the area of the region within $AOB$ but outside of $\omega_1$ and $\omega_2$. [u]Set 4[/u] [b]Z16.[/b] Integers $a, b, c$ form a geometric sequence with an integer common ratio. If $c = a + 56$, find $b$. [b]Z17 / D24.[/b] In parallelogram $ABCD$, $\angle A \cdot \angle C - \angle B \cdot \angle D = 720^o$ where all angles are in degrees. Find the value of $\angle C$. [b]Z18.[/b] Steven likes arranging his rocks. A mountain formation is where the sequence of rocks to the left of the tallest rock increase in height while the sequence of rocks to the right of the tallest rock decrease in height. If his rocks are $1, 2, . . . , 10$ inches in height, how many mountain formations are possible? For example: the sequences $(1-3-5-6-10-9-8-7-4-2)$ and $(1-2-3-4-5-6-7-8-9-10)$ are considered mountain formations. [b]Z19.[/b] Find the smallest $5$-digit multiple of $11$ whose sum of digits is $15$. [b]Z20.[/b] Two circles, $\omega_1$ and $\omega_2$, have radii of $2$ and $8$, respectively, and are externally tangent at point $P$. Line $\ell$ is tangent to the two circles, intersecting $\omega_1$ at $A$ and $\omega_2$ at $B$. Line $m$ passes through $P$ and is tangent to both circles. If line $m$ intersects line $\ell$ at point $Q$, calculate the length of $P Q$. [u]Set 5[/u] [b]Z21.[/b] Sen picks a random $1$ million digit integer. Each digit of the integer is placed into a list. The probability that the last digit of the integer is strictly greater than twice the median of the digit list is closest to $\frac{1}{a}$, for some integer $a$. What is $a$? [b]Z22.[/b] Let $6$ points be evenly spaced on a circle with center $O$, and let $S$ be a set of $7$ points: the $6$ points on the circle and $O$. How many equilateral polygons (not self-intersecting and not necessarily convex) can be formed using some subset of $S$ as vertices? [b]Z23.[/b] For a positive integer $n$, define $r_n$ recursively as follows: $r_n = r^2_{n-1} + r^2_{n-2} + ... + r^2_0$,where $r_0 = 1$. Find the greatest integer less than $$\frac{r_2}{r^2_1}+\frac{r_3}{r^2_2}+ ...+\frac{r_{2023}}{r^2_{2022}}.$$ [b]Z24.[/b] Arnav starts at $21$ on the number line. Every minute, if he was at $n$, he randomly teleports to $2n^2$, $n^2$, or $\frac{n^2}{4}$ with equal chance. What is the probability that Arnav only ever steps on integers? [b]Z25.[/b] Let $ABCD$ be a rectangle inscribed in circle $\omega$ with $AB = 10$. If $P$ is the intersection of the tangents to $\omega$ at $C$ and $D$, what is the minimum distance from $P$ to $AB$? PS. You should use hide for answers. D.1-15 / Z.1-8 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916240p26045561]here [/url]and D.16-30/Z.9-14, 17, 26-30 [url=https://artofproblemsolving.com/community/c3h2916250p26045695]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a<b<c$ be three positive integers. Prove that among any $2c$ consecutive positive integers there exist three different numbers $x,y,z$ such that $abc$ divides $xyz$.

Let $a_1, a_2,...,a_{2005}, b_1, b_2,...,b_{2005}$ be real numbers such that $(a_ix - b_i)^2 \ge \sum_{j\ne i,j=1}^{2005} (a_jx - b_j)$ for all real numbers x and every integer $i$ with $1 \le i \le 2005$. What is maximal number of positive $a_i$'s and $b_i$'s?

Karl starts with $n$ cards labeled $1,2,3,\dots,n$ lined up in a random order on his desk. He calls a pair $(a,b)$ of these cards [i]swapped[/i] if $a>b$ and the card labeled $a$ is to the left of the card labeled $b$. For instance, in the sequence of cards $3,1,4,2$, there are three swapped pairs of cards, $(3,1)$, $(3,2)$, and $(4,2)$. He picks up the card labeled 1 and inserts it back into the sequence in the opposite position: if the card labeled 1 had $i$ card to its left, then it now has $i$ cards to its right. He then picks up the card labeled $2$ and reinserts it in the same manner, and so on until he has picked up and put back each of the cards $1,2,\dots,n$ exactly once in that order. (For example, the process starting at $3,1,4,2$ would be $3,1,4,2\to 3,4,1,2\to 2,3,4,1\to 2,4,3,1\to 2,3,4,1$.) Show that, no matter what lineup of cards Karl started with, his final lineup has the same number of swapped pairs as the starting lineup.

In triangle ABC, let $P$ and $Q$ be two interior points such that $\angle ABP = \angle QBC$ and $\angle ACP = \angle QCB$. Point $D$ lies on segment $BC$. Prove that $\angle APB + \angle DPC = 180^\circ$ if and only if $\angle AQC + \angle DQB = 180^\circ$.

If $A, B, C$, and $D$ are four distinct points in space, prove that there is a plane $P$ on which the orthogonal projections of $A, B, C$, and $D$ form a parallelogram (possibly degenerate).

Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to factorials of some positive integers. Proposed by [i]Nikola Velov, Macedonia[/i]

Let $a, b \in \Gamma$, $a < b$ and the differentiable function $f : [a, b] \to \Gamma$, such that $f (a) = a$ and $f (b) = b$. Prove that $$\int \limits_{a}^{b} \left(f'(x)\right)^2dx \geq b-a$$

a) Prove that every sub-group $(A,+)$ of group $(\mathbb{Z},+)$ is in the form $A=n \cdot \mathbb{Z}$ for some $n \in \mathbb{Z}$ where $n \cdot \mathbb{Z}=\{n \cdot x/x\in\mathbb{Z}\}$. b) Using problem (a) , prove that the greatest common divisor $d$ of non zero integers $a_1, a_2,... ,a_n$ is given by relation $d=\lambda_1a_1+\lambda_2 a_2+...\lambda_n a_n$ with $\lambda_i\in\mathbb{Z}, \,\, i=1,2,...,n$

Fill in each space of the grid with one of the numbers $1,2,\dots,30$, using each number once. For $1\le{}n\le29$, the two spaces containing $n$ and $n+1$ must be in either the same row or the same column. Some numbers have been given to you. [asy] unitsize(32); int[][] a = { {29, 000, 000, 000, 000, 000}, {000, 19, 000, 000, 17, 000}, {13, 000, 000, 21, 000, 8}, {000, 4, 000, 15, 000, 24}, {10, 000, 000, 000, 26, 000}}; for (int i = 0; i < 6; ++i) { for (int j = 0; j < 5; ++j) { draw((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle); if (a[j][i] > 0 && a[j][i] < 999) label(string(a[j][i]), (i+0.5, -j-0.5), fontsize(30pt)); } } [/asy] You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)

Find all pairs of positive integers $(x, y)$ so that $$(xy - 6)^2 | x^2 + y^2$$

On the sides $AB$ and $AC$ of the right $\triangle ABC$ ($\angle A=90^{\circ}$) are chosen points $C_{1}$ and $B_{1}$ respectively. Prove that if $M=CC_{1}\cap BB_{1}$ and $AC_{1}=AB_{1}=AM$, then $[AB_{1}MC_{1}]+[AB_{1}C_{1}]=[BMC]$.

Let $M$ be a finite subset of the plane such that for any two different points $A,B\in M$ there is a point $C\in M$ such that $ABC$ is equilateral. What is the maximal number of points in $M?$

In triangle $ ABC$, $ AB \equal{} 13$, $ BC \equal{} 14$, and $ AC \equal{} 15$. Let $ D$ denote the midpoint of $ \overline{BC}$ and let $ E$ denote the intersection of $ \overline{BC}$ with the bisector of angle $ BAC$. Which of the following is closest to the area of the triangle $ ADE$? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2.5 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 3.5 \qquad \textbf{(E)}\ 4$

Prove that if $ A, B, C $ are angles of a triangle, then $$ 1 < \cos A + \cos B + \cos C \leq \frac{3}{2}.$$

Given a circle, let $AB$ be a chord that is not a diameter, and let $C$ be a point on the longer arc $AB$. Let $K$ and $L$ denote the reflections of $A$ and $B$, respectively, about lines $BC$ and $AC$, respectively. Prove that the distance between the midpoint of $AB$ and the midpoint of $KL$ is independent of the choice of $C$.

We have a $ (n+2)\times n $ rectangle and we’ve divided it into $ n(n+2) \ \ 1\times1 $ squares. $ n(n+2) $ soldiers are standing on the intersection points ($ n+2 $ rows and $ n $ columns). The commander shouts and each soldier stands on its own location or gaits one step to north, west, east or south so that he stands on an adjacent intersection point. After the shout, we see that the soldiers are standing on the intersection points of a $ n\times(n+2) $ rectangle ($ n $ rows and $ n+2 $ columns) such that the first and last row are deleted and 2 columns are added to the right and left (To the left $1$ and $1$ to the right). Prove that $ n $ is even.

We are given $N$ lines ($N > 1$ ) in a plane, no two of which are parallel and no three of which have a point in common. Prove that it is possible to assign, to each region of the plane determined by these lines, a non-zero integer of absolute value not exceeding $N$ , such that the sum of the integers o n either side of any of the given lines is equal to $0$ . (S . Fomin, Leningrad)

For any positive integer $m\ge2$ define $A_m=\{m+1, 3m+2, 5m+3, 7m+4, \ldots, (2k-1)m + k, \ldots\}$. (1) For every $m\ge2$, prove that there exists a positive integer $a$ that satisfies $1\le a<m$ and $2^a\in A_m$ or $2^a+1\in A_m$. (2) For a certain $m\ge2$, let $a, b$ be positive integers that satisfy $2^a\in A_m$, $2^b+1\in A_m$. Let $a_0, b_0$ be the least such pair $a, b$. Find the relation between $a_0$ and $b_0$.

Let $\vartriangle A_1B_1C_1$ and $l_1, m_1, n_1$ be the trisectors closest to $A_1B_1$, $B_1C_1$, $C_1A_1$ of the angles $A_1, B_1, C_1$ respectively. Let $A_2 = l_1 \cap n_1$, $B_2 = m_1 \cap l_1$, $C_2 = n_1 \cap m_1$. So on we create triangles $\vartriangle A_nB_nC_n$ . If $\vartriangle A_1B_1C_1$ is equilateral prove that exists $n \in N$, such that all the sides of $\vartriangle A_nB_nC_n$ are parallel to the sides of $\vartriangle A_1B_1C_1$.

Point $O$ is the foot of the altitude of a quadrilateral pyramid. A sphere with center $O$ is tangent to all lateral faces of the pyramid. Points $A,B,C,D$ are taken on successive lateral edges so that segments $AB$, $BC$, and $CD$ pass through the three corresponding tangency points of the sphere with the faces. Prove that the segment $AD$ passes through the fourth tangency point

The numbers from $1$ to $64$ must be written on the small squares of a chessboard, with a different number in each small square. Consider the $112$ numbers you can make by adding the numbers in two small squares which have a common edge. Is it possible to write the numbers in the squares so that these $112$ sums are all different?