An increasing arithmetic progression consists of one hundred positive integers. Is it possible that every two of them are relatively prime?

Let $H(D)$ denote the space of functions holomorphic on the disc $D=\{ z\colon |z|<1 \}$, endowed with the topology of uniform convergence on each compact subset of $D$. If $f(z)=\sum_{n=0}^{\infty} a_nz^n$, then we shall denote $S_n(f,z)=\sum_{k=0}^n a_kz^k$. A function $f\in H(D)$ is called [i]universal[/i] if, for every continuous function $g\colon\partial D\rightarrow \mathbb{C}$ and for every $\varepsilon >0$, there are partial sums $S_{n(j)}(f,z)$ approximating $g$ uniformly on the arc $\{ e^{it} \colon 0\le t\le 2\pi - \varepsilon\}$. Prove that the set of universal functions contains a dense $G_{\delta}$ subset of $H(D)$.

A $25$ foot ladder is placed against a vertical wall of a building. The foot of the ladder is $7$ feet from the base of the building. If the top of the ladder slips $4$ feet, then the foot of the ladder will slide: $\textbf{(A)}\ 9\text{ ft} \qquad \textbf{(B)}\ 15\text{ ft} \qquad \textbf{(C)}\ 5\text{ ft} \qquad \textbf{(D)}\ 8\text{ ft} \qquad \textbf{(E)}\ 4\text{ ft}$

In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$, let $M$ be the midpoint of $\overline{BC}$. Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$. Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$. Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$, and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$. Prove that $A$, $Y$, and $Z$ are collinear. [i]Tiger Zhang[/i]

Prove that if in the tetrahedron $ ABCD $ we have $ AB = CD $, $ AC = BD $, $ AD = BC $, then all faces of the tetrahedron are acute-angled triangles.

Let S be a set of positive integers such that: min { lcm (x, y) : x, y ∈ S, $x \neq y$ } $\ge$ 2 + max S. Prove that $\displaystyle\sum\limits_{x \in S} \frac{1}{x} \le \frac{3}{2} $.

Prove that: i) $$\sum_{k=1}^{n-1}(1+\ln k)\le n^2-n+1$$ ii) $$\sum_{k=1}^{n-1}\sqrt{\ln k}\le\frac{n^2-n+1}2$$ [i]Proposed by Laurențiu Modan[/i]

The diagonals of a quadrangle inscribed in a circle intersect at point $K$. The projections of the point $ K$ onto the subsequent sides of this quadrangle are points $M, N, P, Q$. Prove that these lines $KM$, $KN$, $KP$, $KQ$ are the angle bisectors of the quadrangle $MNPQ$.

Determine all the functions $f:\mathbb R\mapsto\mathbb R$ satisfies the equation $f(a^2 +ab+ f(b^2))=af(b)+b^2+ f(a^2)\,\forall a,b\in\mathbb R $

A smooth function $f:I\to \mathbb{R}$ is said to be [i]totally convex[/i] if $(-1)^k f^{(k)}(t) > 0$ for all $t\in I$ and every integer $k>0$ (here $I$ is an open interval). Prove that every totally convex function $f:(0,+\infty)\to \mathbb{R}$ is real analytic. [b]Note[/b]: A function $f:I\to \mathbb{R}$ is said to be [i]smooth[/i] if for every positive integer $k$ the derivative of order $k$ of $f$ is well defined and continuous over $\mathbb{R}$. A smooth function $f:I\to \mathbb{R}$ is said to be [i]real analytic[/i] if for every $t\in I$ there exists $\epsilon> 0$ such that for all real numbers $h$ with $|h|<\epsilon$ the Taylor series \[\sum_{k\geq 0}\frac{f^{(k)}(t)}{k!}h^k\] converges and is equal to $f(t+h)$.

Let S be a set of $n\geq 3$ points in space. We color some line segments having two points in $S$ as their endpoints red, let $r$ be the number of edges colored red (color $r$ edges red). We know no two colored segmentes have the same length. Proof there is a path of red edges increasing in size of length at least $\Bigg\lceil\frac{2r}{n}\Bigg\rceil$.

Let $p \geq 5$ be a prime number. Prove that there exists an integer $a$ with $1 \leq a \leq p-2$ such that neither $a^{p-1}-1$ nor $(a+1)^{p-1}-1$ is divisible by $p^2$.

Let $O$ and $O'$ designate two dìagonally opposite vertices of a cube. Bisect those edges of the cube that contain neither of the points $O$ and $O'$. Prove that these midpoints of edges lie in a plane and form the vertices of a regular hexagon

Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.

When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. The smallest possible value of $S$ is $ \textbf{(A)}\ 333 \qquad\textbf{(B)}\ 335 \qquad\textbf{(C)}\ 337 \qquad\textbf{(D)}\ 339 \qquad\textbf{(E)}\ 341 $

Let $a, b$ and $c$ be positive integers such that $a^2 + b^2 + 1 = c^2$ . Prove that $[a/2] + [c / 2]$ is even. Note: $[x]$ is the integer part of $x$.

Let $BC$ be a fixed segment in the plane, and let $A$ be a variable point in the plane not on the line $BC$. Distinct points $X$ and $Y$ are chosen on the rays $CA^\to$ and $BA^\to$, respectively, such that $\angle CBX = \angle YCB = \angle BAC$. Assume that the tangents to the circumcircle of $ABC$ at $B$ and $C$ meet line $XY$ at $P$ and $Q$, respectively, such that the points $X$, $P$, $Y$ and $Q$ are pairwise distinct and lie on the same side of $BC$. Let $\Omega_1$ be the circle through $X$ and $P$ centred on $BC$. Similarly, let $\Omega_2$ be the circle through $Y$ and $Q$ centred on $BC$. Prove that $\Omega_1$ and $\Omega_2$ intersect at two fixed points as $A$ varies. [i]Daniel Pham Nguyen, Denmark[/i]

Let $a_0,a_1,\ldots$ be a sequence of positive integers with $a_0=1$, $a_1=2$ and \[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\] for all $n\geq 2$. Show that if $p$ is a prime less than $2^k$ for some positive integer $k$, then there exists $n\leq k+1$ such that $p\mid a_n$.

Find the minimum value of the following function $f(x) $ defined at $0<x<\frac{\pi}{2}$. \[f(x)=\int_0^x \frac{d\theta}{\cos \theta}+\int_x^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta}\]

[u]Round 5[/u] [b]p13.[/b] Five different schools are competing in a tournament where each pair of teams plays at most once. Four pairs of teams are randomly selected and play against each other. After these four matches, what is the probability that Chad's and Jordan's respective schools have played against each other, assuming that Chad and Jordan come from different schools? [b]p14.[/b] A square of side length $1$ and a regular hexagon are both circumscribed by the same circle. What is the side length of the hexagon? [b]p15.[/b] From the list of integers $1,2, 3,...,30$ Jordan can pick at least one pair of distinct numbers such that none of the $28$ other numbers are equal to the sum or the difference of this pair. Of all possible such pairs, Jordan chooses the pair with the least sum. Which two numbers does Jordan pick? [u]Round 6[/u] [b]p16.[/b] What is the sum of all two-digit integers with no digit greater than four whose squares also have no digit greater than four? [b]p17.[/b] Chad marks off ten points on a circle. Then, Jordan draws five chords under the following constraints: $\bullet$ Each of the ten points is on exactly one chord. $\bullet$ No two chords intersect. $\bullet$ There do not exist (potentially non-consecutive) points $A, B,C,D,E$, and $F$, in that order around the circle, for which $AB$, $CD$, and $EF$ are all drawn chords. In how many ways can Jordan draw these chords? [b]p18.[/b] Chad is thirsty. He has $109$ cubic centimeters of silicon and a 3D printer with which he can print a cup to drink water in. He wants a silicon cup whose exterior is cubical, with five square faces and an open top, that can hold exactly $234$ cubic centimeters of water when filled to the rim in a rectangular-box-shaped cavity. Using all of his silicon, he prints a such cup whose thickness is the same on the five faces. What is this thickness, in centimeters? [u]Round 7[/u] [b]p19.[/b] Jordan wants to create an equiangular octagon whose side lengths are exactly the first $8$ positive integers, so that each side has a different length. How many such octagons can Jordan create? [b]p20.[/b] There are two positive integers on the blackboard. Chad computes the sum of these two numbers and tells it to Jordan. Jordan then calculates the sum of the greatest common divisor and the least common multiple of the two numbers, and discovers that her result is exactly $3$ times as large as the number Chad told her. What is the smallest possible sum that Chad could have said? [b]p21.[/b] Chad uses yater to measure distances, and knows the conversion factor from yaters to meters precisely. When Jordan asks Chad to convert yaters into meters, Chad only gives Jordan the result rounded to the nearest integer meters. At Jordan's request, Chad converts $5$ yaters into $8$ meters and $7$ yaters into $12$ meters. Given this information, how many possible numbers of meters could Jordan receive from Chad when requesting to convert $2014$ yaters into meters? [u]Round 8[/u] [b]p22.[/b] Jordan places a rectangle inside a triangle with side lengths $13$, $14$, and $15$ so that the vertices of the rectangle all lie on sides of the triangle. What is the maximum possible area of Jordan's rectangle? [b]p23.[/b] Hoping to join Chad and Jordan in the Exeter Space Station, there are $2014$ prospective astronauts of various nationalities. It is given that $1006$ of the astronaut applicants are American and that there are a total of $64$ countries represented among the applicants. The applicants are to group into $1007$ pairs with no pair consisting of two applicants of the same nationality. Over all possible distributions of nationalities, what is the maximum number of possible ways to make the $1007$ pairs of applicants? Express your answer in the form $a \cdot b!$, where $a$ and $b$ are positive integers and $a$ is not divisible by $b + 1$. Note: The expression $k!$ denotes the product $k \cdot (k - 1) \cdot ... \cdot 2 \cdot 1$. [b]p24.[/b] We say a polynomial $P$ in $x$ and $y$ is $n$-[i]good [/i] if $P(x, y) = 0$ for all integers $x$ and $y$, with $x \ne y$, between $1$ and $n$, inclusive. We also define the complexity of a polynomial to be the maximum sum of exponents of $x$ and $y$ across its terms with nonzero coeffcients. What is the minimal complexity of a nonzero $4$-good polynomial? In addition, give an example of a $4$-good polynomial attaining this minimal complexity. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2915803p26040550]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given is $2022\times 2022$ cells table. We can select $4$ cells, such that they make the figure $L$ (rotations, symmetric still count) (left one) and put a ball in each of them, or select $4$ cell which makes up the right figure (rotations, symmetric still count) and get one ball from each of them. For which $k$ is it possible in a given moment to be exactly $k$ points in each of the cells

Let $A_1A_2\dots A_{2n}$ be a regular $2n$-gon inscribed in circle $\omega$. Let $P$ be any point on the circle $\omega$. Let $H_1,H_2,\dots, H_n$ be orthocenters of triangles $PA_1A_2, PA_3A_4,\dots, PA_{2n-1}A_{2n}$ respectively. Prove that $H_1H_2\dots H_n$ is a regular $n$-gon.

Jason and Jared take turns placing blocks within a game board with dimensions $3 \times 300$, with Jason going first, such that no two blocks can overlap. The player who cannot place a block within the boundaries loses. Jason can only place $2 \times 100$ blocks, and Jared can only place $2 \times n$ blocks where $n$ is some positive integer greater than 3. Find the smallest integer value of $n$ that still guarantees Jason a win (given both players are playing optimally).

You are given a deck of $n \cdot m$ different cards where $n$ and $m$ are fixed numbers both between $10^{99}$ and $10^{100}$. You perform an [i]$m$-perfect shuffle[/i] for some times. In a single $m$-perfect shuffle, you divide the deck into $m$ piles with $n$ consecutive cards in each pile. You take one card from each pile, in order of the piles, for $n$ times to form the new deck. (The $m$-perfect shuffle is deterministic) For example, if the cards are labeled 12345678 where $n=4$ and $m=2$, you divide the deck into 1234 and 5678, and after one $2$-perfect shuffle you get 15263748. In another example, if the cards are labeled 123456789 where $n=3$ and $m=3$, you divide the deck into 123, 456, and 789, and after one $3$-perfect shuffle you get 147258369. Find an algorithm that, in at most $k$ steps, outputs the smallest positive number of $m$-perfect shuffle after which the deck is exactly the same as the original deck. In each step, you can do one arithmetic operation in $\{+, -, *, /, \bmod\}$, do one comparison, break out of a loop, or store one number to a specific location of an array. You can use the following precomputed numbers of steps in your solution: [list] [*] Checking if $a$ divides $b$ for any two integers $a$ and $b$ takes 2 steps because you need to compute $b \bmod a$ then compare with $0$. [*]A loop over $k$ iterations takes $2k$ steps because you need to increment the loop index by $1$ $k$ times and check the loop guard $k$ times. [*]Simulating one "$m$-perfect shuffle" takes $7nm$ steps because there is one loop index increment, four arithmetic operations, and one store in each iteration of the loop. [/list] [b]Scoring:[/b] An algorithm that completes in at most $k$ steps will be awarded: [list] [*] 1 pt for $k > 10^{10^{10^{10}}}$ [*] 10 pts for $k = 10^{10^{10^{10}}}$ [*] 20 pts for $k = 10^{420}$ [*] 30 pts for $k = 10^{360}$ [*] 50 pts for $k = 10^{240}$ [*] 70 pts for $k = 10^{202}$ [*] 80 pts for $k = 10^{201}$ [*] 95 pts for $k = 10^{120}$ [*] 98 pts for $k = 10^{102}$ [*] 100 pts for $k = 10^{101}$ [/list] [i]Proposed by Mingkuan Xu[/i]

Ten coins are placed in a circle, showing “heads” (the tails are down). Two moves are allowed: (a) turn over four consecutively placed coins; (b) turn over four coins placed as $XX OXX$ ($X$ is one of the coins to be turned over, $O$ is not touched). Is it possible to have all ten coins showing “tails” after a finite sequence of such moves? (A Tolpygo)