Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.

You plan to organize your birthday party, which will be attended either by exactly $m$ persons or by exactly $n$ persons (you are not sure at the moment). You have a big birthday cake and you want to divide it into several parts (not necessarily equal), so that you are able to distribute the whole cake among the people attending the party with everybody getting cake of equal mass (however, one may get one big slice, while others several small slices - the sizes of slices may differ). What is the minimal number of parts you need to divide the cake, so that it is possible, regardless of the number of guests.

Find all primes $p \ge 3$ such that $p- \lfloor p/q \rfloor q$ is a square-free integer for any prime $q<p$.

[b]p1.[/b] There exist real numbers $B$, $M$, and $T$ such that $B + M + T = 23$ and $B - M - T = 20$. Compute $M + T$. [b]p2.[/b] Kaity has a rectangular garden that measures $10$ yards by $12$ yards. Austin’s triangular garden has side lengths $6$ yards, $8$ yards, and $10$ yards. Compute the ratio of the area of Kaity’s garden to the area of Austin’s garden. [b]p3.[/b] Nikhil’s mom and brother both have ages under $100$ years that are perfect squares. His mom is $33$ years older than his brother. Compute the sum of their ages. [b]p4.[/b] Madison wants to arrange $3$ identical blue books and $2$ identical pink books on a shelf so that each book is next to at least one book of the other color. In how many ways can Madison arrange the books? [b]p5.[/b] Two friends, Anna and Bruno, are biking together at the same initial speed from school to the mall, which is $6$ miles away. Suddenly, $1$ mile in, Anna realizes that she forgot her calculator at school. If she bikes $4$ miles per hour faster than her initial speed, she could head back to school and still reach the mall at the same time as Bruno, assuming Bruno continues biking towards the mall at their initial speed. In miles per hour, what is Anna and Bruno’s initial speed, before Anna has changed her speed? (Assume that the rate at which Anna and Bruno bike is constant.) [b]p6.[/b] Let a number be “almost-perfect” if the sum of its digits is $28$. Compute the sum of the third smallest and third largest almost-perfect $4$-digit positive integers. [b]p7.[/b] Regular hexagon $ABCDEF$ is contained in rectangle $PQRS$ such that line $\overline{AB}$ lies on line $\overline{PQ}$, point $C$ lies on line $\overline{QR}$, line $\overline{DE}$ lies on line $\overline{RS}$, and point $F$ lies on line $\overline{SP}$. Given that $PQ = 4$, compute the perimeter of $AQCDSF$. [img]https://cdn.artofproblemsolving.com/attachments/6/7/5db3d5806eaefa00d7fc90fb786a41c0466a90.png[/img] [b]p8.[/b] Compute the number of ordered pairs $(m, n)$, where $m$ and $n$ are relatively prime positive integers and $mn = 2520$. (Note that positive integers $x$ and $y$ are relatively prime if they share no common divisors other than $1$. For example, this means that $1$ is relatively prime to every positive integer.) [b]p9.[/b] A geometric sequence with more than two terms has first term $x$, last term $2023$, and common ratio $y$, where $x$ and $y$ are both positive integers greater than $1$. An arithmetic sequence with a finite number of terms has first term $x$ and common difference $y$. Also, of all arithmetic sequences with first term $x$, common difference $y$, and no terms exceeding $2023$, this sequence is the longest. What is the last term of the arithmetic sequence? [b]p10.[/b] Andrew is playing a game where he must choose three slips, uniformly at random and without replacement, from a jar that has nine slips labeled $1$ through $9$. He wins if the sum of the three chosen numbers is divisible by $3$ and one of the numbers is $1$. What is the probability Andrew wins? [b]p11.[/b] Circle $O$ is inscribed in square $ABCD$. Let $E$ be the point where $O$ meets line segment $\overline{AB}$. Line segments $\overline{EC}$ and $\overline{ED}$ intersect $O$ at points $P$ and $Q$, respectively. Compute the ratio of the area of triangle $\vartriangle EPQ$ to the area of triangle $\vartriangle ECD$. [b]p12.[/b] Define a recursive sequence by $a_1 = \frac12$ and $a_2 = 1$, and $$a_n =\frac{1 + a_{n-1}}{a_{n-2}}$$ for n ≥ 3. The product $a_1a_2a_3 ... a_{2023}$ can be expressed in the form $a^b \cdot c^d \cdot e^f$ , where $a$, $b$, $c$, $d$, $e$, and $f$ are positive (not necessarily distinct) integers, and a, c, and e are prime. Compute $a + b + c + d + e + f$. [b]p13.[/b] An increasing sequence of $3$-digit positive integers satisfies the following properties: $\bullet$ Each number is a multiple of $2$, $3$, or $5$. $\bullet$ Adjacent numbers differ by only one digit and are relatively prime. (Note that positive integers x and y are relatively prime if they share no common divisors other than $1$.) What is the maximum possible length of the sequence? [b]p14.[/b] Circles $O_A$ and $O_B$ with centers $A$ and $B$, respectively, have radii $3$ and $8$, respectively, and are internally tangent to each other at point $P$. Point $C$ is on circle $O_A$ such that line $\overline{BC}$ is tangent to circle $OA$. Extend line $\overline{PC}$ to intersect circle $O_B$ at point $D \ne P$. Compute $CD$. [b]p15.[/b] Compute the product of all real solutions $x$ to the equation $x^2 + 20x - 23 = 2 \sqrt{x^2 + 20x + 1}$. [b]p16.[/b] Compute the number of divisors of $729, 000, 000$ that are perfect powers. (A perfect power is an integer that can be written in the form $a^b$, where $a$ and $b$ are positive integers and $b > 1$.) [b]p17.[/b] The arithmetic mean of two positive integers $x$ and $y$, each less than $100$, is $4$ more than their geometric mean. Given $x > y$, compute the sum of all possible values for $x + y$. (Note that the geometric mean of $x$ and $y$ is defined to be $\sqrt{xy}$.) [b]p18.[/b] Ankit and Richard are playing a game. Ankit repeatedly writes the digits $2$, $0$, $2$, $3$, in that order, from left to right on a board until Richard tells him to stop. Richard wins if the resulting number, interpreted as a base-$10$ integer, is divisible by as many positive integers less than or equal to $12$ as possible. For example, if Richard stops Ankit after $7$ digits have been written, the number would be $2023202$, which is divisible by $1$ and $2$. Richard wants to win the game as early as possible. Assuming Ankit must write at least one digit, after how many digits should Richard stop Ankit? [b]p19.[/b] Eight chairs are set around a circular table. Among these chairs, two are red, two are blue, two are green, and two are yellow. Chairs that are the same color are identical. If rotations and reflections of arrangements of chairs are considered distinct, how many arrangements of chairs satisfy the property that each pair of adjacent chairs are different colors? [b]p20.[/b] Four congruent spheres are placed inside a right-circular cone such that they are all tangent to the base and the lateral face of the cone, and each sphere is tangent to exactly two other spheres. If the radius of the cone is $1$ and the height of the cone is $2\sqrt2$, what is the radius of one of the spheres? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine the sets $S{}$ of positive integers satisfying the following two conditions: [list=a] [*]For any positive integers $a, b, c{}$, if $ab + bc + ca{}$ is in $S$, then so are $a + b + c{}$ and $abc$; and [*]The set $S{}$ contains an integer $N \geqslant 160$ such that $N-2$ is not divisible by $4$. [/list] [i]Bogdan Blaga, United Kingdom[/i]

Two of the altitudes of the scalene triangle $ABC$ have length $4$ and $12$. If the length of the third altitude is also an integer, what is the biggest it can be? $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ \text{none of these} $

Several fleas sit on the squares of a $10\times 10$ chessboard (at most one fea per square). Every minute, all fleas simultaneously jump to adjacent squares. Each fea begins jumping in one of four directions (up, down, left, right), and keeps jumping in this direction while it is possible; otherwise, it reverses direction on the opposite. It happened that during one hour, no two fleas ever occupied the same square. Find the maximal possible number of fleas on the board.

Let $a$ and $b$ be distinct real numbers for which \[\dfrac ab+\dfrac{a+10b}{b+10a}=2.\] Find $\dfrac ab$. $\textbf{(A) }0.6\qquad\textbf{(B) }0.7\qquad\textbf{(C) }0.8\qquad\textbf{(D) }0.9\qquad\textbf{(E) }1$

The median $BM$ is drawn in the triangle $ABC$. It is known that $\angle ABM = 40 {} ^ \circ$ and $\angle CBM = 70 {} ^ \circ $ Find the ratio $AB: BM$.

Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.

Given a tetrahedron $A_1A_2A_3A_4$, define an $A_1$-exsphere such a sphere that is tangent to all planes given by faces of the tetrahedron and the vertex $A_1$ and the sphere are separated by the plane $A_2A_3A_4.$ Denote $\varrho_1,\ldots,\varrho_4$ of all four exspheres. Furthermore, denote $v_i, i=1,\ldots,4$ the distance of the vertex $A_i$ from the opposite face. Show that \[2\left(\frac{1}{v_1}+\frac{1}{v_2}+\frac{1}{v_3}+\frac{1}{v_4}\right)=\frac{1}{\varrho_1}+\frac{1}{\varrho_2}+\frac{1}{\varrho_3}+\frac{1}{\varrho_4}.\]

Six points $A, B, C, D, E,$ and $F$ lie in a straight line in that order. Suppose that $G$ is a point not on the line and that $AC=26, BD=22, CE=31, DF=33, AF=73, CG=40,$ and $DG=30.$ Find the area of $\triangle BGE.$

Find all pairs $(k,n)$ of positive integers such that \[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \] [i]Proposed by Gabriel Chicas Reyes, El Salvador[/i]

A baseball league has $64$ people, each with a different $6$-digit binary number whose base-$10$ value ranges from $0$ to $63$. When any player bats, they do the following: for each pitch, they swing if their corresponding bit number is a $1$, otherwise, they decide to wait and let the ball pass. For example, the player with the number $11$ has binary number $001011$. For the first and second pitch, they wait; for the third, they swing, and so on. Pitchers follow a similar rule to decide whether to throw a splitter or a fastball, if the bit is $0$, they will throw a splitter, and if the bit is $1$, they will throw a fastball. If a batter swings at a fastball, then they will score a hit; if they swing on a splitter, they will miss and get a “strike.” If a batter waits on a fastball, then they will also get a strike. If a batter waits on a splitter, then they get a “ball.” If a batter gets $3$ strikes, then they are out; if a batter gets $4$ balls, then they automatically get a hit. For example, if player $11$ pitched against player $6$ (binary is $000110$), the batter would get a ball for the first pitch, a ball for the second pitch, a strike for the third pitch, a strike for the fourth pitch, and a hit for the fifth pitch; as a result, they will count that as a “hit.” If player $11$ pitched against player $5$ (binary is $000101$), however, then the fifth pitch would be the batter’s third strike, so the batter would be “out.” Each player in the league plays against every other player exactly twice; once as batter, and once as pitcher. They are then given a score equal to the number of outs they throw as a pitcher plus the number of hits they get as a batter. What is the highest score received?

Let $g : \mathbb{N} \to \mathbb{N}$ be a function satisfying: [list] [*] $g(xy) = g(x)g(y)$ for all $x, y \in \mathbb{N}$, [*] $g(g(x)) = x$ for all $x \in \mathbb{N}$, and [*] $g(x) \neq x$ for $2 \leq x \leq 2018$. [/list] Find the minimum possible value of $g(2)$.

Let $a,b,c,d,e,f$ be positive numbers such that $a,b,c,d$ is an arithmetic progression, and $a,e,f,d$ is a geometric progression. Prove that $bc\ge ef$.

A triangle $ABC$ is given. The point $K$ is the base of the external bisector of angle $A$. The point $M$ is the midpoint of the arc $AC$ of the circumcircle. The point $N$ on the bisector of angle $C$ is such that $AN \parallel BM$. Prove that the points $M,N,K$ are collinear. [i](Proposed by Ilya Bogdanov)[/i]

Malmer Pebane, Fames Jung, and Weven Dare are perfect logicians that always tell the truth. Malmer decides to pose a puzzle to his friends: he tells them that the day of his birthday is at most the number of the month of his birthday. Then Malmer announces that he will whisper the day of his birthday to Fames and the month of his birthday to Weven, and he does exactly that. After Malmer whispers to both of them, Fames thinks a bit, then says “Weven cannot know what Malmer’s birthday is.” After that, Weven thinks a bit, then says “Fames also cannot know what Malmer’s birthday is.” This exchange repeats, with Fames and Weven speaking alternately and each saying the other can’t know Malmer’s birthday. However, at one point, Weven instead announces “Fames and I can now know what Malmer’s birthday is. Interestingly, that was the longest conversation like that we could have possibly had before both figuring out Malmer’s birthday.” Find Malmer’s birthday.

Let $f(x)=x^2+x$ for all real $x$. There exist positive integers $m$ and $n$, and distinct nonzero real numbers $y$ and $z$, such that $f(y)=f(z)=m+\sqrt{n}$ and $f(\frac{1}{y})+f(\frac{1}{z})=\frac{1}{10}$. Compute $100m+n$. [i]Proposed by Luke Robitaille[/i]

Let $a, b, c \in R,$ $a \ne 0$, such that $a$ and $4a+3b+2c$ have the same sign. Show that the equation $ax^2+bx+c=0$ cannot have both roots in the interval $(1,2)$.

The shortest distance from an integral point to line $y=\frac{5}{3}x+\frac{4}{5}$ is $\text{(A)}\frac{\sqrt{34}}{170}\qquad\text{(B)}\frac{\sqrt{34}}{85}\qquad\text{(C)}\frac{1}{20}\qquad\text{(D)}\frac{1}{30}$

[b]p1.[/b] Evaluate $6^4 +5^4 +3^4 +2^4$. [b]p2.[/b] What digit is most frequent between $1$ and $1000$ inclusive? [b]p3.[/b] Let $n = gcd \, (2^2 \cdot 3^3 \cdot 4^4,2^4 \cdot 3^3 \cdot 4^2)$. Find the number of positive integer factors of $n$. [b]p4.[/b] Suppose $p$ and $q$ are prime numbers such that $13p +5q = 91$. Find $p +q$. [b]p5.[/b] Let $x = (5^3 -5)(4^3 -4)(3^3 -3)(2^3 -2)(1^3 -1)$. Evaluate $2018^x$ . [b]p6.[/b] Liszt the lister lists all $24$ four-digit integers that contain each of the digits $1,2,3,4$ exactly once in increasing order. What is the sum of the $20$th and $18$th numbers on Liszt’s list? [b]p7.[/b] Square $ABCD$ has center $O$. Suppose $M$ is the midpoint of $AB$ and $OM +1 =OA$. Find the area of square $ABCD$. [b]p8.[/b] How many positive $4$-digit integers have at most $3$ distinct digits? [b]p9.[/b] Find the sumof all distinct integers obtained by placing $+$ and $-$ signs in the following spaces $$2\_3\_4\_5$$ [b]p10.[/b] In triangle $ABC$, $\angle A = 2\angle B$. Let $I$ be the intersection of the angle bisectors of $B$ and $C$. Given that $AB = 12$, $BC = 14$,and $C A = 9$, find $AI$ . [b]p11.[/b] You have a $3\times 3\times 3$ cube in front of you. You are given a knife to cut the cube and you are allowed to move the pieces after each cut before cutting it again. What is the minimumnumber of cuts you need tomake in order to cut the cube into $27$ $1\times 1\times 1$ cubes? p12. How many ways can you choose $3$ distinct numbers fromthe set $\{1,2,3,...,20\}$ to create a geometric sequence? [b]p13.[/b] Find the sum of all multiples of $12$ that are less than $10^4$ and contain only $0$ and $4$ as digits. [b]p14.[/b] What is the smallest positive integer that has a different number of digits in each base from $2$ to $5$? [b]p15.[/b] Given $3$ real numbers $(a,b,c)$ such that $$\frac{a}{b +c}=\frac{b}{3a+3c}=\frac{c}{a+3b},$$ find all possible values of $\frac{a +b}{c}$. [b]p16.[/b] Let S be the set of lattice points $(x, y, z)$ in $R^3$ satisfying $0 \le x, y, z \le 2$. How many distinct triangles exist with all three vertices in $S$? [b]p17.[/b] Let $\oplus$ be an operator such that for any $2$ real numbers $a$ and $b$, $a \oplus b = 20ab -4a -4b +1$. Evaluate $$\frac{1}{10} \oplus \frac19 \oplus \frac18 \oplus \frac17 \oplus \frac16 \oplus \frac15 \oplus \frac14 \oplus \frac13 \oplus \frac12 \oplus 1.$$ [b]p18.[/b] A function $f :N \to N$ satisfies $f ( f (x)) = x$ and $f (2f (2x +16)) = f \left(\frac{1}{x+8} \right)$ for all positive integers $x$. Find $f (2018)$. [b]p19.[/b] There exists an integer divisor $d$ of $240100490001$ such that $490000 < d < 491000$. Find $d$. [b]p20.[/b] Let $a$ and $b$ be not necessarily distinct positive integers chosen independently and uniformly at random from the set $\{1,2, 3, ... ,511,512\}$. Let $x = \frac{a}{b}$ . Find the probability that $(-1)^x$ is a real number. [b]p21[/b]. In $\vartriangle ABC$ we have $AB = 4$, $BC = 6$, and $\angle ABC = 135^o$. $\angle ABC$ is trisected by rays $B_1$ and $B_2$. Ray $B_1$ intersects side $C A$ at point $F$, and ray $B_2$ intersects side $C A$ at point $G$. What is the area of $\vartriangle BFG$? [b]p22.[/b] A level number is a number which can be expressed as $x \cdot \lfloor x \rfloor \cdot \lceil x \rceil$ where $x$ is a real number. Find the number of positive integers less than or equal to $1000$ which are also level numbers. [b]p23.[/b] Triangle $\vartriangle ABC$ has sidelengths $AB = 13$, $BC = 14$, $C A = 15$ and circumcenter $O$. Let $D$ be the intersection of $AO$ and $BC$. Compute $BD/DC$. [b]p24.[/b] Let $f (x) = x^4 -3x^3 +2x^2 +5x -4$ be a quartic polynomial with roots $a,b,c,d$. Compute $$\left(a+1 +\frac{1}{a} \right)\left(b+1 +\frac{1}{b} \right)\left(c+1 +\frac{1}{c} \right)\left(d+1 +\frac{1}{d} \right).$$ [b]p25.[/b] Triangle $\vartriangle ABC$ has centroid $G$ and circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $BC$. If $AD = 2018$, $BD =20$, and $CD = 18$, find the area of triangle $\vartriangle DOG$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an acute triangle ${ABC}$, erect triangles ${ABD}$ and ${ACE}$ externally, so that ${\angle ADB= \angle AEC=90^o}$ and ${\angle BAD= \angle CAE}$. Let ${{A}_{1}}\in BC,{{B}_{1}}\in AC$ and ${{C}_{1}}\in AB$ be the feet of the altitudes of the triangle ${ABC}$, and let $K$ and ${K,L}$ be the midpoints of $[ B{{C}_{1}} ]$ and ${BC_1, CB_1}$, respectively. Prove that the circumcenters of the triangles $AKL,{{A}_{1}}{{B}_{1}}{{C}_{1}}$ and ${DEA_1}$ are collinear. (Bulgaria)

Suppose that a sequence $\{a_n\}_{n\ge 1}$ satisfies $0 < a_n \le a_{2n} + a_{2n+1}$ for all $n\in \mathbb{N}$. Prove that the series$\sum_{n=1}^{\infty} a_n$ diverges.

Determine the number of polynomials of degree $ 3 $ that are irreducible over the field of integers modulo a prime.