Show that real numbers, $p, q, r$ satisfy the condition $p^4(q-r)^2 + 2p^2(q+r) + 1 = p^4$ if and only if the quadratic equations $x^2 + px + q = 0$ and $y^2 - py + r = 0$ have real roots (not necessarily distinct) which can be labeled by $x_1,x_2$ and $y_1,y_2$, respectively, in such a way that $x_1y_1 - x_2y_2 = 1$.

The expression $ x^3 \plus{} . . . x^2 \plus{} . . . x \plus{} ... \equal{} 0$ is written on the blackboard. Two pupils alternately replace the dots by real numbers. The first pupil attempts to obtain an equation having exactly one real root. Can his opponent spoil his efforts?

Arnav is placing three rectangles into a $3 \times 3$ grid of unit squares. He has a $1\times 3$ rectangle, a $1\times 2$ rectangle, and a $1\times 1$ rectangle. He must place the rectangles onto the grid such that the edges of the rectangles align with the gridlines of the grid. If he is allowed to rotate the rectangles, how many ways can he place the three rectangles into the grid, without overlap? [i]Proposed by William Yue[/i]

Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$

Determine all functions $f: \mathbb N\to \mathbb N$ such that for every positive integer $n$ we have: \[ 2n+2001\leq f(f(n))+f(n)\leq 2n+2002. \]

Positive reals $p$ and $q$ are such that the graph of $y = x^2 - 2px + q$ does not intersect the $x$-axis. Find $q$ if there is a unique pair of points $A, B$ on the graph with $AB$ parallel to the $x$-axis and $\angle AOB = \frac{\pi}{2}$, where $O$ is the origin.

A positive integer $M$ has been represented as a product of primes. Each of these primes is increased by 1 . The product $N$ of the new multipliers is divisible by $M$ . Prove that if we represent $N$ as a product of primes and increase each of them by 1 then the product of the new multipliers will be divisible by $N$ . Alexandr Gribalko

Prove that there exist infinitely many positive integer triples $(a,b,c)$ such that $a ,b,c$ are pairwise relatively prime ,and $ab+c ,bc+a ,ca+b$ are pairwise relatively prime .

Let $P$ be an arbitrary interior point of the equilateral triangle $ABC$. From $P$ draw parallel to the sides: $A'_1A_1 \parallel AB$, $B' _1B_1 \parallel BC$ and $C'_1C_1 \parallel CA$. Prove that the sum of legths $| AC_1 | + | BA_1 | + | CB_1 |$ is independent of the choice of point $P$. [img]https://cdn.artofproblemsolving.com/attachments/5/a/15b06706c09e2458fb5938807b9f3833ffb62e.png[/img]

Let a,b,c>0 with ab+bc+ca=1. Prove that: $\frac{b^3c}{a^2+b^2}+\frac{c^3a}{b^2+c^2}+\frac{a^3b}{c^2+a^2}\ge\frac{1}{2}.$

Six numbers are placed on a circle. For every number $A$ we have: $A$ equals the absolute value of $(B- C)$, where $B$ and $C$ follow $A$ clockwise. The total sum of the numbers equals $1$. Find all the numbers. (Folklore)

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?