[hide=E stands for Euclid, L stands for Lobachevsky]they had two problem sets under those two names[/hide] [b]E1.[/b] What is the perimeter of a rectangle if its area is $24$ and one side length is $6$? [b]E2.[/b] John moves 3 miles south, then $2$ miles west, then $7$ miles north, and then $5$ miles east. What is the length of the shortest path, in miles, from John's current position to his original position? [b]E3.[/b] An equilateral triangle $ABC$ is drawn with side length $2$. The midpoints of sides $AB$, $BC$, and $CA$ are constructed, and are connected to form a triangle. What is the perimeter of the newly formed triangle? [b]E4.[/b] Let triangle $ABC$ have sides $AB = 74$ and $AC = 5$. What is the sum of all possible integral side lengths of BC? [b]E5.[/b] What is the area of quadrilateral $ABCD$ on the coordinate plane with $A(1, 0)$, $B(0, 1)$, $C(1, 3)$, and $D(5, 2)$? [b]E6 / L1.[/b] Let $ABCD$ be a square with side length $30$. A circle centered at the center of $ABCD$ with diameter $34$ is drawn. Let $E$ and $F$ be the points at which the circle intersects side $AB$. What is $EF$? [b]E7 / L2.[/b] What is the area of the quadrilateral bounded by $|2x| + |3y| = 6$? [b]E8.[/b] A circle $O$ with radius $2$ has a regular hexagon inscribed in it. Upon the sides of the hexagon, equilateral triangles of side length $2$ are erected outwards. Find the area of the union of these triangles and circle $O$. [b]L3.[/b] Right triangle $ABC$ has hypotenuse $AB$. Altitude $CD$ divides $AB$ into segments $AD$ and $DB$, with $AD = 20$ and $DB = 16$. What is the area of triangle $ABC$? [b]L4.[/b] Circle $O$ has chord $AB$. Extend $AB$ past $B$ to a point $C$. A ray from $C$ is drawn, and this ray intersects circle $O$. Let point $D$ be the point of intersection of the ray and the circle that is closest to point $C$. Given $AB = 20$, $BC = 16$, and $OA = \frac{201}{6}$ , find the longest possible length of $CD$. [b]L5.[/b] Consider a circular cone with vertex $A$. The cone's height is $4$ and the radius of its base is $3$. Inscribe a sphere inside the cone. Find the ratio of the volume of the cone to the volume of the sphere. [b]L6.[/b] A disk of radius $\frac12$ is randomly placed on the coordinate plane. What is the probability that it contains a lattice point (point with integer coordinates)? [b]L7.[/b] Let $ABC$ be an equilateral triangle of side length $2$. Let $D$ be the midpoint of $BC$, and let $P$ be a variable point on $AC$. By moving $P$ along $AC$, what is the minimum perimeter of triangle $BDP$? [b]L8.[/b] Let $ABCD$ be a rectangle with $AB = 8$ and $BC = 9$. Let $DEFG$ be a rhombus, where $G$ is on line $BC$ and $A$ is on line $EF$. If $m\angle EFG = 30^o, what is $DE$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that, if $\alpha$, $\beta$ and $\gamma$ are angles of an arbitrary triangle, $$\text{sin } \frac{\alpha}{2} \text{ sin } \frac{\beta}{2} \text{ sin } \frac{\gamma}{2} < \frac14.$$.

In the convex quadrilateral $ABCD, BC$ is parallel to $AD$. Two circular arcs $\omega_1$ and $\omega_3$ pass through $A$ and $B$ and are on the same side of $AB$. Two circular arcs $\omega_2$ and $\omega_4$ pass through $C$ and $D$ and are on the same side of $CD$. The measures of $\omega_1, \omega_2, \omega_3$ and $\omega_4$ are $\alpha, \beta,\beta$  and $\alpha$ respectively. If $\omega_1$ and $\omega_2$ are tangent to each other externally, prove that so are $\omega_3$ and $\omega_4$.

Let $S$ be a set of $a+b+3$ points on a sphere, where $a$, $b$ are nonnegative integers and no four points of $S$ are coplanar. Determine how many planes pass through three points of $S$ and separate the remaining points into $a$ points on one side of the plane and $b$ points on the other side.

The sequence $\{a_i\}_{i \ge 1}$ is defined by $a_1 = 1$ and \[ a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor \] for all $n \ge 2$. Compute the eighth perfect square in the sequence. [i]Proposed by Lewis Chen[/i]

Find all functions $f : R_{>0} \to R_{>0}$ with the following property: $$f \left( \frac{x}{y + 1}\right) = 1 - xf(x + y)$$ for all $x > y > 0$ .

Let $ABC$ be a triangle, and let $P$ be a point on side $BC$ such that $\frac{BP}{PC}=\frac{1}{2}$. If $\measuredangle ABC$ $=$ $45^{\circ}$ and $\measuredangle APC$ $=$ $60^{\circ}$, determine $\measuredangle ACB$ without trigonometry.

For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0, t_1, \ldots, t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties: (a) $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1, \ldots, t_n$; (b) $f\left(t_0\right)=1 / 2$; (c) $\lim _{t \rightarrow t_k^{+}} f^{\prime}(t)=0$ for $0 \leq k \leq n$; (d) For $0 \leq k \leq n-1$, we have $f^{\prime \prime}(t)=k+1$ when $t_k<t<t_{k+1}$, and $f^{\prime \prime}(t)=n+1$ when $t>t_n$. Considering all choices of $n$ and $t_0, t_1, \ldots, t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f\left(t_0+T\right)=2023$?

In triangle $ABC$, point $M$ is the midpoint of $BC$, $P$ the point of intersection of the tangents at points $B$ and $C$ of the circumscribed circle of $ABC$, $N$ is the midpoint of the segment $MP$. The segment $AN$ meets the circumcircle $ABC$ at the point $Q$. Prove that $\angle PMQ = \angle MAQ$.

Five guys join five girls for a night of bridge. Bridge games are always played by a team of two guys against a team of two girls. The guys and girls want to make sure that every guy and girl play against each other an equal number of times. Given that at least one game is played, what is the least number of games necessary to accomplish this?

Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is [asy] draw((0,0)--(3,0)--(3,1)--(0,1)--cycle); draw((1,-1)--(2,-1)--(2,2)--(1,2)--cycle);[/asy] $ \text{(A)}\ 20\qquad\text{(B)}\ 21\qquad\text{(C)}\ 22\qquad\text{(D)}\ 24\qquad\text{(E)}\ 30 $

Each member of the sequence $112002, 11210, 1121, 117, 46, 34,\ldots$ is obtained by adding five times the rightmost digit to the number formed by omitting that digit. Determine the billionth ($10^9$th) member of this sequence.

For each positive integer $n$, denote by $\omega(n)$ the number of distinct prime divisors of $n$ (for example, $\omega(1)=0$ and $\omega(12)=2$). Find all polynomials $P(x)$ with integer coefficients, such that whenever $n$ is a positive integer satisfying $\omega(n)>2023^{2023}$, then $P(n)$ is also a positive integer with \[\omega(n)\ge\omega(P(n)).\] Greece (Minos Margaritis - Iasonas Prodromidis)

a) Let $x_1,x_2,x_3,y_1,y_2,y_3\in \mathbb{R}$ and $a_{ij}=\sin(x_i-y_j),\ i,j=\overline{1,3}$ and $A=(a_{ij})\in \mathcal{M}_3$ Prove that $\det A=0$. b) Let $z_1,z_2,\ldots,z_{2n}\in \mathbb{C}^*,\ n\ge 3$ such that $|z_1|=|z_2|=\ldots=|z_{n+3}|$ and $\arg z_1\ge \arg z_2\ge \ldots\ge \arg(z_{n+3})$. If $b_{ij}=|z_i-z_{j+n}|,\ i,j=\overline{1,n}$ and $B=(b_{ij})\in \mathcal{M}_n$, prove that $\det B=0$.

If $a,b,c$ are the sides and $\alpha,\beta,\gamma$ the corresponding angles of a triangle, prove the inequality $$\frac{\cos\alpha}{a^3}+\frac{\cos\beta}{b^3}+\frac{\cos\gamma}{c^3}\ge\frac3{2abc}.$$

Let $ \mathcal P$ be a parabola, and let $ V_1$ and $ F_1$ be its vertex and focus, respectively. Let $ A$ and $ B$ be points on $ \mathcal P$ so that $ \angle AV_1 B \equal{} 90^\circ$. Let $ \mathcal Q$ be the locus of the midpoint of $ AB$. It turns out that $ \mathcal Q$ is also a parabola, and let $ V_2$ and $ F_2$ denote its vertex and focus, respectively. Determine the ratio $ F_1F_2/V_1V_2$.

Six squares are colored, front and back, (R = red, B = blue, O = orange, Y = yellow, G = green, and W = white). They are hinged together as shown, then folded to form a cube. The face opposite the white face is [asy] draw((0,2)--(1,2)--(1,1)--(2,1)--(2,0)--(3,0)--(3,1)--(4,1)--(4,2)--(2,2)--(2,3)--(0,3)--cycle); draw((1,3)--(1,2)--(2,2)--(2,1)--(3,1)--(3,2)); label("R",(.5,2.3),N); label("B",(1.5,2.3),N); label("G",(1.5,1.3),N); label("Y",(2.5,1.3),N); label("W",(2.5,.3),N); label("O",(3.5,1.3),N);[/asy] $ \text{(A)}\ \text{B}\qquad\text{(B)}\ \text{G}\qquad\text{(C)}\ \text{O}\qquad\text{(D)}\ \text{R}\qquad\text{(E)}\ \text{Y} $

How many tuples of integers $(a_0,a_1,a_2,a_3,a_4)$ are there, with $1\leq a_i\leq 5$ for each $i$, so that $a_0<a_1>a_2<a_3>a_4$?

Prove that the equation $x^3+11^3=y^3$ has no solution in positive integers $x$ and $y$.

For integers $n \ge k \ge 0$ we define the [i]bibinomial coefficient[/i] $\left( \binom{n}{k} \right)$ by \[ \left( \binom{n}{k} \right) = \frac{n!!}{k!!(n-k)!!} .\] Determine all pairs $(n,k)$ of integers with $n \ge k \ge 0$ such that the corresponding bibinomial coefficient is an integer. [i]Remark: The double factorial $n!!$ is defined to be the product of all even positive integers up to $n$ if $n$ is even and the product of all odd positive integers up to $n$ if $n$ is odd. So e.g. $0!! = 1$, $4!! = 2 \cdot 4 = 8$, and $7!! = 1 \cdot 3 \cdot 5 \cdot 7 = 105$.[/i]

Let $ABC$ be a right-angled triangle $\left(\angle A=90^{\circ}\right)$ and $M$ be the midpoint of $BC$. $\omega_1$ is a circle which passes through $B,M$ and touchs $AC$ at $X$. $\omega_2$ is a circle which passes through $C,M$ and touchs $AB$ at $Y$ ($X,Y$ and $A$ are in the same side of $BC$). Prove that $XY$ passes through the midpoint of arc $BC$ (does not contain $A$) of the circumcircle of $ABC$.

Let $ ABCD $ be a parallelogram with $ BC = 17 $. Let $ M $ be the midpoint of $ \overline{BC} $ and let $ N $ be the point such that $ DANM $ is a parallelogram. What is the length of segment $ \overline{NC} $?

In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $E$. If $AB=BE=5$, $EC=CD=7$, and $BC=11$, compute $AE$.

Let $m, n$ be positive integers. Consider a sequence of positive integers $a_1, a_2, ... , a_n$ that satisfies $m = a_1 \ge a_2\ge ... \ge a_n \ge 1$. Then define, for $1\le  i\le  m$, $b_i =$ # $\{ j \in \{1, 2, ... , n\}: a_j \ge i\}$, so $b_i$ is the number of terms $a_j $ of the given sequence for which $a_j  \ge i$. Similarly, we define, for $1\le   j \le  n$, $c_j=$ # $\{ i \in \{1, 2, ... , m\}: b_i \ge j\}$ , thus $c_j$ is the number of terms bi in the given sequence for which $b_i \ge j$. E.g.: If $a$ is the sequence $5, 3, 3, 2, 1, 1$ then $b$ is the sequence $6, 4, 3, 1, 1$. (a) Prove that $a_j = c_j $ for $1  \le j  \le n$. (b) Prove that for $1\le  k \le m$: $\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j$.

Every nonnegative periodic decimal fraction represents a rational number, also in the form $\frac{p}{q}$ can be represented ($p$ and $q$ are natural numbers and coprime, $p\ge 0$, $q > 0)$. Now let $a_1$, $a_2$, $a_3$ and $a_4$ be digits to represent numbers in the decadic system. Let $a_1 \ne a_3$ or $a_2 \ne a_4$.Prove that it for the numbers: $z_1 = 0, \overline{a_1a_2a_3a_4} = 0,a_1a_2a_3a_4a_1a_2a_3a_4...$ $z_2 = 0, \overline{a_4a_1a_2a_3}$ $z_3 = 0, \overline{a_3a_4a_1a_2}$ $z_4 = 0, \overline{a_2a_3a_4a_1}$ In the above representation $p/q$ always have the same denominator. [hide=original wording]Jeder nichtnegative periodische Dezimalbruch repr¨asentiert eine rationale Zahl, die auch in der Form p/q dargestellt werden kann (p und q nat¨urliche Zahlen und teilerfremd, p >= 0, q > 0). Nun seien a1, a2, a3 und a4 Ziffern zur Darstellung von Zahlen im dekadischen System. Dabei sei a1 $\ne$ a3 oder a2 $\ne$ a4. Beweisen Sie! Die Zahlen z1 = 0, a1a2a3a4 = 0,a1a2a3a4a1a2a3a4... z2 = 0, a4a1a2a3 z3 = 0, a3a4a1a2 z4 = 0, a2a3a4a1 haben in der obigen Darstellung p/q stets gleiche Nenner.[/hide]