The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet? $\textbf{(A) } 41 \qquad\textbf{(B) } 47 \qquad\textbf{(C) } 59 \qquad\textbf{(D) } 61 \qquad\textbf{(E) } 66 $

Let $ABCDEF$ be a regular hexagon of area $1$. Let $M$ be the midpoint of $DE$. Let $X$ be the intersection of $AC$ and $BM$, let $Y$ be the intersection of $BF$ and $AM$, and let $Z$ be the intersection of $AC$ and $BF$. If $[P]$ denotes the area of a polygon $P$ for any polygon $P$ in the plane, evaluate $[BXC] + [AYF] + [ABZ] - [MXZY]$.

Let $\gamma$ be the circumcircle of the acute triangle $ABC$. Let $P$ be the midpoint of the minor arc $BC$. The parallel to $AB$ through $P$ cuts $BC, AC$ and $\gamma$ at points $R,S$ and $T$, respectively. Let $K \equiv AP \cap BT$ and $L \equiv BS \cap AR$. Show that $KL$ passes through the midpoint of $AB$ if and only if $CS = PR$.

Let $ABCD$ be a quadrilateral with each interior angle less than $180^o$. Show that if $A, B, C, D$ do not lie on a circle, then $AB \cdot CD + AD\cdot BC > AC \cdot BD$

Jordan owns 15 pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the rest are low-top. The red high-top sneakers make up a fraction of the collection. What is the least possible value of this fraction? [img]https://wiki-images.artofproblemsolving.com//thumb/a/a2/2024_AMC_8_-19.png/1200px-2024_AMC_8_-19.png[/img] $\textbf{(A) } 0\qquad\textbf{(B) } \dfrac{1}{5} \qquad\textbf{(C) } \dfrac{4}{15} \qquad\textbf{(D) } \dfrac{1}{3} \qquad\textbf{(E) } \dfrac{2}{5}$

In a triangle $ABC$, $\angle BAC = 100^{\circ}, AB = AC$. A point $D$ is chosen on the side $AC$ such that $\angle ABD = \angle CBD$. Prove that $AD + DB = BC.$

$ O $ is the circumcentre of $ ABC $ and $ A_1\neq A $ is the point on $ AO $ and the circumcircle of $ ABC. $ The centers of mass of $ ABC, A_1BC $ are $ G,G_1, $ respectively, and $ P $ is the intersection of $ AG_1 $ with $ OG. $ Show that $ \frac{PG}{PO}=\frac{2}{3} . $ [i]Gabriel Popa, Paul Georgescu[/i]

The numbers from $1$ to $2017$ are written on a board. Deka and Farid play the following game : each of them, on his turn, erases one of the numbers. Anyone who erases a multiple of $2, 3$ or $5$ loses and the game is over. Is there a winning strategy for Deka ?

[b]E[/b]milia wishes to create a basic solution with 7% hydroxide (OH) ions. She has three solutions of different bases available: 10% rubidium hydroxide (Rb(OH)), 8% cesium hydroxide (Cs(OH)), and 5% francium hydroxide (Fr(OH)). (The Rb(OH) solution has both 10% Rb ions and 10% OH ions, and similar for the other solutions.) Since francium is highly radioactive, its concentration in the final solution should not exceed 2%. What is the highest possible concentration of rubidium in her solution?

Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system \[ x_5+x_2=yx_1 \] \[ x_1+x_3=yx_2 \] \[ x_2+x_4=yx_3 \] \[ x_3+x_5=yx_4 \] \[ x_4+x_1=yx_5 \] where $y$ is a parameter.

Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$. [i]Proposed by Evan Chen[/i]

[b]8.1[/b] On the median drawn from the vertex of the triangle to the base, point $A$ is taken. The sum of the distances from $A$ to the sides of the triangle is equal to $s$. Find the distances from $A$ to the sides if the lengths of the sides are equal to $x$ and $y$. [b]8.2[/b] Fraction $0, abc...$ is composed according to the following rule: $a$ and $c$ are arbitrary digits, and each next digit is equal to the remainder of the sum of the previous two digits when divided by $10$. Prove that this fraction is purely periodic. [b]8.3[/b] Two convex polygons with $m$ and $n$ sides are drawn on the plane ($m>n$). What is the greatest possible number of parts, they can break the plane? [b]8.4 [/b]The sum of three integers that are perfect squares is divisible by $9$. Prove that among them, there are two numbers whose difference is divisible by $9$. [b]8.5 / 9.5[/b] Given $k+2$ integers. Prove that among them there are two integers such that either their sum or their difference is divisible by $2k$. [b]8.6[/b] A right angle rotates around its vertex. Find the locus of the midpoints of the segments connecting the intersection points sides of an angle and a given circle. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].

A toymaker has $k$ dice at his disposal, each with $6$ blank sides. On each side of each of these dice, the toymaker must draw one of the digits $0, 1, 2, \ldots , 9$. Determine (in terms of $k$) the largest integer $n$ such that the toymaker can draw digits on the $k$ dice such that, for any positive integer $r \leq n$, it is possible to choose some of the $k$ dice and form with them the decimal representation of $r$. [b]Note:[/b] The digits 6 and 9 are distinguishable: they appear as [u]6[/u] and [u]9[/u].

Al has three bags, each with three marbles each. Bag $1$ has two blue marbles and one red marble, Bag $2$ has one blue marble and two red marbles, and Bag $3$ has three red marbles. He chooses two distinct bags at random, then one marble at random from each of the chosen bags. What is the probability that he chooses two blue marbles?

Find the largest number $n$ that for which there exists $n$ positive integers such that non of them divides another one, but between every three of them, one divides the sum of the other two. [i]Proposed by Morteza Saghafian[/i]

A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \ge 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n)) = 2$.

Let \(n>1\) be a natural number, and let \(K\) be the square of side length \(n\) subdivided into \(n^2\) unit squares. Determine for which values of \(n\) it is possible to dissect \(K\) into \(n\) connected regions of equal area using only the diagonals of those unit squares, subject to the condition that from each unit square at most one of its diagonals is used (some unit squares may have neither diagonal).

How many ways are there to arrange the numbers $1$ through $8$ into a $2$ by $4$ grid such that the sum of the numbers in each of the two rows are all multiples of $6,$ and the sum of the numbers in each of the four columns are all multiples of $3$?

The heights of a triangular pyramid intersect at one point. Prove that all flat angles at any vertex of the surface are either acute, or right, or obtuse.

The perimeter of $\triangle ABC$ is $2$ and it's sides are $BC=a, CA=b,AB=c$. Prove that $$abc+\frac{1}{27}\ge ab+bc+ca-1\ge abc. $$

Let $x; y; m; n$ be integers greater than $1$ such that

A grid measuring $10 \times 11$ is given. How many "crosses" covering five unit squares can be placed on the grid? (pictured right) so that no two of them cover the same square? [img]https://cdn.artofproblemsolving.com/attachments/a/7/8a5944233785d960f6670e34ca7c90080f0bd6.png[/img]

[b]a)[/b] Show that the sum of the squares of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point. [b]b)[/b] Show that the sum of the cubes of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point. [i]Alexandru Sergiu Alamă[/i]

Suppose $x$ is a real number such that $\sin(1 + \cos^2 x + \sin^4 x) = \tfrac{13}{14}$. Compute $\cos(1 + \sin^2 x + \cos^4 x)$.

Let $ M$ be a point in triangle $ ABC$ such that $ \angle AMC\equal{}90^{\circ}$, $ \angle AMB\equal{}150^{\circ}$, $ \angle BMC\equal{}120^{\circ}$. The centers of circumcircles of triangles $ AMC,AMB,BMC$ are $ P,Q,R$, respectively. Prove that the area of $ \triangle PQR$ is greater than the area of $ \triangle ABC$.