[b]p1. [/b] A clock has a long hand for minutes and a short hand for hours. A placement of those hands is [i]natural [/i] if you will see it in a correctly functioning clock. So, having both hands pointing straight up toward $12$ is natural and so is having the long hand pointing toward $6$ and the short hand half-way between $2$ and $3$. A natural placement of the hands is symmetric if you get another natural placement by interchanging the long and short hands. One kind of symmetric natural placement is when the hands are pointed in exactly the same direction. Are there symmetric natural placements of the hands in which the two hands are not pointed in exactly the same direction? If so, describe one such placement. If not, explain why none are possible. [b]p2.[/b] Let $\frac{m}{n}$ be a fraction such that when you write out the decimal expansion of $\frac{m}{n}$ , it eventually ends up with the four digits $2001$ repeated over and over and over. Prove that $101$ divides $n$. [b]p3.[/b] Consider the following two questions: Question $1$: I am thinking of a number between $0$ and $15$. You get to ask me seven yes-or-no questions, and I am allowed to lie at most once in answering your questions. What seven questions can you ask that will always allow you to determine the number? Note: You need to come up with seven questions that are independent of the answers that are received. In other words, you are not allowed to say, "If the answer to question $1$ is yes, then question $2$ is XXX; but if the answer to question $1$ is no, then question $2$ is YYY." Question $2$: Consider the set $S$ of all seven-tuples of zeros and ones. What sixteen elements of $S$ can you choose so that every pair of your chosen seven-tuples differ in at least three coordinates? a. These two questions are closely related. Show that an answer to Question $1$ gives an answer to Question $2$. b. Answer either Question $1$ or Question $2$. [b]p4.[/b] You may wish to use the angle addition formulas for the sine and cosine functions: $\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ $\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ a) Prove the identity $(\sin x)(1 + 2 \cos 2x) = \sin (3x)$. b) For any positive integer $n$, prove the identity $$(sin x)(1 + 2 \cos 2x + 2\cos 4x + ... +2\cos 2nx) = \sin ((2n +1)x)$$ [b]p5.[/b] Define the set $\Omega$ in the $xy$-plane as the union of the regions bounded by the three geometric figures: triangle $A$ with vertices $(0.5, 1.5)$, $(1.5, 0.5)$ and $(0.5,-0.5)$, triangle $B$ with vertices $(-0.5,1.5)$, $(-1.5,-0.5)$ and $(-0.5, 0.5)$, and rectangle $C$ with corners $(0.5, 1.0)$, $(-0.5, 1.0)$, $(-0.5,-1.0)$, and $(0.5,-1.0)$. a. Explain how copies of $\Omega$ can be used to cover the $xy$-plane. The copies are obtained by translating $\Omega$ in the $xy$-plane, and copies can intersect only along their edges. b. We can define a transformation of the plane as follows: map any point $(x, y)$ to $(x + G, x + y + G)$, where $G = 1$ if $y < -2x$, $G = -1$ if $y > -2x$, and $G = 0$ if $y = -2x$. Prove that every point in $\Omega$ is transformed into another point in $\Omega$, and that there are at least two points in $\Omega$ that are transformed into the same point. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For postive integer $n$ consider the hyperplane \[ R_0^n = {x=(x_1x_2...x_n)\in\mathbb{R}^n : \sum\limits^n_{i=1}x_i=0} \] and the lattice \[ Z_0^n = \{y \in R^n_0 : \ (\forall i: y_i \in \mathbb{N})\} \] Define the quasi-norm in $\mathbb{R}^n$ by $\|x\|_p= \sqrt[p]{\sum\limits^{n}_{i=1}|x_i|^p}$ if $0<p<\infty$ and $\|x\|_{\infty} = \max\limits_i |x_i|$. (a) If $x\in R^n_0$ so that $\max x_i - \min x_i \le 1$ then prove that $\forall p \in [1,\infty], \forall y \in Z^n_0$ we have $\|x\|_p\le\|x+y\|_p$ (b) Prove that for every $p\in ]0,1[$, there exist $n \in \mathbb{N}, x\in R^n_0, y\in Z^n_0$ with $\max x_i - \min x_i \le 1$ and $\|x\|_p>\|x+y\|_p$

Decide if there is a permutation $a_1,a_2,\cdots,a_{6666}$ of the numbers $1,2,\cdots,6666$ with the property that the sum $k+a_k$ is a perfect square for all $k=1,2,\cdots,6666$

We are given the natural numbers $1=a_1,\, \, a_2,...,a_n$, for which $a_i\leq a_{i+1}\leq 2a_i$ for $i=1,2,...,n-1$ and the sum $\sum_{i=1}^n a_i$ is even. Prove that these numbers can be partitioned into two groups with equal sum.

How many perfect squares are divisors of the product $ 1!\cdot 2!\cdot 3!\cdots 9!$? $ \textbf{(A)}\ 504 \qquad \textbf{(B)}\ 672 \qquad \textbf{(C)}\ 864 \qquad \textbf{(D)}\ 936 \qquad \textbf{(E)}\ 1008$

In an acute-angled triangle $ABC$, $A_1$ and $B_1$ are the feet of the altitudes from $A$ and $B$ respectively, and $M$ is the midpoint of $AB$. a) Prove that $MA_1$ is tangent to the circumcircle of triangle $A_1B_1C$. b) Prove that the circumcircles of triangles $A_1B_1C,BMA_1$, and $AMB_1$ have a common point.

Nine points are placed on a circle. Show that it is possible to colour the $36$ chords connecting them using four colours so that for any set of four points, each of the four colours is used for at least one of the six chords connecting the given points

Isosceles trapezoid $ABCD$ with bases $AB$ and $CD$ has a point $P$ on $AB$ with $AP=11, BP=27,$ $CD=34,$ and $\angle{CPD}=90^\circ.$ Compute the height of isosceles trapezoid $ABCD.$

Let $a,b,c\ge 0$ and $a+b+c=\sqrt2$. Show that \[\frac1{\sqrt{1+a^2}}+\frac1{\sqrt{1+b^2}}+\frac1{\sqrt{1+c^2}} \ge 2+\frac1{\sqrt3}\] [hide] In general if $a_1, a_2, \cdots , a_n \ge 0$ and $\sum_{i=1}^n a_i=\sqrt2$ we have \[\sum_{i=1}^n \frac1{\sqrt{1+a_i^2}} \ge (n-1)+\frac1{\sqrt3}\] [/hide]

Kobar and Borah are playing on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board, either from $P$ and $Q$ to $2P-Q$ and $2Q-P$, or from $P$ and $Q$ to $5P-4Q$ and $5Q-4P$. The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner. At the beginning of the game, the two numbers on the board are $2024$ and $A$. If it is known that Kobar does not lose on his first move, determine the largest possible value of $A$ so that Borah can win this game.

Prove that in a convex quadrilateral inscribed in a circle, straight lines passing through the midpoints of the sides and perpendicular to the opposite sides intersect at one point.

The pressure $ (P)$ of wind on a sail varies jointly as the area $ (A)$ of the sail and the square of the velocity $ (V)$ of the wind. The pressure on a square foot is $ 1$ pound when the velocity is $ 16$ miles per hour. The velocity of the wind when the pressure on a square yard is $ 36$ pounds is: $ \textbf{(A)}\ 10\frac {2}{3} \text{ mph} \qquad\textbf{(B)}\ 96 \text{ mph} \qquad\textbf{(C)}\ 32\text{ mph} \qquad\textbf{(D)}\ 1\frac {2}{3} \text{ mph} \qquad\textbf{(E)}\ 16 \text{ mph}$

Find $\displaystyle n \in \mathbb N$, $\displaystyle n \geq 2$, and the digits $\displaystyle a_1,a_2,\ldots,a_n$ such that \[ \displaystyle \sqrt{\overline{a_1 a_2 \ldots a_n}} - \sqrt{\overline{a_1 a_2 \ldots a_{n-1}}} = a_n . \]

We define the weight $W$ of a positive integer as follows: $W(1) = 0$, $W(2) = 1$, $W(p) = 1 + W(p + 1)$ for every odd prime $p$, $W(c) = 1 + W(d)$ for every composite $c$, where $d$ is the greatest proper factor of $c$. Compute the greatest possible weight of a positive integer less than 100.

Find all prime numbers $p$ for which $\frac{2^{p-1} -1}{p}$ is a perfect square.

Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\tfrac{12}5\sqrt2$. What is the volume of the tetrahedron? $\textbf{(A) }3\sqrt2\qquad\textbf{(B) }2\sqrt5\qquad\textbf{(C) }\dfrac{24}5\qquad\textbf{(D) }3\sqrt3\qquad\textbf{(E) }\dfrac{24}5\sqrt2$

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)

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

Patrick tosses four four-sided dice, each numbered $1$ through $4$. What's the probability their product is a multiple of four?

For how many integers $1\leq n \leq 2010$, $2010$ divides $1^2-2^2+3^2-4^2+\dots+(2n-1)^2-(2n)^2$? $ \textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 5 $

Three outlaws entered an inn one evening and ordered baked potatoes. They agreed that the first outlaw would eat half of all the potatoes, the second would eat a third, and the third would eat a sixth. However, being tired, they fell asleep. After the potatoes were served, the first outlaw woke up in the middle of the night, ate half the potatoes, and went back to sleep. Then the second outlaw woke up, ate a third of the remaining potatoes, and also went back to sleep. Finally, near morning, the third outlaw woke up, ate a sixth of the remaining potatoes, and went back to sleep. In the morning, they saw that $10$ potatoes were left on the table. How many potatoes did they originally order?

We are given $m,n \in \mathbb{Z}^+.$ Show the number of solution $4-$tuples $(a,b,c,d)$ of the system \begin{align*} ab + bc + cd - (ca + ad + db) &= m\\ 2 \left(a^2 + b^2 + c^2 + d^2 \right) - (ab + ac + ad + bc + bd + cd) &= n \end{align*} is divisible by 10.

Three doctors, four nurses, and three patients stand in a line in random order. The probability that there is at least one doctor and at least one nurse between each pair of patients is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

For some positive integer $n>m$, it turns out that it is representable as sum of $2021$ non-negative integer powers of $m$, and that it is representable as sum of $2021$ non-negative integer powers of $m+1$. Find the maximal value of the positive integer $m$.

Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled $0, 1, \dots, 9$. First he rolls two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that the sum of the first two rolls equals the third roll?