A uniform disk ($ I = \dfrac {1}{2} MR^2 $) of mass 8.0 kg can rotate without friction on a fixed axis. A string is wrapped around its circumference and is attached to a 6.0 kg mass. The string does not slip. What is the tension in the cord while the mass is falling? [asy] size(250); pen p=linewidth(3), dg=gray(0.25), llg=gray(0.90), lg=gray(0.75),g=grey; void f(path P, pen p, pen q) { filldraw(P,p,q); } path P=CR((0,0),1); D((1,0)--(1,-2.5),p+lg); f(P,g,p); P=scale(0.4)*P; f(P,lg,p); path Q=arc((0,0),0.3,-40,130)--arc((-1,-1),0.5,130,320)--cycle; f(Q,llg,dg+p); P=scale(0.45)*P; f(P,llg,dg+p); P=shift((0.15,0.15))*((-1,-1)--(-1,-2)--(-1.1,-2)--(-1.1,-1.1)--(-2,-1.1)--(-2,-1)--cycle); f(P,llg,lg+p); P=shift((1.55,1.55))*scale(3)*P; f(P,llg,g+p); unfill((-1.23,-1.23)--(-1.23,-5)--(-5,-1.23)--cycle); clip((-3.8,-3.8)--(-3.8,3.8)--(3.8,3.8)--(3.8,-3.8)--cycle); P=(0.2,-2.5)--(1.8,-2.5)--(1.8,-4.1)--(0.2,-4.1)--cycle; f(P,llg,lg+p); MP("m",(1,-3.3),(0,0),fontsize(16)); MP("M",(0,-1),fontsize(16));[/asy] $ \textbf {(A) } \text {20.0 N} \qquad \textbf {(B) } \text {24.0 N} \qquad \textbf {(C) } \text {34.3 N} \qquad \textbf {(D) } \text {60.0 N} \qquad \textbf {(E) } \text {80.0 N} $

There are $10000$ cities in country, and roads between some cities. Every city has $<100$ roads. Every cycle route with odd number of road consists of $\geq 101 $ roads. Prove that we can divide all cities in $100$ groups with $100$ cities, such that every road leads from one group to other.

Solve the set of simultaneous equations \begin{align*} v^2+ w^2+ x^2+ y^2 &= 6 - 2u, \\ u^2+ w^2+ x^2+ y^2 &= 6 - 2v, \\ u^2+ v^2+ x^2+ y^2 &= 6- 2w, \\ u^2+ v^2+ w^2+ y^2 &= 6 - 2x, \\ u^2+ v^2+ w^2+ x^2 &= 6- 2y. \end{align*}

Let $ a,n$ be positive integers such that $ a\ge(n\minus{}1)!$. Prove that there exist $ n$ [i]distinct[/i] prime numbers $ p_1,\ldots,p_n$ so that $ p_i|a\plus{}i$, for all $ i\equal{}\overline{1,\ldots,n}$.

A line $r$ contains the points $A$, $B$, $C$, $D$ in that order. Let $P$ be a point not in $r$ such that $\angle{APB} = \angle{CPD}$. Prove that the angle bisector of $\angle{APD}$ intersects the line $r$ at a point $G$ such that: $\frac{1}{GA} + \frac{1}{GC} = \frac{1}{GB} + \frac{1}{GD}$

Let $\Delta$ denote the set of all triangles in a plane. Consider the function $f: \Delta\to(0,\infty)$ defined by $f(ABC) = \min \left( \dfrac ba, \dfrac cb \right)$, for any triangle $ABC$ with $BC=a\leq CA=b\leq AB = c$. Find the set of values of $f$.

Let $M$ be a set and $A,B,C$ given subsets of $M$. Find a necessary and sufficient condition for the existence of a set $X\subset M$ for which $(X\cup A)\backslash(X\cap B)=C$. Describe all such sets.

Find the greatest prime factor of $20!+20!+21!$.

Let $p$ be a sufficiently large prime. Show that the number of distinct residues taken by the set $$\{1 + \frac12 + ... + \frac{1}{n}: n = 1, 2,..., p - 1\}$$ modulo $p$ has at least $\sqrt[4]{p}$ elements. (Carlo Sanna)

The New York Public Library requires patrons to choose a 4-digit Personal Identification Number (PIN) to access its online system. (Leading zeros are allowed.) The PIN is not allowed to contain either of the following two forbidden patterns: * A digit that is repeated 3 or more times in a row. For example, 0001 and 5555 are not PINs, but 0010 is a PIN. * A pair of digits that is duplicated. For example, 1212 and 6363 are not PINs, but 1221 and 6633 are PINs. How many distinct possible PINs are there?

Let $\ell_A$ and $\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m$ lie on $\ell_A$, and distinct points $B_1, B_2, B_3, \ldots, B_n$ lie on $\ell_B$. Additionally, when segments $\overline{A_iB_j}$ are drawn for all $i=1,2,3,\ldots, m$ and $j=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n$, no point strictly between $\ell_A$ and $\ell_B$ lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when $m=7$ and $n=5$. The figure shows that there are 8 regions when $m=3$ and $n=2$. [asy] import geometry; size(10cm); draw((-2,0)--(13,0)); draw((0,4)--(10,4)); label("$\ell_A$",(-2,0),W); label("$\ell_B$",(0,4),W); point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2); draw(B1--A1--B2); draw(B1--A2--B2); draw(B1--A3--B2); label("$A_1$",A1,S); label("$A_2$",A2,S); label("$A_3$",A3,S); label("$B_1$",B1,N); label("$B_2$",B2,N); label("1",centroid(A1,B1,I1)); label("2",centroid(B1,I1,I3)); label("3",centroid(B1,B2,I3)); label("4",centroid(A1,A2,I1)); label("5",(A2+I1+I2+I3)/4); label("6",centroid(B2,I2,I3)); label("7",centroid(A2,A3,I2)); label("8",centroid(A3,B2,I2)); dot(A1); dot(A2); dot(A3); dot(B1); dot(B2); [/asy]

There are $n \ge 3$ distinct positive real numbers. Show that there are at most $n-2$ different integer power of three that can be written as the sum of three distinct elements from these $n$ numbers.

The Dinky is a train connecting Princeton to the outside world. It runs on an odd schedule: the train arrive once every one-hour block at some uniformly random time (once at a random time between $\text{9am}$ and $\text{10am}$, once at a random time between $\text{10am}$ and $\text{11am}$, and so on). One day, Emilia arrives at the station, at some uniformly random time, and does not know the time. She expects to wait for $y$ minutes for the next train to arrive. After waiting for an hour, a train has still not come. She now expects to wait for $z$ minutes. Find $yz$.

Let $k>1$ be an odd integer. For every positive integer n, let $f(n)$ be the greatest positive integer for which $2^{f(n)}$ divides $k^n-1$. Find $f(n)$ in terms of $k$ and $n$.

Determine the smallest natural number $n > 2$, or show that no such natural numbers $n$ exists, that satisfy the following condition: There exists natural numbers $a_1, a_2, \dots, a_n$ such that \[ \gcd(a_1, a_2, \dots, a_n) = \sum_{k = 1}^{n - 1} \underbrace{\left( \frac{1}{\gcd(a_k, a_{k + 1})} + \frac{1}{\gcd(a_k, a_{k + 2})} + \dots + \frac{1}{\gcd(a_k, a_n)} \right)}_{n - k \ \text{terms}} \]

Show that \[ \left(1 + a + a^2\right)^2 < 3\left(1 + a^2 + a^4\right) \] for real $a \neq 1$.

Prove the inequality: $$(1+\sin^2\alpha)^n+(1+\cos^2\alpha)^n\ge2\left(\frac32\right)^n$$is true for every natural number $n$. When does equality hold?

Let $a_1,a_2,...$ be a sequence of natural numbers such that for each $n$, the product $(a_n - 1)(a_n- 2)...(a_n - n^2)$ is a positive integral multiple of $n^{n^2-1}$. Prove that for any finite set $P$ of prime numbers the following inequality holds: $$\sum_{p\in P}\frac{1}{\log_p a_p}< 1$$

An accurate 12-hour analog clock has an hour hand, a minute hand, and a second hand that are aligned at 12:00 o’clock and make one revolution in 12 hours, 1 hour, and 1 minute, respectively. It is well known, and not difficult to prove, that there is no time when the three hands are equally spaced around the clock, with each separating angle $ \frac{2 \cdot \pi}{3}.$ Let $ f(t), g(t), h(t)$ be the respective absolute deviations of the separating angles from \frac{2 \cdot \pi}{3} at $ t$ hours after 12:00 o’clock. What is the minimum value of $ max\{f(t), g(t), h(t)\}?$

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 $AB \cdot CD + AD\cdot BC > AC \cdot BD$

Let $a$, $b$, $c$ be real numbers such that $a^2+b=c^2$, $b^2+c=a^2$, $c^2+a=b^2$. Find all possible values of $abc$.

Prove that if every line connecting any two points of some finite set of points of the plane contains at least one more point of this set, then all points of the set lie on one straight line.

A prime number $p$ and integers $x, y, z$ with $0 < x < y < z < p$ are given. Show that if the numbers $x^3, y^3, z^3$ give the same remainder when divided by $p$, then $x^2 + y^2 + z^2$ is divisible by $x + y + z.$

The LHS Math Team wants to play Among Us. There are so many people who want to play that they are going to form several games. Each game has at most 10 people. People are $\textit{happy}$ if they are in a game that has at least 8 people in it. What is the largest possible number of people who would like to play Among Us such that it is impossible to make everyone $\textit{happy}$? [i]Proposed by Sammy Charney[/i]

Which of the following numbers is a perfect square? $ \textbf{(A)}\ \dfrac{14!15!}2\qquad\textbf{(B)}\ \dfrac{15!16!}2\qquad\textbf{(C)}\ \dfrac{16!17!}2\qquad\textbf{(D)}\ \dfrac{17!18!}2\qquad\textbf{(E)}\ \dfrac{18!19!}2 $