On a large wooden block there are four twelve-hour analog clocks of varying accuracy. At 7PM on April 3, 2015, they all correctly displayed the time. The first clock is accurate, the second clock is two times as fast as the first clock, the third clock is three times as fast as the first clock, and the last clock doesn't move at all. How many hours must elapse (from 7PM) before the times displayed on the clocks coincide again? (The clocks do not distinguish between AM and PM.) [asy] import olympiad; import cse5; size(12cm); defaultpen(linewidth(0.9)+fontsize(11pt)); picture clock(real hh, real mm, string nn) { picture p; draw(p, unitcircle); for(int i=1;i<=12;i=i+1) { // draw(p, 0.9*dir(90-30*i)--dir(90-30*i)); label(p, "$"+(string) i+"$",0.84*dir(90-30*i), fontsize(9pt)); } dot(p, origin); pair hpoint = 0.5 * dir(90 - 30 * (hh + mm/60)); pair mpoint = 0.75 * dir(90 - 6 * mm); draw(p, origin--hpoint, EndArrow(HookHead, 3)); draw(p, origin--mpoint, EndArrow(HookHead, 5)); string tlabel; if (mm > 10) { tlabel = (string) hh + ":" + (string) mm; } else { tlabel = (string) hh + ":0" + (string) mm; } label(p, tlabel, dir(90)*1.2, dir(90)); label(p, tlabel, dir(90)*1.2, dir(90)); label(p, nn, dir(-90)*1.1, dir(-90)); return p; } // The block real h = 1; filldraw( (-1.2,-1)--(8.4,-1)--(8.4,-1-h)--(-1.2,-1-h)--cycle, 0.7*lightgrey, black); add(shift((0.0,0)) * clock(10,22, "I")); add(shift((2.4,0)) * clock( 1,44, "II")); add(shift((4.8,0)) * clock( 5,06, "III")); add(shift((7.2,0)) * clock( 7,00, "IV")); label("\emph{Omnes vulnerant, postuma necat}", (3.6, -1.8), origin); [/asy] [i]Proposed by Evan Chen[/i]

Given a sequence of positive integers $a_1, a_2, a_3, \dots, a_{n}$, define the \emph{power tower function} \[f(a_1, a_2, a_3, \dots, a_{n})=a_1^{a_2^{a_3^{\mathstrut^{ .^{.^{.^{a_{n}}}}}}}}.\] Let $b_1, b_2, b_3, \dots, b_{2017}$ be positive integers such that for any $i$ between 1 and 2017 inclusive, \[f(a_1, a_2, a_3, \dots, a_i, \dots, a_{2017})\equiv f(a_1, a_2, a_3, \dots, a_i+b_i, \dots, a_{2017}) \pmod{2017}\] for all sequences $a_1, a_2, a_3, \dots, a_{2017}$ of positive integers greater than 2017. Find the smallest possible value of $b_1+b_2+b_3+\dots+b_{2017}$. [i]Proposed by Yannick Yao

Prove that the sum of the squares of $1984$ consecutive positive integers cannot be the square of an integer.

In decimal representation $$\text {34!=295232799039a041408476186096435b0000000}.$$ Find the numbers $a$ and $b$.

Pick a $3$-digit number $abc$, which contains no $0$'s. The probability that this is a winning number is $\frac1a\cdot\frac1b\cdot\frac1c$. However, the BMT problem writer tries to balance out the chances for the numbers in the following ways: [list] [*] For the lowest digit $n$ in the number, he rolls an $n$-sided die for each time that the digit appears, and gives the number $0$ probability of winning if an $n$ is rolled. [*] For the largest digit $m$ in the number, he rolls an $m$-sided die once and scales the probability of winning by that die roll. [/list] If you choose optimally, what is the probability that your number is a winning number?

The following spiral sequence of squares is drawn on an infinite blackboard: The $1$st square $(1 \times 1)$ has a common vertical side with the $2$nd square (also $1\times 1$) drawn on the right side of it; the 3rd square $(2 \times 2)$ is drawn on the upper side of the $1$st and 2nd ones; the $4$th square $(3 \times 3)$ is drawn on the left side of the $1$st and $3$rd ones; the $5$th square $(5 \times 5)$ is drawn on the bottom side of the $4$th, 1st and $2$nd ones; the $6$th square $(8 \times 8)$ is drawn on the right side, and so on. Each of the squares has a common side with the rectangle consisting of squares constructed earlier. Prove that the centres of all the squares except the $1$st lie on two straight lines. (A Andjans, Riga)

Prove that one can find 100 distinct pairs of integers such that every digit of each number is no less than 6 and the product of the numbers in each pair is also a number with all its digits being no less than 6. [i](4 points)[/i]

Compute the number of positive integers $n$ between $2017$ and $2017^2$ such that $n^n \equiv 1$ (mod $2017$). ($2017$ is prime.)

Emily draws six dots on a piece of paper such that no three lie on a straight line, then draws a line segment connecting each pair of dots. She then colors five of these segments red. Her coloring is said to be $\emph{red-triangle-free}$ if for every set of three points from her six drawn points there exists an uncolored segment connecting two of the three points. In how many ways can Emily color her drawing such that it is red-triangle-free?

Let $a$ be a real number. Evaluate \[\int _{-\pi+a}^{3\pi+a} |x-a-\pi|\sin \left(\frac{x}{2}\right)dx\]

For any given non-negative integer $m$, let $f(m)$ be the number of $1$'s in the base $2$ representation of $m$. Let $n$ be a positive integer. Prove that the integer $$\sum^{2^n - 1}_{m = 0} \Big( (-1)^{f(m)} \cdot 2^m \Big)$$ contains at least $n!$ positive divisors.

Given a triangle $ABC$ with $AB>BC$, let $ \Omega $ be the circumcircle. Let $M$, $N$ lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. Let $K$ be the intersection of $MN$ and $AC$. Let $P$ be the incentre of the triangle $AMK$ and $Q$ be the $K$-excentre of the triangle $CNK$. If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$. [i]M. Kungodjin[/i]

Let $P$ be the set of positive integers that are prime numbers. Find the number of subsets of $P$ that have the property that the sum of their elements is $34$ such as $\{3, 31\}$.

Find all nonnegative integers $x,y,z$ such that $(x+1)^{y+1} + 1= (x+2)^{z+1}$.

The points $D$ and $E$ lie on the side $(BC)$ of the triangle $ABC$ such that $D$ is between $B$ and $E.$ A point $R$ on the segment $(AE)$ is called [i]remarkable[/i] if the lines $PQ$ and $BC$ are parallel, where $\{P\}=DR \cap AC$ and $\{Q\}=CR \cap AB.$ A point $R'$ on the segment $(AD)$ is called [i]remarkable[/i] if the lines $P'Q'$ and $BC$ are parallel, where $\{P'\}=BR' \cap AC$ and $\{Q'\}=ER' \cap AB.$ a) If there exists a remarkable point on the segment $(AE),$ prove that any point of the segment $(AE)$ is remarkable. b) If each of the segments $(AD)$ and $(AE)$ contains a remarkable point, prove that $BD=CE=\varphi \cdot DE,$ where $\varphi= \frac{1+\sqrt{5}}{2}$ is the golden ratio.

$16.$ In an acute angle triangle $ABC,$ let $D$ be the foot of the altitude from $A,$ and $E$ be the midpoint of $BC.$ Let $F$ be the midpoint of $AC.$ Suppose $\angle{BAE}=40^o. $ If $\angle{DAE}=\angle{DFE},$ What is the magnitude of $\angle{ADF}$ in degrees $?$

Given is a triangle $ABC$ with circumcentre $O$. The circumcircle of triangle $AOC$ intersects side $BC$ at $D$ and side $AB$ at $E$. Prove that the triangles $BDE$ and $AOC$ have circumradiuses of equal length.

Let $n$ be a positive integer; and $S(n)$ denote the sum of all digits in the decimal representation of $n$. A positive integer obtained by removing one or several digits from the right hand end of the decimal representation of $n$ is called the [i]truncation[/i] of $n$. The sum of all truncations of $n$ is denoted as $T(n)$. Prove that $S(n)+9T(n)=n$

Triangle $ABC$ has incircle $\omega$ and $A$-excircle $\omega_A.$ Circle $\gamma_B$ passes through $B$ and is externally tangent to $\omega$ and $\omega_A.$ Circle $\gamma_C$ passes through $C$ and is externally tangent to $\omega$ and $\omega_A.$ If $\gamma_B$ intersects line $BC$ again at $D,$ and $\gamma_C$ intersects line $BC$ again at $E,$ prove that $BD=EC.$

In the usual coordinate system $ xOy$ a line $ d$ intersect the circles $ C_1:$ $ (x\plus{}1)^2\plus{}y^2\equal{}1$ and $ C_2:$ $ (x\minus{}2)^2\plus{}y^2\equal{}4$ in the points $ A,B,C$ and $ D$ (in this order). It is known that $ A\left(\minus{}\frac32,\frac{\sqrt3}2\right)$ and $ \angle{BOC}\equal{}60^{\circ}$. All the $ Oy$ coordinates of these $ 4$ points are positive. Find the slope of $ d$.

Given a triangle $ABC$ and external points $X, Y$ , and $Z$ such that $\angle BAZ = \angle CAY , \angle CBX = \angle ABZ$, and $\angle ACY = \angle BCX$, prove that $AX,BY$ , and $CZ$ are concurrent.

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i] [b][i]Original Formulation[/i][/b] Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes. [i]Proposed by Soviet Union. [/i]

[b]p1.[/b] Find the largest k such that the equation $x^2 - 2x + k = 0$ has at least one real root. [b]p2.[/b] Indiana Jones needs to cross a flimsy rope bridge over a mile long gorge. It is so dark that it is impossible to cross the bridge without a flashlight. Furthermore, the bridge is so weak that it can only support the weight of two people. The party has only one flashlight, which has a weak beam so whenever two people cross, they are constrained to walk together, at the speed of the slower person. Indiana Jones can cross the bridge in $5$ minutes. His girlfriend can cross in $10$ minutes. His father needs $20$ minutes, and his father’s side kick needs $25$ minutes. They need to get everyone across safely in on hour to escape the bay guys. Can they do it? [b]p3.[/b] There are ten big bags with coins. Nine of them contain fare coins weighing $10$ g. each, and one contains counterfeit coins weighing $9$ g. each. By one weighing on a digital scale find the bag with counterfeit coins. [b]p4.[/b] Solve the equation: $\sqrt{x^2 + 4x + 4} = x^2 + 5x + 5$. [b]p5.[/b] (a) In the $x - y$ plane, analytically determine the length of the path $P \to A \to C \to B \to P$ around the circle $(x - 6)^2 + (y - 8)^2 = 25$ from the point $P(12, 16)$ to itself. [img]https://cdn.artofproblemsolving.com/attachments/f/b/24888b5b478fa6576a54d0424ce3d3c6be2855.png[/img] (b) Determine coordinates of the points $A$ and $B$. [b]p6.[/b] (a) Let $ABCD$ be a convex quadrilateral (it means that diagonals are inside the quadrilateral). Prove that $$Area\,\, (ABCD) \le \frac{|AB| \cdot |AD| + |BC| \cdot |CD|}{2}$$ (b) Let $ABCD$ be an arbitrary quadrilateral (not necessary convex). Prove the same inequality as in part (a). (c) For an arbitrary quadrilateral $ABCD$ prove that $Area\,\, (ABCD) \le \frac{|AB| \cdot |CD| + |BC| \cdot |AD|}{2}$ PS. You should use hide for answers.

Let $(\Omega, \mathcal A, P)$ be a probability space, and let $(X_n, \mathcal F_n)$ be an adapted sequence in $(\Omega, \mathcal A, P)$ (that is, for the $\sigma$-algebras $\mathcal F_n$, we have $\mathcal F_1\subseteq \mathcal F_2\subseteq \dots \subseteq \mathcal A$, and for all $n$, $X_n$ is an $\mathcal F_n$-measurable and integrable random variable). Assume that $$\mathrm E (X_{n+1} \mid \mathcal F_n )=\frac12 X_n+\frac12 X_{n-1}\,\,\,\,\, (n=2, 3, \ldots )$$ Prove that $\mathrm{sup}_n \mathrm{E}|X_n|<\infty$ implies that $X_n$ converges with probability one as $n\to\infty$. [I. Fazekas]

Suppose $x$, $y$, and $z$ are nonzero complex numbers such that $(x+y+z)(x^2+y^2+z^2)=x^3+y^3+z^3$. Compute \[(x+y+z)\left(\dfrac1x+\dfrac1y+\dfrac1z\right).\]