Found problems: 85335
Hamilton Avenue has eight houses. On one side of the street are the houses
numbered 1,3,5,7 and directly opposite are houses 2,4,6,8 respectively. An
eccentric postman starts deliveries at house 1 and delivers letters to each of
the houses, finally returning to house 1 for a cup of tea. Throughout the
entire journey he must observe the following rules. The numbers of the houses
delivered to must follow an odd-even-odd-even pattern throughout, each house
except house 1 is visited exactly once (house 1 is visited twice) and the postman
at no time is allowed to cross the road to the house directly opposite. How
many different delivery sequences are possible?
If I roll three fair $4$-sided dice, what is the probability that the sum of the resulting numbers is relatively prime to the product of the resulting numbers?
What is the sum of all integer solutions to $1<(x-2)^2<25$?
$ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 25 $
Pedro has $111$ blue chips and $88$ white chips. There is a machine that for every $14$ blue chips , it gives $11$ white pieces and for every $7$ white chips, it gives $13$ blue pieces. Decide if Pedro can achieve, through successive operations with the machine, increase the total number of chips by $33$, so that the number of blue chips equals $\frac53$ of the amount of white chips. If possible, indicate how to do it. If not, indicate why.
Let $n$ be a positive integer. Denote $M = \{(x, y)|x, y \text{ are integers }, 1 \leq x, y \leq n\}$. Define function $f$ on $M$ with the following properties:
[b]a.)[/b] $f(x, y)$ takes non-negative integer value; [b]
b.)[/b] $\sum^n_{y=1} f(x, y) = n - 1$ for $1 \eq x \leq n$;
[b]c.)[/b] If $f(x_1, y_1)f(x2, y2) > 0$, then $(x_1 - x_2)(y_1 - y_2) \geq 0.$
Find $N(n)$, the number of functions $f$ that satisfy all the conditions. Give the explicit value of $N(4)$.
Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$.
(Australia)
[b]7.1 . [/b] The area of the quadrilateral is $3$ cm$^2$ , and the lengths of its diagonals are $6$ cm and $2$ cm. Find the angle between the diagonals.
[b]7.2[/b] Prove that the number $1 + 2^{3456789}$ is composite.
[b]7.3[/b] $20$ people took part in the chess tournament. The participant who took clear (undivided) $19$th place scored $9.5$ points. How could they distribute points among other participants?
[b]7.4[/b] The sum of the distances between the midpoints of opposite sides of a quadrilateral is equal to its semi-perimeter. Prove that this quadrilateral is a parallelogram.
[b]7.5[/b] $40$ people travel on a bus without a conductor passengers carrying only coins in denominations of $10$, $15$ and $20$ kopecks. Total passengers have $ 49$ coins. Prove that passengers will not be able to pay the required amount of money to the ticket office and pay each other correctly. (Cost of a bus ticket in 1963 was 5 kopecks.)
[b]7.6[/b] Some natural number $a$ is divided with a remainder by all natural numbers less than $a$. The sum of all the different (!) remainders turned out to be equal to $a$. Find $a$.
[b]7.7[/b] Two squares were cut out of a chessboard. In what case is it possible and in what case not to cover the remaining squares of the board with dominoes (i.e., figures of the form $2\times 1$) without overlapping?
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].
Which of the following is equal to the product
\[ \frac {8}{4}\cdot\frac {12}{8}\cdot\frac {16}{12}\cdots\frac {4n \plus{} 4}{4n}\cdots\frac {2008}{2004}?
\]$ \textbf{(A)}\ 251 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 2008 \qquad \textbf{(E)}\ 4016$
Suppose there are $n$ ordered pairs of positive integers $(a_i,b_i)$ such that $a_i+b_i=2020$ and $a_ib_i$ is a multiple of $2020$, where $1\le i \le n$. Compute the sum \[\sum_{i=1}^{n} a_i+b_i.\]
Let $\phi = \frac{1+\sqrt{5}}{2}$. A [i]base-$\phi$ number[/i] $(a_n a_{n-1} ... a_1 {a_0})_{\phi}$, where $0 \le a_n, a_{n-1}, ..., a_0 \le 1$ are integers, is defined by \[ (a_n a_{n-1} ... a_1 {a_0})_{\phi} = a_n \cdot \phi^n + a_{n-1} \cdot \phi^{n-1} + ... + a_1 \cdot \phi^1 + a_0. \]
Compute the number of base-$\phi$ numbers $(b_jb_{j-1}... b_1{b_0})_\phi$ which satisfy $b_j \ne 0$ and
\[ (b_jb_{j-1}... b_1{b_0})_\phi = \underbrace{(100 ... 100)_\phi}_{\text{Twenty}\ 100's}. \][i]Proposed by Yang Liu[/i]
The four circles in the figure determine 10 bounded regions. $10$ numbers summing to $100$ are written in these regions, one in each region. The sum of the numbers contained in each circle is equal to $S$ (the same quantity for each of the four circles). Determine the greatest and smallest possible values of $S$.
Edward’s formula for the stock market predicts correctly that the price of HMMT is directly proportional to a secret quantity $x$ and inversely proportional to $y$, the number of hours he slept the night before. If the price of HMMT is $\$12$ when $x = 8$ and $y = 4$, how many dollars does it cost when $x = 4$ and $y = 8$?
Find all positive integers \( m \) for which there exists an infinite subset \( A \) of the positive integers such that: for any pairwise distinct positive integers \( a_1, a_2, \cdots, a_m \in A \), the sum \( a_1 + a_2 + \cdots + a_m \) and the product \( a_1a_2 \cdots a_m \) are both square-free.
Kelvin the Frog lives in the 2-D plane. Each day, he picks a uniformly random direction (i.e. a uniformly random bearing $\theta\in [0,2\pi]$) and jumps a mile in that direction. Let $D$ be the number of miles Kelvin is away from his starting point after ten days. Determine the expected value of $D^4$.
Consider five points $ A$, $ B$, $ C$, $ D$ and $ E$ such that $ ABCD$ is a parallelogram and $ BCED$ is a cyclic quadrilateral. Let $ \ell$ be a line passing through $ A$. Suppose that $ \ell$ intersects the interior of the segment $ DC$ at $ F$ and intersects line $ BC$ at $ G$. Suppose also that $ EF \equal{} EG \equal{} EC$. Prove that $ \ell$ is the bisector of angle $ DAB$.
[i]Author: Charles Leytem, Luxembourg[/i]
Which triangles satisfy the equation $\frac{c^2-a^2}{b}+\frac{b^2-c^2}{a}=b-a$ when $a, b$ and $c$ are sides of a triangle?
An infinite (in both directions) sequence of rooms is situated on one side of an infinite hallway. The rooms are numbered by consecutive integers and each contains a grand piano. A finite number of pianists live in these rooms. (There may be more than one of them in some of the rooms.) Every day some two pianists living in adjacent rooms (the Arth and ($k +1$)st) decide that they interfere with each other’s practice, and they move to the ($k - 1$)st and ($k + 2$)nd rooms, respectively. Prove that these moves will cease after a finite number of days.
(VG Ilichev)
Let $\triangle ABC $ be an acute triangle.$O$ denote its circumcenter.Points $D$,$E$,$F$ are the midpoints of the sides $BC$,$CA$,and $AB$.Let $M$ be a point on the side $BC$ . $ AM \cap EF = \big\{ N \big\} $ . $ON \cap \big( ODM \big) = \big\{ P \big\} $ Prove that $M'$ lie on $\big(DEF\big)$ where $M'$ is the symmetrical point of $M$ thought the midpoint of $DP$.
Given coprime positive integers $p,q>1$, call all positive integers that cannot be written as $px+qy$(where $x,y$ are non-negative integers) [i]bad[/i], and define $S(p,q)$ to be the sum of all bad numbers raised to the power of $2019$. Prove that there exists a positive integer $n$, such that for any $p,q$ as described, $(p-1)(q-1)$ divides $nS(p,q)$.
Given positive numbers $a,b,c$, find the minimum of the function $f(x) = \sqrt{a^2 +x^2} +\sqrt{(b-x)^2 +c^2}$.
The convex $1976$-gon is divided into $1975$ triangles. Prove that there is a straight line separating one of these triangles from the rest.
Two players A and B play the following game. Initially, there are $ m$ equal positive integers $ n$ written on a blackboard. A begins and the players move alternately. The player to move chooses one of the non-zero numbers on the board. If this number k is the smallest among all positive integers on the board, the player replaces it with $ k\minus{}1$; if not, the player replaces it with the smallest positive number on the board. The player who first turns all the numbers into zeroes, wins. Who wins if both players use their best strategies?
Prove that there exists a polynomial $P(x)$ with real coefficients and degree greater than 1 such that both of the following conditions are true
$\cdot$ $P(a)+P(b)+P(c)\ge 2021$ holds for all nonnegative real numbers $a,b,c$ such that $a+b+c=3$
$\cdot$ There are infinitely many triples $(a,b,c)$ of nonnegative real numbers such that $a+b+c=3$ and $P(a)+P(b)+P(c)= 2021$
Suppose that $\omega$ is a primitive $2007^{\text{th}}$ root of unity. Find $\left(2^{2007}-1\right)\displaystyle\sum_{j=1}^{2006}\dfrac{1}{2-\omega^j}$.
Let $LEX INGT_1ONMAT_2H$ be a regular $13$-gon. Find $\angle LMT_1$, in degrees.
[i]Proposed by Edwin Zhao[/i]