Found problems: 85335
$n$ is a natural number larger than $3$ and denote all positive coprime numbers with $n$ as $1= b_1 < b_2 < \cdots b_k$. For a positive integer $m$ which is larger than $3$ and is coprime with $n$, let $A$ be the set of tuples $(a_1,a_2, \cdots a_k)$ satisfying the condition.
$$\textbf{Condition}: \text{For all integers } i, 0 \le a_i < m \text{ and } a_1b_1 + a_2b_2 + \cdots a_kb_k \text{ is a mutiple of } n$$
For elements of $A$, show that the difference of number of elements such that $a_1 = 1$ and the number of elements such that $a_2 = 2$ maximum $1$
In the adjoining figure $AB$ and $BC$ are adjacent sides of square $ABCD$; $M$ is the midpoint of $AB$; $N$ is the midpoint of $BC$; and $AN$ and $CM$ intersect at $O$. The ratio of the area of $AOCD$ to the area of $ABCD$ is
[asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--(2,1)--(2,2)--(1,0));
label("A", (0,0), S);
label("B", (2,0), S);
label("C", (2,2), N);
label("D", (0,2), N);
label("M", (1,0), S);
label("N", (2,1), E);
label("O", (1.2, .8));
[/asy]
$ \textbf{(A)}\ \frac{5}{6} \qquad\textbf{(B)}\ \frac{3}{4} \qquad\textbf{(C)}\ \frac{2}{3} \qquad\textbf{(D)}\ \frac{\sqrt{3}}{2} \qquad\textbf{(E)}\ \frac{(\sqrt{3}-1)}{2} $
When Meena turned 16 years old, her parents gave her a cake with $n$ candles, where $n$ has exactly 16 different positive integer divisors. What is the smallest possible value of $n$?
The sequence $a_1,a_2,...,a_n$ satisfies the following conditions: $$a_1=0, |a_2|=|a_1+1|, ..., |a_n|=|a_{n-1}+1|.$$ Prove that $$(a_1+a_2+...+a_n)/n \ge -1/2$$
Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\]
Define the set $A$ by
\[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\]
Prove that, if $A$ is not empty, then
\[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]
Determine all four-digit numbers $\overline{abcd}$ which are perfect squares and for which the equality holds:
$\overline{ab}=3 \cdot \overline{cd} + 1$.
Starting at $(0, 0)$, Richard takes $2n+1$ steps, with each step being one unit either East, North, West, or South. For each step, the direction is chosen uniformly at random from the four possibilities. Determine the probability that Richard ends at $(1, 0)$.
Find the value of $x$ where the graph of $$y=\log_3(\sqrt{x^2+729}+x)-2\log_3(\sqrt{x^2+729}-x)$$ crosses the $x$-axis.
Some numbers are written along the ring. If inequality $(a-d)(b-c) < 0$ is held for the four arbitrary numbers in sequence $a,b,c,d$, you have to change the numbers $b$ and $c$ places. Prove that you will have to do this operation finite number of times.
Two treasure-hunters found a treasure containing coins of value $a_1< a_2 < ... < a_{2003}$ (the quantity of coins of each value is unlimited). The first treasure-hunter forms all the possible sets of different coins containing odd number of elements, and takes the most valuable coin of each such set. The second treasure-hunter forms all the possible sets of different coins containing even number of elements, and takes the most valuable coin of each such set. Which one of them is going to have more money and how much more?
(H. Nestra)
$ (6?3)+4-(2-1) = 5. $ To make this statement true, the question mark between the 6 and the 3 should be replaced by
$ \text{(A)}\div\qquad\text{(B)}\ \times\qquad\text{(C)}+\qquad\text{(D)}\ -\qquad\text{(E)}\ \text{None of these} $
A [i]regular octahedron[/i] has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of $Q$?
[asy]
// Note: This diagram was not made by me.
import graph;
// The Solid
// To save processing time, do not use three (dimensions)
// Project (roughly) to two
size(15cm);
pair Fr, Lf, Rt, Tp, Bt, Bk;
Lf=(0,0);
Rt=(12,1);
Fr=(7,-1);
Bk=(5,2);
Tp=(6,6.7);
Bt=(6,-5.2);
draw(Lf--Fr--Rt);
draw(Lf--Tp--Rt);
draw(Lf--Bt--Rt);
draw(Tp--Fr--Bt);
draw(Lf--Bk--Rt,dashed);
draw(Tp--Bk--Bt,dashed);
label(rotate(-8.13010235)*slant(0.1)*"$Q$", (4.2,1.6));
label(rotate(21.8014095)*slant(-0.2)*"$?$", (8.5,2.05));
pair g = (-8,0); // Define Gap transform
real a = 8;
draw(g+(-a/2,1)--g+(a/2,1), Arrow()); // Make arrow
// Time for the NET
pair DA,DB,DC,CD,O;
DA = (6.92820323028,0);
DB = (3.46410161514,6);
DC = (DA+DB)/3;
CD = conj(DC);
O=(0,0);
transform trf=shift(3g+(0,3));
path NET = O--(-2*DA)--(-2DB)--(-DB)--(2DA-DB)--DB--O--DA--(DA-DB)--O--(-DB)--(-DA)--(-DA-DB)--(-DB);
draw(trf*NET);
label("$7$",trf*DC);
label("$Q$",trf*DC+DA-DB);
label("$5$",trf*DC-DB);
label("$3$",trf*DC-DA-DB);
label("$6$",trf*CD);
label("$4$",trf*CD-DA);
label("$2$",trf*CD-DA-DB);
label("$1$",trf*CD-2DA);
[/asy]
$\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}~5\qquad$
i) A transformation of the plane into itself preserves all rational distances. Prove that it preserves all distances.
ii) Show that the corresponding statement for the line is false.
2. Each of $n$ members of a club is given a different item of information. The members are allowed to share the information, but, for security reasons, only in the following way: A pair may communicate by telephone. During a telephone call only one member may speak. The member who speaks may tell the other member all the information (s)he knows. Determine the minimal number of phone calls that are required to convey all the information to each of the members.
Hi, from my sketches I'm thinking the answer is $2n-2$ but I dont know how to prove that this number of calls is the smallest. Can anyone enlighten me? Thanks
Bartek patiently performs operations on fractions. In each move, he adds its inverse to the current result, obtaining a new result. Bartek starts with the number $1$: after the first move, he receives the result 2, after the second move, the result is $\frac{5}{2}$, after the third move $\frac{29}{10}$, etc. After $300$ moves, Bartek receives the result $x$. Determine the largest integer not greater than $x$.
In the plane, a straight line $ m $ is given and points $ A $ and $ B $ lie on opposite sides of the straight line $ m $. Find a point $ M $ on the line $ m $ such that the difference in distances of this point from points $ A $ and $ B $ is as large as possible.
A strictly increasing sequence $(a_n)$ has the property that $\gcd(a_m,a_n) = a_{\gcd(m,n)}$ for all $m,n\in \mathbb{N}$. Suppose $k$ is the least positive integer for which there exist positive integers $r < k < s$ such that $a_k^2 = a_ra_s$. Prove that $r | k$ and $k | s$.
Let $a$, $b$, $c$ be side length of a triangle. Prove the inequality
\begin{align*} \sqrt{a^2 + ab + b^2} + \sqrt{b^2 + bc + c^2} + \sqrt{c^2 + ca + a^2} \leq \sqrt{5a^2 + 5b^2 + 5c^2 + 4ab + 4 bc + 4ca}.\end{align*}
Let $a, b, c$ be positive reals. Show that $$1 +\frac{3}{ab + bc + ca}\ge \frac{6}{a + b + c}$$
There is a $4 \times 4$ array of integers $A$, all initially equal to $0$. An operation may be performed on the array for any row or column such that every number in that row or column has $1$ added to it, and then is replaced with its remainder modulo $3$. Given a random $4 \times 4$ array of integers between $0$ and $2$ not identical to $A$, the probability that it can be reached through a series of operations on $A$ is $\frac{p}{q},$ where $p,q$ are relatively prime positive integers. Find $p$.
Two players $A$ and $B$ have one stone each on a $100 \times 100$ chessboard. They move their stones one after the other, and a move means moving one's stone to a neighbouring field (horizontally or vertically, not diagonally). At the beginning of the game, the stone of $A$ lies in the lower left corner, and the one of $B$ in the lower right corner. Player $A$ starts.
Prove: Player $A$ is, independently from that what $B$ does, able to reach, after finitely many steps, the field $B$'s stone is lying on at that moment.
Let $n = 6901$. There are $6732$ positive integers less than or equal to $n$ that are also relatively prime to $n$. Find the sum of the distinct prime factors of $n$.
Does there exist an arithmetic progression with $2017$ terms such that each term is not a perfect power, but the product of all $2017$ terms is?
Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C. within a population of men. For each of the three factors, the probability that a randomly selected man in the population as only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability that a randomly selected man has exactly two of these two risk factors (but not the third) is 0.14. The probability that a randomly selected man has all three risk factors, given that he has A and B is $\tfrac{1}{3}$. The probability that a man has none of the three risk factors given that he does not have risk factor A is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Let $\alpha_a$ denote the greatest odd divisor of a natural number $a$, and let $S_b=\sum_{a=1}^b\frac{\alpha_a}a$ Prove that the sequence $S_b/b$ has a finite limit when $b\to\infty$, and find this limit.