Found problems: 85335
In convex pentagon $ABCDE$ points $A_1$, $B_1$, $C_1$, $D_1$, $E_1$ are intersections of pairs of diagonals $(BD, CE)$, $(CE, DA)$, $(DA, EB)$, $(EB, AC)$ and $(AC, BD)$ respectively. Prove that if four of quadrilaterals $AB_{1}A_{1}B$, $BC_{1}B_{1}C$, $CD_{1}C_{1}D$, $DE_{1}D_{1}E$ and $EA_{1}E_{1}A$ are cyclic then the fifth one is also cyclic.
Let $a$ be such an integer, that $3a$ can be written in the form $x^2 + 2y^2$, with integers $x$ and $y$.
Prove that the number $a$ can also be written in this form.
[b]Additional problems:[/b]
[b]a)[/b] Find a general (necessary and sufficent) criterion for an integer $n$ to be of that form.
[b]b)[/b] In how many ways can the integer $n$ be represented in that way?
Find the number of pairs $(m,n)$ of positive integers with $1\le m<n\le 30$ such that there exists a real number $x$ satisfying $$\sin(mx)+\sin(nx)=2.$$
The twenty- first century began on January $1$, $2001$ and runs through December $31$, $2100$. Note that March $1$, $2014$ fell on Saturday, so there were fi ve Mondays in March $2014$. In how many years of the twenty- first century does March have fi ve Mondays?
Problem 9. The numbers 1,2,…,〖97〗^2 are written in the cells of a 97×97 board. In the center of each cell, there is a tower with the height equal to the number of that cell. Is it possible to see the top of any tower from the top of any other tower? (one point A can see the other point B, iff there is no other point on the segment AB).
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
The origami club meets once a week at a fixed time, but this week, the club had to reschedule the meeting to a different time during the same day. However, the room that they usually meet has $5$ available time slots, one of which is the original time the origami club meets. If at any given time slot, there is a $30$ percent chance the room is not available, what is the probability the origami club will be able to meet at that day?
A natural number $n$ is said to be $good$ if $n$ is the sum or $r$ consecutive positive integers, for some $r \geq 2 $. Find the number of good numbers in the set $\{1,2 \dots , 100\}$.
A collection of $n$ positive numbers, where repeats are allowed, adds to $500$. They can be split into $20$ groups each adding to $25$, and can also be split into $25$ groups each adding to $20$. (A group is allowed to contain any amount of integers, even just one integer.) What is the least possible value of $n$?
[i]Aaron Wang[/i]
In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b \geq 2$. A Martian student writes down
\begin{align*}3 \log(\sqrt{x}\log x) &= 56\\\log_{\log (x)}(x) &= 54
\end{align*}
and finds that this system of equations has a single real number solution $x > 1$. Find $b$.
Suppose three boba drinks and four burgers cost $28$ dollars, while two boba drinks and six burgers cost $\$ 37.70$. If you paid for one boba drink using only pennies, nickels, dimes, and quarters, determine the least number of coins you could use.
What is the remainder when $128^{2023}$ is divided by $126$?
For positive integers $m$ and $n$, find the smalles possible value of $|2011^m-45^n|$.
[i](Swiss Mathematical Olympiad, Final round, problem 3)[/i]
A positive integer is called [i]digit-reduced[/i] if at most nine different digits occur in its decimal representation (leading $0$s are omitted.) Let $M$ be a finite set of [i]digit-reduced[/i] numbers. Show that the sum of the reciprocals of the elements in $M$ is less than $180$.
Determine the value of $ab$ if $\log_8 a + \log_4 b^2 = 5$ and $\log_8 b + \log_4 a^2 = 7$.
The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).
Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
It is known that among several banknotes of pairwise distinct face values (which are positive integers) there are exactly $N{}$ fakes. In a single test, a detector determines the sum of the face values of all real banknotes in an arbitrary set we have selected. Prove that by using the detector $N{}$ times, all fake banknotes can be identified, if a) $N=2$ and b) $N=3$.
[i]Proposed by S. Tokarev[/i]
On a chessboard $5 \times 9$ squares, the following game is played.
Initially, a number of frogs are randomly placed on some of the squares, no square containing more than one frog. A turn consists of moving all of the frogs subject to the following rules:
$\bullet$ Each frog may be moved one square up, down, left, or right;
$\bullet$ If a frog moves up or down on one turn, it must move left or right on the next turn, and vice versa;
$\bullet$ At the end of each turn, no square can contain two or more frogs.
The game stops if it becomes impossible to complete another turn. Prove that if initially $33$ frogs are placed on the board, the game must eventually stop. Prove also that it is possible to place $32$ frogs on the board so that the game can continue forever.
A positive integer $N$ has the digits $1, 2, 3, 4, 5, 6$ and $7$, so that each digit $i$, $i \in \{1, 2, 3, 4, 5, 6, 7\}$ occurs $4i$ times in the decimal representation of $N$. Prove that $N$ is not a perfect square.
Let $P$, $A$, $B$ and $C$ be points on a line $r$, in that order, so that $AB = BC$. Let $H$ be a point that does not belong to this line and let $S$ be the other intersection of the circles $(HPB)$ and $(HAC)$. Let $I$ be the second intersection of the circle with diameter $HB$ and $(HAC)$. Show that the points $P$, $H$, $I$ lie on the same line if and only if $HS$ is perpendicular to $r$.
Let $ABC$ be a scalene triangle and let $P$ and $Q$ be two distinct points in its interior. Suppose that the angle bisectors of $\angle PAQ,\,\angle PBQ,$ and $\angle PCQ$ are the altitudes of triangle $ABC$. Prove that the midpoint of $\overline{PQ}$ lies on the Euler line of $ABC$.
(The Euler line is the line through the circumcenter and orthocenter of a triangle.)
[i]Proposed by Holden Mui[/i]
Let $p>5$ be a prime number. Show that $p-4$ cannot be the fourth power of a prime number.
$a_1\geq a_2\geq... \geq a_{100n}>0$
If we take from $(a_1,a_2,...,a_{100n})$ some $2n+1$ numbers $b_1\geq b_2 \geq ... \geq b_{2n+1}$ then $b_1+...+b_n > b_{n+1}+...b_{2n+1}$ Prove, that
$$(n+1)(a_1+...+a_n)>a_{n+1}+a_{n+2}+...+a_{100n}$$
Let $P$ and $Q$ be two distinct points in the plane. Let us denote by $m(PQ)$ the segment bisector of $PQ$. Let $S$ be a finite subset of the plane, with more than one element, that satisfies the following properties:
(i) If $P$ and $Q$ are in $S$, then $m(PQ)$ intersects $S$.
(ii) If $P_1Q_1, P_2Q_2, P_3Q_3$ are three diferent segments such that its endpoints are points of $S$, then, there is non point in $S$ such that it intersects the three lines $m(P_1Q_1)$, $m(P_2Q_2)$, and $m(P_3Q_3)$.
Find the number of points that $S$ may contain.