Found problems: 85335
Determine all triples $(x,y, p)$ with $x$, $y$ positive integers and $p$ a prime number verifying the equation $p^x -y^p = 1$.
A simple polygon in the plane is a figure bounded by a closed nonself-intersecting broken line.
(a) Do there exist two congruent simple $7$-gons in the plane such that all the seven vertices of one of the $7$-gons are the vertices of the other one and yet these two $7$-gons have no common sides?
(b) Do there exist three such $7$-gons?
(V Proizvolov)
For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. The first term in the series is:
${{ \textbf{(A)}\ -1221 \qquad\textbf{(B)}\ -21.5 \qquad\textbf{(C)}\ -20.5 \qquad\textbf{(D)}\ 3 }\qquad\textbf{(E)}\ 3.5 } $
With the use of three different weights, namely 1 lb., 3 lb., and 9 lb., how many objects of different weights can be weighed, if the objects is to be weighed and the given weights may be placed in either pan of the scale?
$ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 11\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 7 $
Kunal and Arnab play a game as follows. Initially there are $2$ piles of coins with $x$ and $y$ coins respectively. The game starts with Kunal. In each turn a player chooses one pile and removes as many coins as he wants from that pile. The game goes on and the last one to remove a coin loses. Determine all possible values of $(x,y)$ which ensure Kunal's victory against Arnab given both os them play optimally.
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[i]You are required to find an exhaustive set of solutions[/i]
A lock has $16$ keys arranged in a $4 \times 4$ array, each key oriented either horizontally or vertically. In order to open it, all the keys must be vertically oriented. When a key is switched to another position, all the other keys in the same row and column automatically switch their positions too (see diagram). Show that no matter what the starting
positions are, it is always possible to open this lock. (Only one key at a time can be switched.)
Given a triangle $ABC$, let $I$ be the incenter, and $J$ be the $A$-excenter. A line $\ell$ through $A$ perpendicular to $BC$ intersect the lines $BI$, $CI$, $BJ$, $CJ$ at $P$, $Q$, $R$, $S$ respectively. Suppose the angle bisector of $\angle BAC$ meet $BC$ at $K$, and $L$ is a point such that $AL$ is a diameter in $(ABC)$.
Prove that the line $KL$, $\ell$, and the line through the centers of circles $(IPQ)$ and $(JRS)$, are concurrent.
[i]Proposed by Chuah Jia Herng & Ivan Chan Kai Chin[/i]
Determine the integral part of $A$, where $A =\frac{1}{672}+\frac{1}{673}+... +\frac{1}{2014}$
Call a number $n$ [i]good[/i] if it can be expressed as $2^x+y^2$ for where $x$ and $y$ are nonnegative integers.
(a) Prove that there exist infinitely many sets of $4$ consecutive good numbers.
(b) Find all sets of $5$ consecutive good numbers.
[i]Proposed by Michael Ma[/i]
For a positive integer $m$, let $f(m)$ denote the smallest power of $2024$ not less than $m$ (e.g. $f(1)=1, f(2023)=f(2024)=2024,$ and $f(2025)=2024^2$). Find all positive real numbers $c$ for which there exists a sequence $x_1,x_2,\cdots$ of real numbers in $[0,1]$ such that
$$|x_m-x_n|\geq\frac{c}{f(m)}$$ for all positive integers $m>n\geq1$.
Proposed by Shantanu Nene
Suppose an integer-valued function $f$ satisfies
$$\sum_{k=1}^{2n+1}f(k)=\ln|2n+1|-4\ln|2n-1|\enspace\text{and}\enspace\sum_{k=0}^{2n}f(k)=4e^n-e^{n-1}$$
for all non-negative integers $n$. Determine $\sum_{n=0}^\infty\frac{f(n)}{2^n}$.
A ladder is leaning against a house with its lower end $15$ feet from the house. When the lower end is pulled $9$ feet farther from the house, the upper end slides $13$ feet down. How long is the ladder (in feet)?
Find the smallest positive integer $n$, such that one can color every cell of a $n \times n$ grid in red, yellow or blue with all the following conditions satisfied:
(1) the number of cells colored in each color is the same;
(2) if a row contains a red cell, that row must contain a blue cell and cannot contain a yellow cell;
(3) if a column contains a blue cell, it must contain a red cell but cannot contain a yellow cell.
Prove the identity $ \left[ \frac{3+x}{6} \right] -\left[ \frac{4+x}{6} \right] +\left[ \frac{5+x}{6} \right] =\left[ \frac{1+x}{2} \right] -\left[ \frac{1+x}{3} \right] ,\quad\forall x\in\mathbb{R} , $
where $ [] $ is the integer part.
[i]C. Mortici[/i]
Let be given a triangle $ABC$ with $BC = a, CA = b, AB = c$. Find point $P$ in the plane for which $aAP^{2}+bBP^{2}+cCP^{2}$ is minimum, and compute this minimum.
Each vertex of a cube is assigned an 1 or a -1, and each face is assigned the product of the numbers assigned to its vertices. Determine the possible values the sum of these 14 numbers can attain.
Compute the number of $10$-bit sequences of $0$’s and $1$’s do not contain $001$ as a subsequence.
Let $C=\{ z \in \mathbb{C} : |z|=1 \}$ be the unit circle on the complex plane. Let $z_1, z_2, \ldots, z_{240} \in C$ (not necessarily different) be $240$ complex numbers, satisfying the following two conditions:
(1) For any open arc $\Gamma$ of length $\pi$ on $C$, there are at most $200$ of $j ~(1 \le j \le 240)$ such that $z_j \in \Gamma$.
(2) For any open arc $\gamma$ of length $\pi/3$ on $C$, there are at most $120$ of $j ~(1 \le j \le 240)$ such that $z_j \in \gamma$.
Find the maximum of $|z_1+z_2+\ldots+z_{240}|$.
The city is a rectangle divided onto squares by $m$ streets coming from the West to the East and $n$ streets coming from the North to the South. There are militioners (policemen) on the streets but not on the crossroads. They watch the certain automobile, moving along the closed route, marking the time and the direction of its movement. Its trace is not known in advance, but they know, that it will not pass over the same segment of the way twice. What is the minimal number of the militioners providing the unique determination of the route according to their reports?
Given are a circle and an ellipse lying inside it with focus $C$. Find the locus of the circumcenters of triangles $ABC$, where $AB$ is a chord of the circle touching the ellipse.
The [i]giraffe[/i] is a chess piece that moves $4$ squares in one direction and then a box in a perpendicular direction. What is the smallest value of $n$ such that the giraffe that starts from a corner on an $n \times n$ board can visit all the squares of said board?
A sequence of integers $a_{1}, a_{2}, \cdots$ is called $good$ if:
• $a_{1}=1$, and;
• $a_{i+1}-a_{i}$ is either $1$ or $2$ for all $i \geq 1$.
Find all positive integers n that cannot be written as a sum $n = a_{1} + a_{2} + \cdots + a_{k}$,
such that the integers $a_{1} , a_{2} , \cdots , a_{k}$ forms a good sequence.
Given real numbers $ x_1 < x_2 < \ldots < x_n$ such that every real number occurs at most two times among the differences $ x_j \minus{} x_i$, $ 1\leq i < j \leq n$, prove that there exists at least $ \lfloor n/2\rfloor$ real numbers that occurs exactly one time among such differences.
What is the ones digit of \[222{,}222-22{,}222-2{,}222-222-22-2?\]
$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad\textbf{(E) }8$
Let $m$ be a positive integer. Determine the smallest positive integer $n$ for which there exist real numbers $x_1, x_2,...,x_n \in (-1, 1)$ such that $|x_1| + |x_2| +...+ |x_n| = m + |x_1 + x_2 + ... + x_n|$.