Found problems: 85335
Is there a positive integer $m$ such that the equation \[ {1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c} \] has infinitely many solutions in positive integers $a,b,c$?
There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold:
[i]i.)[/i] Each pair of students are in exactly one club.
[i]ii.)[/i] For each student and each society, the student is in exactly one club of the society.
[i]iii.)[/i] Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is
in exactly $m$ societies.
Find all possible values of $k$.
[i]Proposed by Guihua Gong, Puerto Rico[/i]
Find the number of $4 \times 4$ array whose entries are from the set $\{ 0 , 1, 2, 3 \}$ and which are such that the sum of the numbers in each of the four rows and in each of the four columns is divisible by $4$.
A [i]labyrinth[/i] is a system of $2024$ caves and $2023$ non-intersecting (bidirectional) corridors, each of which connects exactly two caves, where each pair of caves is connected through some sequence of corridors. Initially, Erik is standing in a corridor connecting some two caves. In a move, he can walk through one of the caves to another corridor that connects that cave to a third cave. However, when doing so, the corridor he was just in will magically disappear and get replaced by a new one connecting the end of his new corridor to the beginning of his old one (i.e., if Erik was in a corridor connecting caves $a$ and $b$ and he walked through cave $b$ into a corridor that connects caves $b$ and $c$, then the corridor between caves $a$ and $b$ will disappear and a new corridor between caves $a$ and $c$ will appear).
Since Erik likes designing labyrinths and has a specific layout in mind for his next one, he is wondering whether he can transform the labyrinth into that layout using these moves. Prove that this is in fact possible, regardless of the original layout and his starting position there.
Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$
Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x < a$ and $y < b$. How many of the following inequalities must be true?
(I) $x+y < a+b$
(II) $x-y < a-b$
(III) $xy < ab$
(IV) $\frac{x}{y} < \frac{a}{b}$
${ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 3\qquad\textbf{(E)}\ 4$
Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$. Prove that
[list]
[*]$m = 1$ and $l = 2k$; or
[*]$l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$.
[/list]
Let $P$ be a point in the interior of quadrilateral $ABCD$ such that:
$$\angle BPC=2\angle BAC \ \ ,\ \ \angle PCA = \angle PAD \ \ ,\ \ \angle PDA=\angle PAC$$
Prove that:
$$\angle PBD= \left | \angle BCA - \angle PCA \right |$$
[i]Proposed by Ali Zamani[/i]
Let $ABC$ be an isosceles triangle such that $AB=AC$. Find angles of triangle $ABC$ if
$\frac{AB}{BC}=1+2\cos{\frac{2\pi}{7}}$
In a right triangle $ ABC $, the altitude $ CD $ is drawn from the vertex of the right angle $ C $ and a circle is inscribed in each of the triangles $ ABC $, $ ACD $ and $ BCD $. Prove that the sum of the radii of these circles equals the height $ CD $.
Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 3$. Prove that
$$\frac{x + 3}{y + z} + \frac{y + 3}{z + x} + \frac{z + 3}{x + y} + 3 \ge 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^2}{(x + y + z)^3}.$$
Proposed by [i]Petar Filipovski, Macedonia[/i]
Let $(A,+,\cdot)$ be a ring with unity, having the following property: for all $x\in A$ either $x^2=1$ or $x^n=0$ for some $n\in\mathbb N$. Show that $A$ is a commutative ring.
Each day $289$ students are divided into $17$ groups of $17$. No two students are ever in the same group more than once. What is the largest number of days that this can be done?
In triangular pyramid $S-ABC$, any two of $SA,SB,SC$ are perpendicular. $M$ is the centre of gravity of $\triangle ABC$. $D$ is the midpoint of $AB$, line $DP//SC$. Prove:
[b](a)[/b] $DP$ and $SM$ intersect.
[b](b)[/b] $DP\cap SM=D'$, then $D'$ is the center of circumsphere of $S-ABC$.
An urn was filled with three balls by the following procedure: it was thrown a coin three times, inserting, each time a white ball came up heads, and every time tails came up, a black ball. We draw from this urn, four times consecutive, one ball; we return it to the urn before the next extraction. Which is the probability that in the four extractions a cue ball is obtained?
Let $I \subset \mathbb{R}$ be an open interval and $f:I \to \mathbb{R}$ a twice differentiable function such that $f(x)f''(x)=0,$ for any $x \in I.$ Prove that $f''(x)=0,$ for any $x \in I.$
For a function $f : [0,1] \to [0,1] $ we define $f^1 = f $ and $f^{n+1} (x) = f (f^n(x))$ for $0 \le x \le 1$ and $n \in N$. Given that there is a $n$ such that $|f^n(x) - f^n(y)| < |x - y| $ for all distinct $x, y \in [0,1]$, prove that there is a unique $x_0 \in [0,1]$ such that $f (x_0) = x_0$.
A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ 6 $
Given cubic polynomial with integer coefficients and three irrational roots. Show that none of these roots can be root of quadratic equation with integer coefficients.
Let $p$ be a prime number. We construct a directed graph of $p$ vertices, labeled with integers from $0$ to $p-1$. There is an edge from vertex $x$ to vertex $y$ if and only if $x^2+1\equiv y \pmod{p}$. Let $f(p)$ denotes the length of the longest directed cycle in this graph. Prove that $f(p)$ can attain arbitrarily large values.
Prove that there are infinite triples $(a,b,c)$ of positive integers $a,b,c>1$, $gcd(a,b)=gcd(b,c)=gcd(c,a)=1$ such that $a+b+c$ divides $a^b+b^c+c^a$.
(a) Find a polygon which can be cut by a straight line into two congruent parts so that one side of the polygon is divided in half while another side at a ratio of $1 : 2$.
(b) Does there exist a convex polygon with this property?
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer.
[i]Laurentiu Panaitopol, Romania[/i]
It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the $x$ and $y$ axes and so that the medians to the midpoints of the legs lie on the lines $y = 3x + 1$ and $y = mx + 2$. The number of different constants $m$ for which such a triangle exists is
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \text{more than 3} $
Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers.