Let $n > 1$ be an integer. Prove that there exists a unique positive integer $A$ that has the following property:
(1) $A < n^2$
(2) $n$ is a divisor of $\left[\frac{n^2}{A}\right] + 1$. Here $[X]$ denotes the greatest integer not exceeding the real number $X$.
Search found 79 matches
- Mon Sep 16, 2013 12:48 am
- Forum: Number Theory
- Topic: An Existence
- Replies: 4
- Views: 3549
- Mon Sep 16, 2013 12:39 am
- Forum: Geometry
- Topic: Where I visualize cyclic ness?
- Replies: 4
- Views: 4421
Where I visualize cyclic ness?
Let $ABC$ be an acute angled triangle with altitudes $AD, BE$ and $CF$. Suppose $EF$ intersects $BC$ at $P$. Let the line through $D$ parallel to $EF$ intersect $AB$ at $R$ and $AC$ at $Q$. Prove that the circumcircle of triangle $PQR$ passes through the midpoint of side $BC$.
- Fri Sep 06, 2013 8:16 pm
- Forum: Geometry
- Topic: [OGC1] General math. theorem 1
- Replies: 1
- Views: 2424
Re: [OGC1] General math. theorem 1
I think stating this as a theorem isn't elegant. The statement can actually be said as a reformulated form of the definition of a straight line and of a right angle.
- Fri Sep 06, 2013 8:09 pm
- Forum: Geometry
- Topic: Two brother triangles
- Replies: 1
- Views: 1963
Two brother triangles
Let ABC be an acute-angled triangle. Two points D and E are taken that has the following property: 1) D and E belongs to side AC and side AB, respectively. 2) <ADE = <ABC 3) The incenter of triangle ABC lies on the circumcircle of triangle ADE. Suppose K is the point of intersection of side BC and t...
- Fri Sep 06, 2013 7:48 pm
- Forum: Geometry
- Topic: The friendship of four circles
- Replies: 0
- Views: 1570
The friendship of four circles
Let ABCD be a convex quadrilateral with O the intersection of diagonals. The circumcircle of the triangles AOB and COD meet at two distinct points O and M. Denote the intersection of the circumcircle of triangle AOD with line OM by S and the intersection of the circumcircle of triangle BOC with line...
- Wed Aug 14, 2013 12:37 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2003 shortlist, tough number theory problem
- Replies: 0
- Views: 2272
IMO 2003 shortlist, tough number theory problem
Let p be a prime number, and A be a set of positive integers. The set A has the following property: 1) Suppose a set S of prime numbers is formed, so that for every q belongs to S; there exists at least one number belongs to A, say it x, such that q divides x. Again, no prime number r, does not belo...
- Tue Aug 06, 2013 9:11 pm
- Forum: Number Theory
- Topic: The Regularity of Mathematician DJ
- Replies: 0
- Views: 1526
The Regularity of Mathematician DJ
Mathematician DJ regularly perform two tasks: 1) He factor some positive integers every day. As he starts factoring a number, he immediately finish the task. 2) He writes a letter to his friend ‘R’ every day , in which DJ mention how many numbers he factored after writing the last letter and before ...
- Sun Jul 28, 2013 5:47 pm
- Forum: Number Theory
- Topic: The game of choices
- Replies: 2
- Views: 2498
The game of choices
The Game of choices is played between $A$ and $B$ as follows: 1) Initially the referee chooses two positive integers $m$ and $x$. 2) Then the player $A$ must choose i) one positive integer $k$, ii) a set of positive integers $D$ with cardinality at most $x$. 3) Next, to win, the player $B$ must choo...
- Thu Jul 04, 2013 10:26 pm
- Forum: Number Theory
- Topic: Multi-exponential diophantine equation
- Replies: 1
- Views: 2650
Multi-exponential diophantine equation
Find all integer solutions $(s,u,v)$ to the diophantine equation;
\[s^3 = u^2 + 3 v^2\]
given that $s$ is odd.
\[s^3 = u^2 + 3 v^2\]
given that $s$ is odd.