Found problems: 85335
Let $N$ be a positive integer such that the sum of the squares of all positive divisors of $N$ is equal to the product $N(N+3)$. Prove that there exist two indices $i$ and $j$ such that $N=F_iF_j$ where $(F_i)_{n=1}^{\infty}$ is the Fibonacci sequence defined as $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$.
[i]Proposed by Alain Rossier, Switzerland[/i]
Let $ABC$ be a triangle with incenter $I$. Let $D$ be the point of intersection between the incircle and the side $BC$, the points $P$ and $Q$ are in the rays $IB$ and $IC$, respectively, such that $\angle IAP=\angle CAD$ and $\angle IAQ=\angle BAD$. Prove that $AP=AQ$.
Show that for any $i=1,2,3$, there exist infinity many positive integer $n$, such that among $n$, $n+2$ and $n+28$, there are exactly $i$ terms that can be expressed as the sum of the cubes of three positive integers.
Let $A,B\in \mathcal{M}_2(\mathbb{C})$ two non-zero matrices such that $AB+BA=O_2$ and $\det(A+B)=0$. Prove $A$ and $B$ have null traces.
Find distinct positive integers $n_1<n_2<\dots<n_7$ with the least possible sum, such that their product $n_1 \times n_2 \times \dots \times n_7$ is divisible by $2016$.
In triangle ABC, consider the sizes $\tan \angle A, \tan \angle B$, and $\tan \angle C$ into another such as the numbers $1, 2$ and $3$. Find the ratio of the sidelenghts $AC$ and $AB$ of the triangle.
Determine all composite positive integers $n$ such that, for each positive divisor $d$ of $n$, there are integers $k\geq 0$ and $m\geq 2$ such that $d=k^m+1$.
At the beginning of a two-player game, the number $2004!$ is written on the blackboard. The players move alternately. In each move, a positive integer smaller than the number on the blackboard and divisible by at most $20$ different prime numbers is chosen. This is subtracted from the number on the blackboard, which is erased and replaced by the difference. The winner is the player who obtains $0$. Does the player who goes first or the one who goes second have a guaranteed win, and how should that be achieved?
Compute \[\lim_{h\to 0}\dfrac{\sin(\frac{\pi}{3}+4h)-4\sin(\frac{\pi}{3}+3h)+6\sin(\frac{\pi}{3}+2h)-4\sin(\frac{\pi}{3}+h)+\sin(\frac{\pi}{3})}{h^4}.\]
In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once and the Europeans teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won?
Let $p$ be an odd prime, and put $N=\frac{1}{4} (p^3 -p) -1.$ The numbers $1,2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N,$ denote $r(n)$ the fraction of integers $\{ 1,2, \dots, n \}$ that are red.
Prove that there exists a positive integer $a \in \{ 1,2, \dots, p-1\}$ such that $r(n) \neq a/p$ for all $n = 1,2, \dots , N.$
[I]Netherlands[/i]
A point $X$ exactly $\sqrt{2}-\frac{\sqrt{6}}{3}$ away from the origin is chosen randomly. A point $Y$ less than $4$ away from the origin is chosen randomly. The probability that a point $Z$ less than $2$ away from the origin exists such that $\triangle XYZ$ is an equilateral triangle can be expressed as $\frac{a\pi + b}{c \pi}$ for some positive integers $a, b, c$ with $a$ and $c$ relatively prime. Find $a+b+c$.
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc \equal{} 1$. Prove that
\[ \frac {1}{a^{3}\left(b \plus{} c\right)} \plus{} \frac {1}{b^{3}\left(c \plus{} a\right)} \plus{} \frac {1}{c^{3}\left(a \plus{} b\right)}\geq \frac {3}{2}.
\]
Let $P,Q,R$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$. Suppose that $BQ$ and $CR$ meet at $A', AP$ and $CR$ meet at $B'$, and $AP$ and $BQ$ meet at $C'$, such that $AB' = B'C', BC' =C'A'$, and $CA'= A'B'$. Compute the ratio of the area of $\triangle PQR$ to the area of $\triangle ABC$.
Tori's mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered $70\%$ of the arithmetic, $40\%$ of the algebra, and $60\%$ of the geometry problems correctly, she did not pass the test because she got less than $60\%$ of the problems right. How many more problems would she have needed to answer correctly to earn a $60\%$ passing grade?
$ \text{(A)}\ 1\qquad\text{(B)}\ 5\qquad\text{(C)}\ 7\qquad\text{(D)}\ 9\qquad\text{(E)}\ 11 $
Given a positive integer $a$, prove that $\sigma(am) < \sigma(am + 1)$ for infinitely many positive integers $m$. (Here $\sigma(n)$ is the sum of all positive divisors of the positive integer number $n$.)
[i]Vlad Matei[/i]
Let be a point $ P $ inside a triangle $ ABC. $ Prove that the following relations are equivalent:
$ \text{(i)} $ Any collinear triple of points $ (E,P,F) $ with $ E,F $ on $ AB,AC, $ respectively, verifies the equality
$$ \frac{1}{AE} +\frac{1}{AF} =\frac{AB+BC+CA}{AB\cdot AC} $$
$ \text{(ii)} P $ is the incircle of $ ABC $
Let $0<\alpha,\beta<\frac{\pi}{2}$ which satisfy
\[(\cos^2\alpha+\cos^2\beta)(1+\tan\alpha\tan\beta)=2\]
Prove that $\alpha+\beta=\frac{\pi}{2}$.
The lengths, $a$ and $b$, of two sides of a triangle are known.
(a) What length should the third side be, in order for the largest angle of the triangle to be of the least possible value?
(b) What length should the third side be in order for the smallest angle of the triangle to be of the greatest possible value?
Consider three different planes and consider also one point on each of them. Give necessary and sufficient conditions for the existence of a quadratic which passes through the given points and whose tangent-plane at each of these points is the respective given plane.
The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$.
[hide="comment"]
Hi guys ,
Here is a nice problem:
Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$
Here are some futher question proposed by me :Prove or disprove that :
1) $gcd(n,a_n)=1$
2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$
Thanks kiu si u
[i]Edited by Orl.[/i]
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Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedron is exactly edge $AB$.
A box contains 900 cards numbered from 100 to 999. Cards are drawn randomly, one at a time, without replacement, and the sum of their digits is recorded. What is the minimum number of cards that must be drawn to guarantee that at least three of these sums are the same?
There are two round tables with $n{}$ dwarves sitting at each table. Each dwarf has only two friends: his neighbours to the left and to the right. A good wizard wants to seat the dwarves at one round table so that each two neighbours are friends. His magic allows him to make any $2n$ pairs of dwarves into pairs of friends (the dwarves in a pair may be from the same or from different tables). However, he knows that an evil sorcerer will break $n{}$ of those new friendships. For which $n{}$ is the good wizard able to achieve his goal no matter what the evil sorcerer does?
[i]Mikhail Svyatlovskiy[/i]
A number is written in each square of a chessboard, so that each number not on the border is the mean of the $4$ neighboring numbers. Show that if the largest number is $N$, then there is a number equal to $N$ in the border squares.