Found problems: 85335
Consider a sequence $F_0=2$, $F_1=3$ that has the property $F_{n+1}F_{n-1}-F_n^2=(-1)^n\cdot2$. If each term of the sequence can be written in the form $a\cdot r_1^n+b\cdot r_2^n$, what is the positive difference between $r_1$ and $r_2$?
A certain country wishes to interconnect $2021$ cities with flight routes, which are always two-way, in the following manner:
• There is a way to travel between any two cities either via a direct flight or via a sequence of connecting flights.
• For every pair $(A, B)$ of cities that are connected by a direct flight, there is another city $C$ such that $(A, C)$ and $(B, C)$ are connected by direct flights.
Show that at least $3030$ flight routes are needed to satisfy the two requirements.
Determine all possible values of $a_1$ for which there exists a sequence $a_1, a_2, \dots$ of rational numbers satisfying $$a_{n+1}^2-a_{n+1}=a_n$$
for all positive integers $n$.
Prove that the line joining the centroid and the incenter of a non-isosceles triangle is perpendicular to the base if and only if the sum of the other two sides is thrice the base.
Find the angles of the cyclic quadrilateral if you know that each of its diagonals is a bisector of one angle and a trisector of the opposite one (the trisector of the angle is one of the two rays that lie in the interior of the angle and divide it into three equal parts).
(Vyacheslav Yasinsky)
Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied:
[list=1]
[*] Each number in the table is congruent to $1$ modulo $n$.
[*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$.
[/list]
Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.
A group of n friends wrote a math contest consisting of eight short-answer problem $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$, and four full-solution problems $F_1, F_2, F_3, F_4$. Each person in the group correctly solved exactly 11 of the 12 problems. We create an 8 x 4 table. Inside the square located in the $i$th row and $j$th column, we write down the number of people who correctly solved both problem $S_i$ and $F_j$. If the 32 entries in the table sum to 256, what is the value of n?
Let $a, b, c$ be distinct real numbers such that $a+b+c=0$ and $$
a^{2}-b=b^{2}-c=c^{2}-a.
$$
Evaluate all the possible values of $a b+a c+b c$.
[i]Proposed by Nguyen Anh Vu, Vietnam[/i]
The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?
Let $ABC$ be a scalene triangle with incenter $I$ and let $AI$ meet the circumcircle of triangle $ABC$ again at $M$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $AB, AC$ at $D, E$, respectively. Let $O$ be the circumcenter of triangle $BDE$ and let $L$ be the intersection of $\omega$ and the altitude from $A$ to $BC$ so that $A$ and $L$ lie on the same side with respect to $DE$. Denote by $\Omega$ a circle centered at $O$ and passing through $L$, and let $AL$ meet $\Omega$ again at $N$.
Prove that the lines $LD$ and $MB$ meet on the circumcircle of triangle $LNE$.
Let $a,b,c$ be positive real numbers such that $a+b+c=2005$. Find the minimum value of the expression:
$$E=a^{2006}+b^{2006}+c^{2006}+\frac{(ab)^{2004}+(bc)^{2004}+(ca)^{2004}}{(abc)^{2004}}$$
For a real number $t$ and positive real numbers $a,b$ we have
\[2a^2-3abt+b^2=2a^2+abt-b^2=0\]
Find $t.$
Find the first digit after the decimal point of the number $\displaystyle \frac1{1009}+\frac1{1010}+\cdots + \frac1{2016}$
Let points $M$ and $N$ lie on sides $AB$ and $BC$ of triangle $ABC$ in such a way that $MN||AC$. Points $M'$ and $N'$ are the reflections of $M$ and $N$ about $BC$ and $AB$ respectively. Let $M'A$ meet $BC$ at $X$, and let $N'C$ meet $AB$ at $Y$. Prove that $A,C,X,Y$ are concyclic.
Can you find three polynomials $P,Q,R$ of three variables $x,y,z$, providing the condition:
a)$P(x-y+z)^3 + Q(y-z-1)^3 +R(z-2x+1)^3 = 1$
b)$P(x-y+z)^3 + Q(y-z-1)^3 +R(z-x+1)^3 = 1$
for all $x,y,z$?
For every $n$ positive integers we denote $$\frac{x_n}{y_n}=\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}}$$
where $x_n, y_n$ are coprime positive integers. Prove that $y_n$ is not divisible by $2^n$ for any positive integers $n$.
Ha Duy Hung, high school specializing in the Ha University of Education, Hanoi, Xuan Thuy, Cau Giay, Hanoi
If $a,b$ are real numbers such that $a^3 +12a^2 + 49a + 69 = 0$ and $ b^3 - 9b^2 + 28b - 31 = 0,$ find $a + b .$
Let $a \in \mathbb{N}$ such that $a + n^2$ can be expressed as the sum of two squares for all $n \in \mathbb{N}$. Prove that $a$ is the square of a natural number.
Show that for every polynomial $W$ in three variables there exist polynomials $U$ and $V$ such that:
$$W(x,y,z) = U(x,y,z)+V(x,y,z),$$
$$U(x,y,z) = U(y,x,z),$$
$$V(x,y,z) = -V(x,z,y).$$
Let $a_0,a_1,\dots,a_n$ be positive real numbers such that $a_{k+1}-a_k \geq 1$ for all $k=0,1,\dots,n-1.$ Prove that
\[1+\frac{1}{a_0} \left( 1+\frac1{a_1-a_0}\right)\cdots\left(1+\frac1{a_n-a_0}\right)\leq \left(1+\frac1{a_0}\right) \left(1+\frac1{a_1}\right)\cdots \left(1+\frac1{a_n}\right).\]
Consider a regular cube with side length $2$. Let $A$ and $B$ be $2$ vertices that are furthest apart. Construct a sequence of points on the surface of the cube $A_1$, $A_2$, $\ldots$, $A_k$ so that $A_1=A$, $A_k=B$ and for any $i = 1,\ldots, k-1$, the distance from $A_i$ to $A_{i+1}$ is $3$. Find the minimum value of $k$.
Some 46 squares are randomly chosen from a $9 \times 9$ chess board and colored in [color=red]red[/color]. Show that there exists a $2\times 2$ block of 4 squares of which at least three are colored in [color=red]red[/color].
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: [list][*] $f(2)=2$, [*] $f(mn)=f(m)f(n)$, [*] $f(n+1)>f(n)$. [/list]
Let $ ABCD$ be a convex quadrilateral. Points $ P,Q$ and $ R$ are the feets of the perpendiculars from point $ D$ to lines $ BC, CA$ and $ AB$, respectively. Prove that $ PQ\equal{}QR$ if and only if the bisectors of the angles $ ABC$ and $ ADC$ meet on segment $ AC$.
Let $P$ be a simple polygon completely in $C$, a circle with radius $1$, such that $P$ does not pass through the center of $C$. The perimeter of $P$ is $36$. Prove that there is a radius of $C$ that intersects $P$ at least $6$ times, or there is a circle which is concentric with $C$ and have at least $6$ common points with $P$.
[i]Proposed by Seyed Reza Hosseini[/i]