Found problems: 85335
Let $a_0=1,a_1=2,$ and $a_n=4a_{n-1}-a_{n-2}$ for $n\ge 2.$
Find an odd prime factor of $a_{2015}.$
Let $ABC$ be a triangle and let $\Gamma$ be its circumcircle. Let $D$ be a point on $AB$ such that $CD$ is parallel to the line tangent to $\Gamma$ at $A$. Let $E$ be the intersection of $CD$ with $\Gamma$ distinct from $C$, and $F$ the intersection of $BC$ with the circumcircle of $\bigtriangleup ADC$ distinct from $C$. Finally, let $G$ be the intersection of the line $AB$ and the internal bisector of $\angle DCF$. Show that $E,\ G,\ F$ and $C$ lie on the same circle.
Let $a$ be number of $n$ digits ($ n > 1$). A number $b$ of $2n$ digits is obtained by writing two copies of $a$ one after the other. If $\frac{b}{a^2}$ is an integer $k$, find the possible values values of $k$.
The first row of a table of size $ 2005\times5$ is filled with 1,2,3,4,5 so that every two neighbouring cells contain distinct numbers. Prove that it is possible to fill four other rows with 1,2,3,4,5 so that any neighbouring cells in them will contain distinct numbers as well as any cells of the same column will contain pairwise distinct numbers.
In acute triangle $ABC$, point $H$ is the intersection point of heights $CE$ on side $AB$ and $BD$ on side $AC$. A circle with diameter $DE$ intersects $AB$ and $AC$ at $F$ and $G$ respectively. $FG$ and $AH$ intersect at $K$. If $BC=25,BD=20, BE=7$, find the length of $AK$.
Let $ p$ be an odd prime number and $ a_1,a_2,...,a_p$ and $ b_1,b_2,...,b_p$ two arbitrary permutations of the numbers $ 1,2,...,p$ . Show that the least positive residues modulo $ p$ of the numbers $ a_1b_1, a_2b_2,...,a_pb_p$ never form a permutation of the numbers $ 1,2,...,p$.
Suppose $n$ is a natural number. In how many ways can we place numbers $1,2,....,n$ around a circle such that each number is a divisor of the sum of it's two adjacent numbers?
Let $ABC$ be a triangle in which $AB=AC$. Suppose the orthocentre of the triangle lies on the incircle. Find the ratio $\frac{AB}{BC}$.
Write $S_n$ for the set $\{1, 2,..., n\}$. Determine all positive integers $n$ for which there exist functions $f : S_n \to S_n$ and $g : S_n \to S_n$ such that for every $x$ exactly one of the equalities $f(g(x)) = x$ and $g(f(x)) = x$ holds.
Ava and Tiffany participate in a knockout tournament consisting of a total of $32$ players. In each of $5$ rounds, the remaining players are paired uniformly at random. In each pair, both players are equally likely to win, and the loser is knocked out of the tournament. The probability that Ava and Tiffany play each other during the tournament is $\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Compute $100a + b.$
You and your friend play a game on a $ 7 \times 7$ grid of buckets. Your friend chooses $5$ “lucky” buckets by marking an “$X$” on the bottom that you cannot see. However, he tells you that they either form a vertical, or horizontal line of length $5$. To clarify, he will select either of the following sets of buckets:
either $\{(a, b),(a, b + 1),(a, b + 2),(a, b + 3),(a, b + 4)\}$,
or $\{(b, a),(b + 1, a),(b + 2, a),(b + 3, a),(b + 4, a)\}$,
with $1\le a \le 7$, and $1 \le b \le 3$. Your friend lets you pick up at most $n$ buckets, and you win if one of the buckets you picked was a “lucky” bucket. What is the minimum possible value of $n$ such that, if you pick your buckets optimally, you can guarantee that at least one is “lucky”?
Find the product of all values of $d$ such that $x^{3} +2x^{2} +3x +4 = 0$ and $x^{2} +dx +3 = 0$ have a common root.
Consider a rectangle $R$ partitioned into $2016$ smaller rectangles such that the sides of each smaller rectangle is parallel to one of the sides of the original rectangle. Call the corners of each rectangle a vertex. For any segment joining two vertices, call it basic if no other vertex lie on it. (The segments must be part of the partitioning.) Find the maximum/minimum possible number of basic segments over all possible partitions of $R$.
Show that the smallest number of colors that is needed for coloring numbers $1, 2,..., 2013$ so that for every two
number $a, b$ which is the same color, $ab$ is not a multiple of $2014$, is $3$ colors.
There are $14$ boys in a class. Each boy is asked how many other boys in the class have his first name, and how many have his last name. It turns out that each number from $0$ to $6$ occurs among the answers.
Prove that there are two boys in the class with the same first name and the same last name.
Let $\{x\}$ denote the fractional part of $x$, which means the unique real $0\leq\{x\}<1$ such that $x-\{x\}$ is an integer. Let $f_{a,b}(x)=\{x+a\}+2\{x+b\}$ and let its range be $[m_{a,b},M_{a,b})$. Find the minimum value of $M_{a,b}$ as $a$ and $b$ range along all real numbers.
Alice and Bob take turns alternatively on a $2020\times2020$ board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees?
[i]Proposed by Seyed Reza Hosseini[/i]
How many cubics in the form $x^3 -ax^2 + (a+d)x -(a+2d)$ for integers $a,d$ have roots that are all non-negative integers?
Find all real solutions to the equation
$$(x+1)^{2001}+(x+1)^{2000}(x-2)+(x+1)^{1999}(x-2)^2+...+(x+1)^2(x-2)^{1999}+(x+1)^{2000}(x-2)+(x+1)^{2001}=0$$
Solve in none-negative integers ${x^3} + 7{x^2} + 35x + 27 = {y^3}$.
Let $a$, $b$, $c$ be positive real numbers for which \[
\frac{5}{a} = b+c, \quad
\frac{10}{b} = c+a, \quad \text{and} \quad
\frac{13}{c} = a+b. \] If $a+b+c = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
[i]Proposed by Evan Chen[/i]
Which one statisfies $n^{29} \equiv 7 \pmod {65}$?
$ \textbf{(A)}\ 37 \qquad \textbf{(B)}\ 39 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 46 \qquad \textbf{(E)}\ 55$
$[x,y]-[x,z]=y-z$ and $x \neq y \neq z \neq x$
Prove, that $x|y,x|z$
Let $p{}$ be a fixed prime number. Juku and Miku play the following game. One of the players chooses a natural number $a$ such that $a>1$ and $a$ is not divisible by $p{}$, his opponent chooses any natural number $n{}$ such that $n>1$. Miku wins if the natural number written as $n{}$ "$1$"s in the positional numeral system with base $a$ is divisible by $p{}$, otherwise Juku wins. Which player has a winning strategy if:
(a) Juku chooses the number $a$, tells it to Miku and then Miku chooses the number $n{}$;
(b) Juku chooses the number $n{}$, tells it to Miku and then Miku chooses the number $a$?
Let $C: y=\ln x$. For each positive integer $n$, denote by $A_n$ the area of the part enclosed by the line passing through two points $(n,\ \ln n),\ (n+1,\ \ln (n+1))$ and denote by $B_n$ that of the part enclosed by the tangent line at the point $(n,\ \ln n)$, $C$ and the line $x=n+1$. Let $g(x)=\ln (x+1)-\ln x$.
(1) Express $A_n,\ B_n$ in terms of $n,\ g(n)$ respectively.
(2) Find $\lim_{n\to\infty} n\{1-ng(n)\}$.