Search found 5 matches
- Thu Dec 03, 2020 9:24 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2020 P5
- Replies: 0
- Views: 6699
APMO 2020 P5
Let $n \geq 3$ be a fixed integer. The number $1$ is written $n$ times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers $a$ and $b$, replacing them with the numbers $1$ and $a+b$, then adding one stone to the first b...
- Thu Dec 03, 2020 9:23 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2020 P4
- Replies: 0
- Views: 6635
APMO 2020 P4
Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property: For any infinite sequence $a_1$, $a_2$, $\cdots$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i < j$ and an inte...
- Thu Dec 03, 2020 9:21 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2020 P3
- Replies: 0
- Views: 6524
APMO 2020 P3
Determine all positive integers $k$ for which there exist a positive integer $m$ and a set $S$ of positive integers such that any integer $n > m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways.
- Thu Dec 03, 2020 9:21 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2020 P2
- Replies: 0
- Views: 6632
APMO 2020 P2
Show that $r = 2$ is the largest real number $r$ which satisfies the following condition: If a sequence $a_1$, $a_2$, $\ldots$ of positive integers fulfills the inequalities \[a_n \leq a_{n+2} \leq\sqrt{a_n^2+ra_{n+1}}\]for every positive integer $n$, then there exists a positive integer $M$ such th...
- Thu Dec 03, 2020 9:19 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2020 P1
- Replies: 1
- Views: 5334
APMO 2020 P1
Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let $D$ be a point on the side $BC$. The tangent to $\Gamma$ at $A$ intersects the parallel line to $BA$ through $D$ at point $E$. The segment $CE$ intersects $\Gamma$ again at $F$. Suppose $B$, $D$, $F$, $E$ are concyclic. Prove that $AC$, $BF$, ...