Found problems: 30
2013 Federal Competition For Advanced Students, Part 1, 2
Solve the following system of equations in rational numbers:
\[ (x^2+1)^3=y+1,\\ (y^2+1)^3=z+1,\\ (z^2+1)^3=x+1.\]
2013 USA Team Selection Test, 3
In a table with $n$ rows and $2n$ columns where $n$ is a fixed positive integer, we write either zero or one into each cell so that each row has $n$ zeros and $n$ ones. For $1 \le k \le n$ and $1 \le i \le n$, we define $a_{k,i}$ so that the $i^{\text{th}}$ zero in the $k^{\text{th}}$ row is the $a_{k,i}^{\text{th}}$ column. Let $\mathcal F$ be the set of such tables with $a_{1,i} \ge a_{2,i} \ge \dots \ge a_{n,i}$ for every $i$ with $1 \le i \le n$. We associate another $n \times 2n$ table $f(C)$ from $C \in \mathcal F$ as follows: for the $k^{\text{th}}$ row of $f(C)$, we write $n$ ones in the columns $a_{n,k}-k+1, a_{n-1,k}-k+2, \dots, a_{1,k}-k+n$ (and we write zeros in the other cells in the row).
(a) Show that $f(C) \in \mathcal F$.
(b) Show that $f(f(f(f(f(f(C)))))) = C$ for any $C \in \mathcal F$.
2013 MTRP Senior, 2
There are 1000 doors $D_1, D_2, . . . , D_{1000}$ and 1000 persons $P_1, P_2, . . . , P_{1000}$.
Initially all the doors were closed. Person $P_1$ goes and opens all the doors.
Then person $P_2$ closes door $D_2, D_4, . . . , D_{1000}$ and leaves the odd numbered doors open. Next $P_3$ changes the state of every third door, that
is, $D_3, D_6, . . . , D_{999}$ . (For instance, $P_3$ closes the open door $D_3$ and opens
the closed door D6, and so on). Similarly, $P_m$ changes the state of the
the doors $D_m, D_{2m}, D_{3m}, . . . , D_{nm}, . . .$ while leaving the other doors untouched. Finally, $P_{1000}$ opens $D_{1000}$ if it was closed or closes it if it were
open. At the end, how many doors will remain open?
2013 Stanford Mathematics Tournament, 1
How many positive three-digit integers $\underline a\, \underline b\,\underline c$ can represent a valid date in $2013$, where either $a$ corresponds to a month and $\underline b\,\underline c$ corresponds to the day in that month, or $\underline a\, \underline b$ corresponds to a month and $c$ corresponds to the day? For example, 202 is a valid representation for February 2nd, and 121 could represent either January 21st or December 1st.
2013 ELMO Problems, 1
Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$?
[i]Proposed by Ray Li[/i]