Found problems: 85335
1953 AMC 12/AHSME, 38
If $ f(a)\equal{}a\minus{}2$ and $ F(a,b)\equal{}b^2\plus{}a$, then $ F(3,f(4))$ is:
$ \textbf{(A)}\ a^2\minus{}4a\plus{}7 \qquad\textbf{(B)}\ 28 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 11$
2018 China Team Selection Test, 2
A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.
2024 BAMO, C/1
Sugar Station sells $44$ different kinds of candies, packaged one to a box. Each box is priced at a positive integer number of cents, and it costs $\$1.51$ to buy one of every kind. (There is no discount based on the number of candies in a purchase.) Unfortunately, Anna only has $\$0.75$.
[list=a]
[*] Show that Anna can buy at least $22$ boxes, each containing a different candy.
[*] Show that Anna can do even better, buying at least $25$ boxes, each containing a different candy.
[/list]
2016 Online Math Open Problems, 15
Two bored millionaires, Bilion and Trilion, decide to play a game. They each have a sufficient supply of $\$ 1, \$ 2,\$ 5$, and $\$ 10$ bills. Starting with Bilion, they take turns putting one of the bills they have into a pile. The game ends when the bills in the pile total exactly $\$1{,}000{,}000$, and whoever makes the last move wins the $\$1{,}000{,}000$ in the pile (if the pile is worth more than $\$1{,}000{,}000$ after a move, then the person who made the last move loses instead, and the other person wins the amount of cash in the pile). Assuming optimal play, how many dollars will the winning player gain?
[i]Proposed by Yannick Yao[/i]
2006 Estonia Team Selection Test, 3
A grid measuring $10 \times 11$ is given. How many "crosses" covering five unit squares can be placed on the grid?
(pictured right) so that no two of them cover the same square?
[img]https://cdn.artofproblemsolving.com/attachments/a/7/8a5944233785d960f6670e34ca7c90080f0bd6.png[/img]
2016 Israel Team Selection Test, 1
Let $a,b,c$ be positive numbers satisfying $ab+bc+ca+2abc=1$. Prove that $4a+b+c \geq 2$.
2013 CentroAmerican, 3
Let $ABCD$ be a convex quadrilateral and let $M$ be the midpoint of side $AB$. The circle passing through $D$ and tangent to $AB$ at $A$ intersects the segment $DM$ at $E$. The circle passing through $C$ and tangent to $AB$ at $B$ intersects the segment $CM$ at $F$. Suppose that the lines $AF$ and $BE$ intersect at a point which belongs to the perpendicular bisector of side $AB$. Prove that $A$, $E$, and $C$ are collinear if and only if $B$, $F$, and $D$ are collinear.
2021 LMT Spring, A 24
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Using the four words
“Hi”, “hey”, “hello”, and “haiku”,
How many haikus
Can somebody make?
(Repetition is allowed,
Order does matter.)
[i]Proposed by Jeff Lin[/i]
2007 Hanoi Open Mathematics Competitions, 7
Find all sequences of integer $x_1,x_2,..,x_n,...$ such that $ij$ divides $x_i+x_j$ for any distinct positive integer $i$, $j$.
1999 Israel Grosman Mathematical Olympiad, 1
For any $16$ positive integers $n,a_1,a_2,...,a_{15}$ we define $T(n,a_1,a_2,...,a_{15}) = (a_1^n+a_2^n+ ...+a_{15}^n)a_1a_2...a_{15}$.
Find the smallest $n$ such that $T(n,a_1,a_2,...,a_{15})$ is divisible by $15$ for any choice of $a_1,a_2,...,a_{15}$.