Difference between revisions of "User:Fura3334"

Revision as of 00:46, 17 February 2024

IF YOU'RE AN ADMIN, PLS DONT DELETE THIS PAGE, IM WORKING ON SUS MOCK AIME (well, if i haven't edited this page for 2 weeks, you can delete it)

Problem 1

Kube the robot completes a task repeatedly, each time taking $t$ minutes. One day, Furaken asks Kube to complete $n$ identical tasks in $20$ hours. If Kube works slower and spends $t+6$ minutes on each task, it will finish $n$ tasks in exactly $20$ hours. If Kube works faster and spends $t-6$ minutes on each task, it can finish $n+1$ tasks in $20$ hours with $12$ minutes to spare. Find $t$ .

Problem 2

Let $x$ , $y$ , $z$ be positive real numbers such that

$xz = 1000$
$z = 100\sqrt{xy}$
$10^{\lg x\lg y + 2\lg y\lg z + 3\lg z\lg x} = 2 \cdot 5^{12}$

Find $\lfloor x+y+z \rfloor$ .

Problem 3

Let $p$ be an odd prime such that $2^{p-7}\equiv3 \pmod{p}$ . Find $p$ .

Problem 4

For triangle $ABC$ , let $M$ be the midpoint of $AC$ . Extend $BM$ to $D$ such that $MD=2.8$ . Let $E$ be the point on $MB$ such that $ME=0.8$ , and let $F$ be the point on $MC$ such that $MF=1$ . Line $EF$ intersects line $CD$ at $P$ such that $\tfrac{DP}{PC}=\tfrac23$ . Given that $EF$ is parallel to $BC$ , the maximum possible area of triangle $ABC$ can be written as $\tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .

Problem 5

Let $1 + \sqrt3 + \sqrt{13}$ be a root of the polynomial $x^4 + ax^3 + bx^2 + cx + d$ where $a$ , $b$ , $c$ , $d$ are integers. Find $d$ .

Problem 6

File:Susmockaimep6diagram

Fly has a large number of red, yellow, green and blue pearls. Fly is making a necklace consisting of $8$ pearls as shown in the diagram. One slot already has a red pearl, and another slot has a green pearl. Find the number of ways to fill the $6$ remaining slots such that any two pearls that are connected directly have different colors.

@@ Line 23: / Line 23: @@
 ==Problem 6==
-[https://latex.artofproblemsolving.com/4/6/1/461d152569bf18e1c406d5b3c3a3afa4201b4245.png diagram]
+[[File:susmockaimep6diagram|center]]
 Fly has a large number of red, yellow, green and blue pearls. Fly is making a necklace consisting of <math>8</math> pearls as shown in the diagram. One slot already has a red pearl, and another slot has a green pearl. Find the number of ways to fill the <math>6</math> remaining slots such that any two pearls that are connected directly have different colors.

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