Difference between revisions of "User:Fura3334"

Revision as of 05:02, 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

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.

Problem X

(I haven't decided the problem number yet) Let $N=109007732774081$ . Given that $N=pq$ where $p$ , $q$ are distinct primes greater than $1000$ , and that $N$ cannot be expressed as the sum of 2 perfect squares, find the remainder when $\frac{\varphi(N)^2}{2} - (\varphi(N)+1)^6$ is divided by 144.

@@ Line 26: / Line 26: @@
 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.
-==Problem ?==
+==Problem X==
+(I haven't decided the problem number yet)
 Let <math>N=109007732774081</math>. Given that <math>N=pq</math> where <math>p</math>, <math>q</math> are distinct primes greater than <math>1000</math>, and that <math>N</math> cannot be expressed as the sum of 2 perfect squares, find the remainder when
 <cmath>\frac{\varphi(N)^2}{2} - (\varphi(N)+1)^6</cmath>
 is divided by 144.

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