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Difference between revisions of "User:Fura3334"

(Created page with "[hide="the 6 problems for sus mock aime"] [b]Problem 1[/b] Kube the robot completes a task repeatedly, each time taking <math>t</math> minutes. One day, Furaken asks Kube to c...")
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Revision as of 00:39, 17 February 2024

[hide="the 6 problems for sus mock aime"] [b]Problem 1[/b] 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 $t$. Answer: [hide]i'm not putting the answer here, send your answer in this thread then i'll see whether yours is same as mine[/hide]

[b]Problem 2[/b] 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 $\lfloor x+y+z \rfloor$. Answer: [hide]i'm not putting the answer here, send your answer in this thread then i'll see whether yours is same as mine[/hide]

[b]Problem 3[/b] Let $p$ be an odd prime such that $2^{p-7}\equiv3 \pmod{p}$. Find $p$. Answer: [hide]i'm not putting the answer here, send your answer in this thread then i'll see whether yours is same as mine[/hide]

[b]Problem 4[/b] [img]https://latex.artofproblemsolving.com/d/4/4/d44b620826b6207a9018efd4b6db89b39b2b9b3c.png[/img] 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 $p+q$. Answer: [hide]i'm not putting the answer here, send your answer in this thread then i'll see whether yours is same as mine[/hide]

[b]Problem 5[/b] 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 $d$. Answer: [hide]i'm not putting the answer here, send your answer in this thread then i'll see whether yours is same as mine[/hide]

[b]Problem 6[/b] [img]https://latex.artofproblemsolving.com/4/6/1/461d152569bf18e1c406d5b3c3a3afa4201b4245.png[/img] 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. Answer: [hide]i'm not putting the answer here, send your answer in this thread then i'll see whether yours is same as mine[/hide][/hide]

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