Difference between revisions of "User:Fura3334"
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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) | 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 <math>t</math> minutes. One day, Furaken asks Kube to complete <math>n</math> identical tasks in <math>20</math> hours. If Kube works slower and spends <math>t+6</math> minutes on each task, it will finish <math>n</math> tasks in exactly <math>20</math> hours. If Kube works faster and spends <math>t-6</math> minutes on each task, it can finish <math>n+1</math> tasks in <math>20</math> hours with <math>12</math> minutes to spare. Find <math>t</math>. | Kube the robot completes a task repeatedly, each time taking <math>t</math> minutes. One day, Furaken asks Kube to complete <math>n</math> identical tasks in <math>20</math> hours. If Kube works slower and spends <math>t+6</math> minutes on each task, it will finish <math>n</math> tasks in exactly <math>20</math> hours. If Kube works faster and spends <math>t-6</math> minutes on each task, it can finish <math>n+1</math> tasks in <math>20</math> hours with <math>12</math> minutes to spare. Find <math>t</math>. | ||
− | |||
− | + | ==Problem 2== | |
Let <math>x</math>, <math>y</math>, <math>z</math> be positive real numbers such that | Let <math>x</math>, <math>y</math>, <math>z</math> be positive real numbers such that | ||
− | <math>xz = 1000</math> | + | * <math>xz = 1000</math> |
− | <math>z = 100\sqrt{xy}</math> | + | * <math>z = 100\sqrt{xy}</math> |
− | <math>10^{\lg x\lg y + 2\lg y\lg z + 3\lg z\lg x} = 2 \cdot 5^{12}</math> | + | * <math>10^{\lg x\lg y + 2\lg y\lg z + 3\lg z\lg x} = 2 \cdot 5^{12}</math> |
Find <math>\lfloor x+y+z \rfloor</math>. | Find <math>\lfloor x+y+z \rfloor</math>. | ||
− | |||
− | + | ==Problem 3== | |
− | + | ||
− | + | ||
+ | ==Problem 4== | ||
+ | [[File:susmockaimep4diagram.png|252px|center]] | ||
− | |||
− | |||
For triangle <math>ABC</math>, let <math>M</math> be the midpoint of <math>AC</math>. Extend <math>BM</math> to <math>D</math> such that <math>MD=2.8</math>. Let <math>E</math> be the point on <math>MB</math> such that <math>ME=0.8</math>, and let <math>F</math> be the point on <math>MC</math> such that <math>MF=1</math>. Line <math>EF</math> intersects line <math>CD</math> at <math>P</math> such that <math>\tfrac{DP}{PC}=\tfrac23</math>. Given that <math>EF</math> is parallel to <math>BC</math>, the maximum possible area of triangle <math>ABC</math> can be written as <math>\tfrac{p}{q}</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | For triangle <math>ABC</math>, let <math>M</math> be the midpoint of <math>AC</math>. Extend <math>BM</math> to <math>D</math> such that <math>MD=2.8</math>. Let <math>E</math> be the point on <math>MB</math> such that <math>ME=0.8</math>, and let <math>F</math> be the point on <math>MC</math> such that <math>MF=1</math>. Line <math>EF</math> intersects line <math>CD</math> at <math>P</math> such that <math>\tfrac{DP}{PC}=\tfrac23</math>. Given that <math>EF</math> is parallel to <math>BC</math>, the maximum possible area of triangle <math>ABC</math> can be written as <math>\tfrac{p}{q}</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | ||
− | |||
− | + | ==Problem 5== | |
Let <math>1 + \sqrt3 + \sqrt{13}</math> be a root of the polynomial <math>x^4 + ax^3 + bx^2 + cx + d</math> where <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> are integers. Find <math>d</math>. | Let <math>1 + \sqrt3 + \sqrt{13}</math> be a root of the polynomial <math>x^4 + ax^3 + bx^2 + cx + d</math> where <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> are integers. Find <math>d</math>. | ||
− | |||
− | + | ==Problem 6== | |
− | [ | + | [[File:susmockaimep6diagram.png|252px|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. | 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 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. |
Latest revision as of 23:49, 14 March 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 minutes. One day, Furaken asks Kube to complete identical tasks in hours. If Kube works slower and spends minutes on each task, it will finish tasks in exactly hours. If Kube works faster and spends minutes on each task, it can finish tasks in hours with minutes to spare. Find .
Problem 2
Let , , be positive real numbers such that
Find .
Problem 3
Problem 4
For triangle , let be the midpoint of . Extend to such that . Let be the point on such that , and let be the point on such that . Line intersects line at such that . Given that is parallel to , the maximum possible area of triangle can be written as where and are relatively prime positive integers. Find .
Problem 5
Let be a root of the polynomial where , , , are integers. Find .
Problem 6
Fly has a large number of red, yellow, green and blue pearls. Fly is making a necklace consisting of 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 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 . Given that where , are distinct primes greater than , and that cannot be expressed as the sum of 2 perfect squares, find the remainder when is divided by 144.