# Difference between revisions of "G285 2021 MC10A"

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==Problem 6== | ==Problem 6== | ||

− | + | If <cmath>k=wv_1+xv_2</cmath><cmath>l=yv_1^2+zv_2^2</cmath><cmath>y=2w , z=2x</cmath> Find <math>v_2</math> in terms of <math>y,z,k,l</math> | |

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[[G285 MC10A Problems/Problem 6|Solution]] | [[G285 MC10A Problems/Problem 6|Solution]] | ||

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==Problem 11== | ==Problem 11== | ||

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[[G285 MC10A Problems/Problem 11|Solution]] | [[G285 MC10A Problems/Problem 11|Solution]] | ||

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+ | ==Problem 12== | ||

+ | Let <math>\phi(n)</math> denote the number of integers less than <math>n</math> such that each is relatively prime to <math>n</math>. Find the number of 3-digit positive integers <math>abc</math> such that <math>\phi(a)+\phi(b) \equiv 1 \pmod{3}</math> and <math>\phi(b)+\phi(c) \equiv 0 \pmod{4}</math> | ||

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+ | [[G285 MC10A Problems/Problem 12|Solution]] | ||

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+ | ==Problem 13== | ||

+ | Let a recursive sequence <math>a_1=1</math> and <math>a_2=13</math> be defined as: <cmath>a_n = \frac{2a_{n-1}+3a_{n-2}}{6n}</cmath> for <math>n \ge 2</math>. Let <math>Q(x)</math> be a monic polynomial with real roots <math>\{r_1,r_2,r_3,r_4 \}</math>. If each root is the reciprocal of the <math>4</math> smallest <math>a_n</math> such that <math>Q(a_n)>0</math>, find the reciprocal of the smallest possible value of <math>\left \lceil Q(-1) \right \rceil</math> | ||

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+ | [[G285 MC10A Problems/Problem 13|Solution]] | ||

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+ | ==Problem 14== | ||

+ | Let an ellipsoid centered at the origin have radii <math>\{x,y,z \} = \{3,18,24 \}</math>. If a cross-section of the figure is taken at an angle of <math>30^o</math> to the horizontal base that lies along the <math>x</math> and <math>y</math> axes, find the area of the cross-section. | ||

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+ | [[G285 MC10A Problems/Problem 14|Solution]] | ||

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+ | ==Problem 15== | ||

+ | Find <cmath>\sum_{j=1}^{50} s^3 \sum_{h=3}^{10} {4h+5}</cmath> | ||

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+ | <math>\textbf{(A)}\ 323400\qquad\textbf{(B)}\ 336600\qquad\textbf{(C)}\ 673200\qquad\textbf{(D)}\ 646800\qquad\textbf{(E)}\ 2124150</math> | ||

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+ | [[G285 MC10A Problems/Problem 15|Solution]] | ||

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+ | ==Problem 23== | ||

+ | Let regular hexagon <math>ABCDEF</math> of side length <math>4</math> be centered at <math>(-2,2- 2\sqrt{3})</math> on the Cartesian Plane, where points <math>P</math>,<math>X</math>, and <math>Y</math> lie in its interior. Let the ratio of the area of <math>\triangle AEP</math> to <math>\triangle BDP</math> is <math>-2+ \sqrt{3}:6- \sqrt{3}</math>, the ratio of the area of <math>AXB</math> to <math>EXD</math> is <math>7:1</math>, <math>PY \parallel AE</math> and <math>PX=PY</math>. Now, suppose <math>\triangle PXY</math> can be rotated about point <math>B</math> <math>m</math> degrees counterclockwise to form a new triangle <math>P'X'Y'</math>, such that if the coordinates of <math>P' = (x_1,y_1)</math>, <math>X' = (x_2,y_2)</math> and <math>Y' = (x_3,y_3)</math>, <cmath>\sum_{i=0}^{3} x_i = - \frac{3 \sqrt{147}}{5} \sum_{j=0}^{3} y_j</cmath><cmath>x_3-x_2=x_2-x_1= -\frac{\sqrt{3}}{2} y_3-y_2 = -\frac{\sqrt{3}}{2} y_1-y_2</cmath> If <math>m+x_1+x_2+x_3+y_1+y_2+y_3</math> can be represented as <math>d-\frac{n \sqrt{a}}{p}</math>, find <math>d+n+a+p</math> | ||

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+ | <math>\textbf{(A)}\ 64\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ 74\qquad\textbf{(D)}\ 81\qquad\textbf{(E)}\ 84</math> | ||

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+ | [[G285 MC10A Problems/Problem 23|Solution]] | ||

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+ | ==Problem 24== | ||

+ | Let <math>v_p (a)</math> denote the number of <math>p</math>'s in the prime factorization of <math>a</math>. If <math>k</math> and <math>n</math> are positive integers such that <math>\sqrt{n} < k < n</math>, find the largest sum <math>k+n</math> such that <cmath>\sum_{i=k}^{n} \binom{n-i}{i} v_p(i) > (1+k)^n</cmath> | ||

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+ | [[G285 MC10A Problems/Problem 24|Solution]] | ||

+ | |||

+ | ==Problem 25== |

## Latest revision as of 18:56, 8 June 2021

## Contents

## Problem 1

What is the smallest value of that minimizes ?

## Problem 2

Suppose the set denotes . Then, a subset of length is chosen. All even digits in the subset are then are put into group , and the odd digits are put in . Then, one number is selected at random from either or with equal chances. What is the probability that the number selected is a perfect square, given ?

## Problem 3

Let be a unit square. If points and are chosen on and respectively such that the area of . What is ?

## Problem 4

What is the smallest value of for which

## Problem 5

Let a recursive sequence be denoted by such that and . Suppose for . Let an infinite arithmetic sequence be such that . If is prime, for what value of will ?

## Problem 6

If Find in terms of

## Problem 7

A regular tetrahedron has length . Suppose on the center of each surface, a hemisphere of diameter is constructed such that the hemisphere falls inside the volume of the figure. If the ratio between the radius of the largest sphere that can be inscribed inside the old tetrahedron and new tetrahedron , where is square free, and . Find .

## Problem 8

If can be expressed as , where is square free and , find if and .

## Problem 9

If a real number is , . If a real number is , . If a number is neither or , it will be . What is the probability that randomly selected numbers from the interval are , , and , in any given order?

## Problem 10

Suppose the area of is equal to the sum of its side lengths. Let point be on the circumcircle of such that is a diameter. If is the center of the circumcircle, and is the center of the incircle of , and , find .

## Problem 11

## Problem 12

Let denote the number of integers less than such that each is relatively prime to . Find the number of 3-digit positive integers such that and

## Problem 13

Let a recursive sequence and be defined as: for . Let be a monic polynomial with real roots . If each root is the reciprocal of the smallest such that , find the reciprocal of the smallest possible value of

## Problem 14

Let an ellipsoid centered at the origin have radii . If a cross-section of the figure is taken at an angle of to the horizontal base that lies along the and axes, find the area of the cross-section.

## Problem 15

Find

## Problem 23

Let regular hexagon of side length be centered at on the Cartesian Plane, where points ,, and lie in its interior. Let the ratio of the area of to is , the ratio of the area of to is , and . Now, suppose can be rotated about point degrees counterclockwise to form a new triangle , such that if the coordinates of , and , If can be represented as , find

## Problem 24

Let denote the number of 's in the prime factorization of . If and are positive integers such that , find the largest sum such that