Difference between revisions of "2021 April MIMC 10"

Revision as of 22:00, 20 April 2021

Problem 1

What is the sum of $2^{3}-(-3^{4})-3^{4}+1$ ?

$\textbf{(A)} ~-155 \qquad\textbf{(B)} ~-153 \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~9 \qquad\textbf{(E)} ~171$

Problem 2

Okestima is reading a $150$ page book. He reads a page every $\frac{2}{3}$ minutes, and he pauses $3$ minutes when he reaches the end of page 90 to take a break. He does not read at all during the break. After, he comes back with food and this slows down his reading speed. He reads one page in $2$ minutes. If he starts to read at $2:30$ , when does he finish the book?

$\textbf{(A)} ~4:33 \qquad\textbf{(B)} ~5:30 \qquad\textbf{(C)} ~5:33 \qquad\textbf{(D)} ~6:30 \qquad\textbf{(E)} ~7:33$

Solution

Problem 3

Find the number of real solutions that satisfy the equation $(x^2+2x+2)^{3x+2}=1$ .

$\textbf{(A)} ~0 \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~2 \qquad\textbf{(D)} ~3 \qquad\textbf{(E)} ~4$

Solution

Problem 4

Stiskwey wrote all the possible permutations of the letters $AABBCCCD$ ( $AABBCCCD$ is different from $AABBCCDC$ ). How many such permutations are there?

$\textbf{(A)} ~420 \qquad\textbf{(B)} ~630 \qquad\textbf{(C)} ~840 \qquad\textbf{(D)} ~1680 \qquad\textbf{(E)} ~5040$

Solution

Problem 5

5. Given $x:y=5:3, z:w=3:2, y:z=2:1$ , Find $x:w$ .

$\textbf{(A)} ~3:1 \qquad\textbf{(B)} ~10:3 \qquad\textbf{(C)} ~5:1 \qquad\textbf{(D)} ~20:3 \qquad\textbf{(E)} ~10:1$

Solution

Problem 6

A worker cuts a piece of wire into two pieces. The two pieces, $A$ and $B$ , enclose an equilateral triangle and a square with equal area, respectively. The ratio of the length of $B$ to the length of $A$ can be expressed as $a\sqrt[b]{c}:d$ in the simplest form. Find $a+b+c+d$ .

$\textbf{(A)} ~9 \qquad\textbf{(B)} ~10 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~14 \qquad\textbf{(E)} ~15$

Solution

Problem 7

Find the least integer $k$ such that $838_k=238_k+1536$ where $a_k$ denotes $a$ in base- $k$ .

$\textbf{(A)} ~12 \qquad\textbf{(B)} ~13 \qquad\textbf{(C)} ~14 \qquad\textbf{(D)} ~15 \qquad\textbf{(E)} ~16$

Solution

@@ Line 36: / Line 36: @@
 ==Problem 6==
-. A worker cuts a piece of wire into two pieces. The two pieces, <math>A</math> and <math>B</math>, enclose an equilateral triangle and a square with equal area, respectively. The ratio of the length of <math>B</math> to the length of <math>A</math> can be expressed as <math>a\sqrt[b]{c}:d</math> in the simplest form. Find <math>a+b+c+d</math>.
+A worker cuts a piece of wire into two pieces. The two pieces, <math>A</math> and <math>B</math>, enclose an equilateral triangle and a square with equal area, respectively. The ratio of the length of <math>B</math> to the length of <math>A</math> can be expressed as <math>a\sqrt[b]{c}:d</math> in the simplest form. Find <math>a+b+c+d</math>.
 <math>\textbf{(A)} ~9 \qquad\textbf{(B)} ~10 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~14 \qquad\textbf{(E)} ~15</math>
-==
+[[2021 April MIMC 10 Problems/Problem 6|Solution]]
+==Problem 7==
+Find the least integer <math>k</math> such that <math>838_k=238_k+1536</math> where <math>a_k</math> denotes <math>a</math> in base-<math>k</math>.
+<math>\textbf{(A)} ~12 \qquad\textbf{(B)} ~13 \qquad\textbf{(C)} ~14 \qquad\textbf{(D)} ~15 \qquad\textbf{(E)} ~16</math>
+[[2021 April MIMC 10 Problems/Problem 7|Solution]]
+==Problem 8==

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Difference between revisions of "2021 April MIMC 10"

Revision as of 22:00, 20 April 2021

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8