AMC 10/12 Math Contest Preparation

Which of the following options will not balance any of the other options?

Note that there are 5 options (not 4).

\(\large \log_{10}{3}\) lies between

\[ \large \frac {315 - x} {101} + \frac {313 - x} {103} + \frac {311 - x} {105} + \frac {309 - x} {107} = -4 \]

Find the positive integer \(x\) that satisfies this equation.

\[\large\left\lfloor \frac { x }{ 1! } \right\rfloor +\left\lfloor \frac { x }{ 2! } \right\rfloor +\left\lfloor \frac { x }{ 3! } \right\rfloor =224\]

Find the integer value of \(x\) that satisfies the equation above.

Note: \(\lfloor x \rfloor\) means the greatest integer that is smaller or equal to \(x\).

How many ordered pairs of real numbers satisfy

\[ 36 ^ { x^2 + y} + 36 ^{ y^2 + x } = \frac{2} {\sqrt{6} } ? \]

If positive numbers \(x\), \(y\) and \(z\) satisfy \(x+y+z=75,\) what is the maximum value of \[\sqrt{x}+5\sqrt{y}+7\sqrt{z}?\]

\[ x(x+1)(x+2)(x+3)=120\]

Find the sum of all the real roots of the equation above.

If \(a\) and \(b\) are positive numbers and \[x^4-38x^2y^2+y^4=(x^2+axy-y^2)(x^2-bxy-y^2),\] what is the value of \(a+b\)?

The polynomial \( x^{9999} + x^{8888} + x^{7777} + \ldots + x^{1111} + 1 \) is divisible by which of the choices?

The non-zero roots of \[ax^2+bx+c=0\] are \(r\) and \(s.\) What are the roots of \[cx^2+bx+a = 0?\]

If the six numbers \[7, k_1, k_2, k_3, k_4, \frac{7}{32}\] form a geometric sequence in this order, what is \(k_2?\)

If the lengths of sides \(AB,\) \(BC\) and \(AC\) in the figure shown form a geometric progression in that order, what is the ratio between \(AC\) and \(AB\) to 3 decimal places?

Evaluate \[\frac{1}{\sqrt{4}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{10}}+\cdots+\frac{1}{\sqrt{397}+\sqrt{400}}.\]

What is the last digit of

\[ 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 ? \]

Find the smallest integer greater than 1 which

• is short of 5 upon division by 6,
• is short of 4 upon division by 5,
• is short of 3 upon division by 4,
• is short of 2 upon division by 3, amd
• is short of 1 upon division by 2.

\[\large 6128704063125000=2^3\times 3^5 \times 5^7 \times 7^9\]

For the number above, find the number of divisors which are perfect squares.

Find the last digit when \[\large \color{blue}1^{\color{blue}1} + \color{green}2^{\color{green}2} + \color{purple}3^{ \color{purple}3} + \ldots + \color{red}9^{ \color{red}9} + \color{violet}{10}^{\color{violet}{10}} \] is written out as an integer.

\[ \dfrac1{24} , \dfrac{2}{24}, \dfrac{3}{24}, \ldots , \dfrac{23}{24}, \dfrac{24}{24} \]

How many of the fractions above cannot be reduced?

Find the tens digit of \[ \Large 2014^{2014^{2014}}. \]

Figure above shows an isosceles right triangle \(ABC\) with \(AB = AC = 6 \). Given that \(SDPF\) is a square, find the area of the square \(SDPF\).

\(O\) is the center of the circle above. Find, in degrees, \(\angle QPR + \angle ORQ.\)

In the above diagram, quadrilateral \(ABCD\) is a parallelogram and \(M\) is the midpoint of \(\overline{DC}.\) If the area of \(\triangle BNC\) is \(42,\) what is the area of \(\triangle MNC?\)

\(ABCD\) is a square with \(AB=13\). Points \(E\) and \(F\) are exterior to \(ABCD\) such that \(BE=DF=5\) and \(AE=CF=12\).

If the length of \(EF\) can be represented as \(a\sqrt b \) where \(a\) and \(b\) are positive integers and \(b\) is not divisible by the square of any prime, then find \(ab\).

Three of the vertices of a square have coordinates of \( (9, 6), ( 12, 10)\), and \( (5, 9 ) \). What are the coordinates of the last vertex?

What is the area enclosed by \((y-x)^{2}=4\) and \((y+x)^{2}=4?\)

Given that

\[ \cos(A)-4\sin(A)=1,\]

what are the possible values of

\[ \sin(A) + 4\cos(A)? \]

\[ \large \frac{\frac{\frac{\frac{\frac{\sin x}{\cos x}}{\tan x}}{\cot x}}{\sec x}}{\csc x}\]

Find the value of this fraction tower if \(0^\circ < x < {90}^\circ\).

How many of the \(14^\text{th}\) roots of unity lie on the axes of the complex plane?

If I only walk West and North, how many ways are there for me to get there?

In a class of students, \(64\text{%}\) of the students passed the physics exam and \(68\text{%}\) of the students passed the math exam. What is the minimum percentage of students in the class that passed both exams?

You have three colors of paint, and you want to paint a regular tetrahedron (a polyhedra with 4 faces that are all equilateral triangles). How many ways are there to do so if you only paint each face one of your three colors and two colored tetrahedrons are considered equivalent if they can be rotated to look the same?

\[\large \left ( \sqrt {71} +1 \right )^{71} - \left ( \sqrt {71} -1 \right )^{71} \]

What is the last digit of the number above?

There are 4 dormitories at your school that you could be assigned to. The three older dormitories each house the same number of students. The newest dorm houses twice the number of students as one of the older dorms.

What is the probability that you will be assigned to the newest dormitory, given that dorm assignments are given at random until all spots are filled?

• If it was Heads \(\rightarrow\) I put in a red marble.
• If it was Tails \(\rightarrow\) I put in a blue marble.

You reach into my bag and randomly take out one of the two marbles. It is red. You put it back in. Then you reach into the bag again. What is the chance that this time, you pull out a blue marble?

There are 3 envelopes, one with $1, one with $2, and one with $4 dollars. You do not know which envelope has which amount of money.

After you pick one at random, you are told which envelope of the remaining 2 that has the lowest value, but not the actual value of that envelope. What is the expected value if you switch to the last envelope remaining (the one that was not yet mentioned)?

Note: You are not told which of the numbers rolled is a 4.