AMC 10/12 Math Contest Preparation

In the diagram to the right, both beams are balanced.

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

Note that there are 5 options (not 4).

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

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

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

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

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

The diagram shows how a mobile will be balanced when left to hang.

What are the relative weights of these shapes?

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.

The polynomial \( x^{9999} + x^{8888} + x^{7777} + \cdots + x^{1111} + 1 \) 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

\[cx^2+bx+a = 0?\]

The numbers of red circles in each group in the above diagram form an arithmetic progression. What is the common difference?

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 \(k_2?\)

In their native language, the Bakairi of South America have an additive system for representing numbers:

1 = tokale
2 = azage
3 = azage tokale
4 = azage azage
5 = azage azage tokale
6 = azage azage azage

What number corresponds to "azage azage azage azage tokale"?

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

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

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

Find the last digit when

\[\color{blue}1^{\color{blue}1} + \color{green}2^{\color{green}2} + \color{purple}3^{ \color{purple}3} + \cdots + \color{red}9^{ \color{red}9} + \color{violet}{10}^{\color{violet}{10}} \]

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 \[ \Large 2014^{2014^{2014}}. \]

\(ABC\) is an isosceles right triangle with \(AB = AC = 6 \).

Given that \(SDPF\) is a square, find the area of the square.

\(O\) is the center of the circle to the right.

Find, in degrees, \(\angle QPR + \angle ORQ.\)

In the diagram to the right, 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?

Suppose that we have two complex numbers \(z_1 = 3 - 2i\) and \(z_2 = 4 - 3i\).

In what quadrant is this complex number \(z_1z_2\) located?

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

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

\[ \huge \frac{\hspace{2mm} \frac{\hspace{2mm} \frac{\hspace{2mm} \frac{\hspace{2mm} \frac{\sin x}{\cos x}\hspace{2mm} }{\tan x}\hspace{2mm} }{\cot x}\hspace{2mm} }{\sec x}\hspace{2mm} }{\csc x}\]

Find the value of the above 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?

As a tourist in NY, I want to go from the Grand Central Station \((42^\text{nd}\) street and \(4^\text{th}\) avenue\()\) to Times Square \((47^\text{th}\) street and \(7^\text{th}\) avenue\().\)

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 all sides of a regular tetrahedron (a polyhedron with 4 equilateral triangles faces).

How many ways are there to do so if you only paint each face one of your three colors? (Rotations of the tetrahedron are considered equivalent).

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

What is the last digit of the number above?

When I put two marbles in this bag, I flipped a coin twice to determine their colors. For each flip,

• 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, another with $2, and the other with $4. 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 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 has not yet been mentioned)?

Two fair dice are rolled, and it is revealed that (at least) one of the numbers rolled was a 4. What is the probability that the other number rolled was a 6?

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