Question bank AMC10

The numbers $3,5,7,a,b$ have arithmetic mean $15$. What is the mean of $a,b$?

Two fair dice are relabeled. One has labels $2,2,4,4,6,6$ and the other has labels $1,1,3,3,5,5$. They are rolled independently. What is the probability that their sum is $7$?

Points $B$ and $C$ lie on $\overline{AD}.$ The length of $\overline{AB}$ is $4$ times the length of $\overline{BD},$ and the length of $\overline{AC}$ is $9$ times the length of $\overline{CD}.$ The length of $\overline{BC}$ is what fraction of the length of $\overline{AD}?$

The sum of two positive integers is five times the difference between the larger and the smaller. Find the ratio of the larger to the smaller.

Isaac writes one positive integer twice and another positive integer three times. The sum of the five numbers is $100$, and one of the two integers is $28$. What is the other integer?

Three distinct points are selected uniformly from the nine points of the square grid shown. What is the probability that the three points are collinear?

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The point $O$ is the center of the circle circumscribed about $\triangle ABC,$ with $\angle BOC = 120^\circ$ and $\angle AOB = 140^\circ,$ as shown. What is the degree measure of $\angle ABC?$

A real number $x$ satisfies $10^x\cdot100^{2x}=1000^5$. Find $x$.

Alice uploads a video to a website. Its score starts at $0$, rises by $1$ for each like vote, and falls by $1$ for each dislike vote. When its score is $90$, $65\%$ of all votes are likes. How many votes have been cast?

A radio quiz consists of three multiple-choice questions, each with three choices and exactly one correct answer. A contestant wins by answering at least two correctly. If each answer is chosen independently at random, what is the probability of winning?

What is the area of the shaded region of the given $8\times5$ rectangle?

The solution interval of $a\le2x+3\le b$ has length 10. Find $b-a$.

What is the value of
$$M=1+2+3-4+5+6+7-8+\cdots+197+198+199-200?$$

A box contains five candies numbered $1,2,3,4,5$. They are drawn randomly one at a time without replacement until the sum of the drawn values exceeds $4$. What is the probability that exactly three draws are required?

Triangles $ABC$ and $ADC$ are isosceles with $AB = BC$ and $AD = DC.$ Point $D$ is inside $\triangle ABC,$ $\angle ABC = 40^\circ,$ and $\angle ADC = 140^\circ.$ What is the degree measure of $\angle BAD?$

Nonzero real numbers $x,y$ satisfy $\frac{3x+y}{x-3y}=-2$. Evaluate $\frac{x+3y}{3x-y}$.

Mary chooses a positive two-digit integer. She multiplies it by $3$ and adds $11$, obtaining a two-digit result. Reversing the two digits of that result gives a number from $71$ to $75$, inclusive. What number did Mary choose?

Among the positive integers below $100$ whose every digit is prime, one is chosen uniformly at random. What is the probability that the selected number is prime?

The $8 \times 18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y?$

Real numbers $w,z$ make the following fractions defined and satisfy $\frac{1/w+1/z}{1/w-1/z}=2014$. Evaluate $\frac{w+z}{w-z}$.