Question: $$\left(x^{2} y-3 y^{2}+5 x y^{2}\right)-\left(-x^{2} y+3 x y^{2}-3 y^{2}\right)$$Which of the following is equivalent to the expression above?

Choices:
- $4 x^{2} y^{2}$
- $8 x y^{2}-6 y^{2}$
- $2 x^{2} y+2 x y^{2}$
- $2 x^{2} y+8 x y^{2}-6 y^{2}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: A software company is selling a new game in a standard edition and a collector's edition. The box for the standard edition has a volume of 20 cubic inches, and the box for the collector's edition has a volume of 30 cubic inches. The company receives an order for 75 copies of the game, and the total volume of the order to be shipped is 1,870 cubic inches. Which of the following systems of equations can be used to determine the number of standard edition games, $s$, and collector's edition games, $c$, that were ordered?

Choices:
- $75-s=c$ $20 s+30 c=1,870$
- $75-s=c$ $30 s+20 c=1,870$
- $\quad s-c=75$ $25(s+c)=1,870$
- $\quad s-c=75$ $30 s+20 c=1,870$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: If $(a x+2)(b x+7)=15 x^{2}+c x+14$ for all values of $x$, and $a+b=8$, what are the two possible values for $c$ ?

Choices:
- 3 and 5
- 6 and 35
- 10 and 21
- 31 and 41

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Alan drives an average of 100 miles each week. His car can travel an average of 25 miles per gallon of gasoline. Alan would like to reduce his weekly expenditure on gasoline by $\$ 5$. Assuming gasoline costs $\$ 4$ per gallon, which equation can Alan use to determine how many fewer average miles, $m$, he should drive each week?

Choices:
- $\frac{25}{4} m=95$
- $\frac{25}{4} m=5$
- $\frac{4}{25} m=95$
- $\frac{4}{25} m=5$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $$\begin{aligned}& y=x^{2}+3 x-7 \\& y-5 x+8=0\end{aligned}$$How many solutions are there to the system of equations above?

Choices:
- There are exactly 4 solutions.
- There are exactly 2 solutions.
- There is exactly 1 solution.
- There are no solutions.

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: If $\frac{2 n}{5}=10$, what is the value of $2 n-1 ?$

Choices:
- 24
- 49
- 50
- 99

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: The sum of three numbers is 855 . One of the numbers, $x$, is $50 \%$ more than the sum of the other two numbers. What is the value of $x$ ?

Choices:
- 570
- 513
- 214
- 155

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: The weights, in pounds, for 15 horses in a stable were reported, and the mean, median, range, and standard deviation for the data were found. The horse with the lowest reported weight was found to actually weigh 10 pounds less than its reported weight. What value remains unchanged if the four values are reported using the corrected weight?

Choices:
- Mean
- Median
- Range
- Standard deviation

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: If Mosteller's and Current's formulas give the same estimate for $A$, which of the following expressions is equivalent to $\sqrt{h w}$ ?

Choices:
- $\frac{4+w}{2}$
- $\frac{4+w}{1,800}$
- $2(4+w)$
- $\frac{(4+w)^{2}}{2}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $$\sqrt{k+2}-x=0$$In the equation above, $k$ is a constant. If $x=9$, what is the value of $k$ ?

Choices:
- 1
- 7
- 16
- 79

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: If $p$ percent of an adult's daily allowance of potassium is provided by $x$ servings of Crunchy Grain cereal per day, which of the following expresses $p$ in terms of $x$ ?

Choices:
- $p=0.5 x$
- $p=5 x$
- $p=(0.05)^{x}$
- $p=(1.05)^{x}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Alma bought a laptop computer at a store that gave a 20 percent discount off its original price. The total amount she paid to the cashier was $p$ dollars, including an 8 percent sales tax on the discounted price. Which of the following represents the original price of the computer in terms of $p$ ?

Choices:
- $0.88 p$
- $\frac{p}{0.88}$
- $(0.8)(1.08) p$
- $\frac{p}{(0.8)(1.08)}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: A customer paid $\$ 53.00$ for a jacket after a 6 percent sales tax was added. What was the price of the jacket before the sales tax was added?

Choices:
- $\$ 47.60$
- $\$ 50.00$
- $\$ 52.60$
- $\$ 52.84$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: In State X, Mr. Camp's eighth-grade class consisting of 26 students was surveyed and 34.6 percent of the students reported that they had at least two siblings. The average eighth-grade class size in the state is 26. If the students in Mr. Camp's class are representative of students in the state's eighth-grade classes and there are 1,800 eighth-grade classes in the state, which of the following best estimates the number of eighth-grade students in the state who have fewer than two siblings?

Choices:
- 16,200
- 23,400
- 30,600
- 46,800 Questions 23 and 24 refer to the following information.

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $$\begin{aligned}& 3 x^{2}-5 x+2 \\& 5 x^{2}-2 x-6\end{aligned}$$Which of the following is the sum of the two polynomials shown above?

Choices:
- $8 x^{2}-7 x-4$
- $8 x^{2}+7 x-4$
- $8 x^{4}-7 x^{2}-4$
- $8 x^{4}+7 x^{2}-4$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $$x+y=75$$The equation above relates the number of minutes, $x$, Maria spends running each day and the number of minutes, $y$, she spends biking each day. In the equation, what does the number 75 represent?

Choices:
- The number of minutes spent running each day
- The number of minutes spent biking each day
- The total number of minutes spent running and biking each day
- The number of minutes spent biking for each minute spent running

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: A company that makes wildlife videos purchases camera equipment for $\$ 32,400$. The equipment depreciates in value at a constant rate for 12 years, after which it is considered to have no monetary value. How much is the camera equipment worth 4 years after it is purchased?

Choices:
- $\$ 10,800$
- $\$ 16,200$
- $\$ 21,600$
- $\$ 29,700$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Lani spent $15 \%$ of her 8-hour workday in meetings. How many minutes of her workday did she spend in meetings?

Choices:
- 1.2
- 15
- 48
- 72

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Which of the following is equivalent to $9^{\frac{3}{4}}$ ?

Choices:
- $\sqrt[3]{9}$
- $\sqrt[4]{9}$
- $\sqrt{3}$
- $3 \sqrt{3}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Line $m$ in the $x y$-plane contains the points $(2,4)$ and $(0,1)$. Which of the following is an equation of line $m$ ?

Choices:
- $y=2 x+3$
- $y=2 x+4$
- $y=\frac{3}{2} x+3$
- $y=\frac{3}{2} x+1$

High--Intermediate Geometry--Equations of Lines and Circles


Question: $$\sqrt{4 x}=x-3$$What are all values of $x$ that satisfy the given equation?I. 1 II. 9

Choices:
- I only
- II only
- I and II
- Neither I nor II

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Washington High School randomly selected freshman, sophomore, junior, and senior students for a survey about potential changes to next year's schedule. Of students selected for the survey, $\frac{1}{4}$ were freshmen and $\frac{1}{3}$ were sophomores. Half of the remaining selected students were juniors. If336 students were selected for the survey, how many were seniors?

Choices:
- 240
- 140
- 120
- 70

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: In planning maintenance for a city's infrastructure, a civil engineer estimates that, starting from the present, the population of the city will decrease by 10 percent every 20 years. If the present population of the city is 50,000, which of the following expressions represents the engineer's estimate of the population of the city $t$ years from now?

Choices:
- $50,000(0.1)^{20 t}$
- $50,000(0.1)^{\frac{t}{20}}$
- $50,000(0.9)^{20 t}$
- $50,000(0.9)^{\frac{t}{20}}$

High--Intermediate Numbers and Algebra--Sequences


Question: If $f(x)=\frac{x^{2}-6 x+3}{x-1}$, what is $f(-1)$ ?

Choices:
- -5
- -2
- 2
- 5

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: The volume of a sphere is given by the formula $V=\frac{4}{3} \pi r^{3}$, where $r$ is the radius of the sphere. Which of the following gives the radius of the sphere in terms of the volume of the sphere?

Choices:
- $\frac{4 \pi}{3 V}$
- $\frac{3 V}{4 \pi}$
- $\sqrt[3]{\frac{4 \pi}{3 V}}$
- $\sqrt[3]{\frac{3 V}{4 \pi}}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: What is the set of all solutions to the equation $\sqrt{x+2}=-x ?$

Choices:
- $\{-1,2\}$
- $\{-1\}$
- $\{2\}$
- There are no solutions to the given equation.

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: If $x$ food calories is equivalent to $k$ kilojoules, of the following, which best represents the relationship between $x$ and $k$ ?

Choices:
- $k=0.24 x$
- $k=4.2 x$
- $x=4.2 k$
- $x k=4.2$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Which of the following complex numbers is equal to $(5+12 i)-\left(9 i^{2}-6 i\right)$, for $i=\sqrt{-1}$ ?

Choices:
- $-14-18 i$
- $-4-6 i$
- $4+6 i$
- $14+18 i$

High--Intermediate Numbers and Algebra--Complex Numbers


Question: In the $x y$-plane, the parabola with equation $y=(x-11)^{2}$ intersects the line with equation $y=25$ at two points, $A$ and $B$. What is the length of $\overline{A B}$ ?

Choices:
- 10
- 12
- 14
- 16

High--Intermediate Geometry--Equations of Lines and Circles


Question: If $\frac{8}{x}=160$, what is the value of $x ?$

Choices:
- 1,280
- 80
- 20
- 0.05

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Which of the following ordered pairs $(x, y)$ satisfies the inequality $5 x-3 y<4$ ?I. $(1,1)$II. $(2,5)$III. $(3,2)$

Choices:
- I only
- II only
- I and II only
- I and III only

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question: $$\begin{aligned}y & >2 x-1 \\2 x & >5\end{aligned}$$Which of the following consists of the $y$-coordinates of all the points that satisfy the system of inequalities above?

Choices:
- $y>6$
- $y>4$
- $y>\frac{5}{2}$
- $y>\frac{3}{2}$

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question: One pound of grapes costs $\$ 2$. At this rate, how many dollars will $c$ pounds of grapes cost?

Choices:
- $2 c$
- $2+c$
- $\frac{2}{c}$
- $\frac{c}{2}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: At Lincoln High School, approximately 7 percent of enrolled juniors and 5 percent of enrolled seniors were inducted into the National Honor Society last year. If there were 562 juniors and 602 seniors enrolled at Lincoln High School last year, which of the following is closest to the total number of juniors and seniors at Lincoln High School last year who were inducted into the National Honor Society?

Choices:
- 140
- 69
- 39
- 30

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question: If $y=k x$, where $k$ is a constant, and $y=24$ when $x=6$, what is the value of $y$ when $x=5$ ?

Choices:
- 6
- 15
- 20
- 23

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $$\frac{1}{2 x+1}+5$$Which of the following is equivalent to the expression above for $x>0$ ?

Choices:
- $\frac{2 x+5}{2 x+1}$
- $\frac{2 x+6}{2 x+1}$
- $\frac{10 x+5}{2 x+1}$
- $\frac{10 x+6}{2 x+1}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Ken is working this summer as part of a crew on a farm. He earned $\$ 8$ per hour for the first 10 hours he worked this week. Because of his performance, his crew leader raised his salary to $\$ 10$ per hour for the rest of the week. Ken saves $90 \%$ of his earnings from each week. What is the least number of hours he must work the rest of the week to save at least $\$ 270$ for the week?

Choices:
- 38
- 33
- 22
- 16

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question: Kathy is a repair technician for a phone company. Each week, she receives a batch of phones that need repairs. The number of phones that she has left to fix at the end of each day can be estimated with the equation $P=108-23 d$, where $P$ is the number of phones left and $d$ is the number of days she has worked that week. What is the meaning of the value 108 in this equation?

Choices:
- Kathy will complete the repairs within 108 days.
- Kathy starts each week with 108 phones to fix.
- Kathy repairs phones at a rate of 108 per hour.
- Kathy repairs phones at a rate of 108 per day.

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: If $\sin x^{\circ}=a$, which of the following must be true for all values of $x$ ?

Choices:
- $\cos x^{\circ}=a$
- $\sin \left(90^{\circ}-x^{\circ}\right)=a$
- $\cos \left(90^{\circ}-x^{\circ}\right)=a$
- $\sin \left(x^{2}\right)^{\circ}=a^{2}$

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: $$2 a x-15=3(x+5)+5(x-1)$$In the equation above, $a$ is a constant. If no value of $x$ satisfies the equation, what is the value of $a$ ?

Choices:
- 1
- 2
- 4
- 8

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question: $$y=a(x-2)(x+4)$$In the quadratic equation above, $a$ is a nonzero constant. The graph of the equation in the $x y$-plane is a parabola with vertex $(c, d)$. Which of the following is equal to $d$ ?

Choices:
- $-9 a$
- $-8 a$
- $-5 a$
- $-2 a$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $$y=x^{2}-a$$In the equation above, $a$ is a positive constant and the graph of the equation in the $x y$-plane is a parabola. Which of the following is an equivalent form of the equation?

Choices:
- $y=(x+a)(x-a)$
- $y=(x+\sqrt{a})(x-\sqrt{a})$
- $y=\left(x+\frac{a}{2}\right)\left(x-\frac{a}{2}\right)$
- $y=(x+a)^{2}$ DIRECTIONS

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: The formula below is often used by project managers to compute $E$, the estimated time to complete a job, where $O$ is the shortest completion time, $P$ is the longest completion time, and $M$ is the most likely completion time.$$E=\frac{O+4 M+P}{6}$$Which of the following correctly gives $P$ in terms of $E, O$, and $M$ ?

Choices:
- $P=6 E-O-4 M$
- $P=-6 E+O+4 M$
- $P=\frac{O+4 M+E}{6}$
- $P=\frac{O+4 M-E}{6}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $$(4 x+4)(a x-1)-x^{2}+4$$In the expression above, $a$ is a constant. If the expression is equivalent to $b x$, where $b$ is a constant, what is the value of $b$ ?

Choices:
- -5
- -3
- 0
- 12

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $$C=75 h+125$$The equation above gives the amount $C$, in dollars, an electrician charges for a job that takes $h$ hours. Ms. Sanchez and Mr. Roland each hired this electrician. The electrician worked 2 hours longer on Ms. Sanchez's job than on Mr. Roland's job. How much more did the electrician charge Ms. Sanchez than Mr. Roland?

Choices:
- $\$ 75$
- $\$ 125$
- $\$ 150$
- $\$ 275$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Survey Results\begin{center}\begin{tabular}{|l|c|}\hlineAnswer & Percent \\\hlineNever & $31.3 \%$ \\\hlineRarely & $24.3 \%$ \\\hlineOften & $13.5 \%$ \\\hlineAlways & $30.9 \%$ \\\hline\end{tabular}\end{center}The table above shows the results of a survey in which tablet users were asked how often they would watch video advertisements in order to access streaming content for free. Based on the table, which of the following is closest to the probability that a tablet user answered "Always," given that the tablet user did not answer "Never"?

Choices:
- 0.31
- 0.38
- 0.45
- 0.69

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: The Downtown Business Association (DBA) in a certain city plans to increase its membership by a total of $n$ businesses per year. There were $b$ businesses in the DBA at the beginning of this year. Which function best models the total number of businesses, $y$, the DBA plans to have as members $x$ years from now?

Choices:
- $y=n x+b$
- $y=n x-b$
- $y=b(n)^{x}$
- $y=n(b)^{x}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: For a polynomial $p(x)$, the value of $p(3)$ is -2 .Which of the following must be true about $p(x)$ ?

Choices:
- $x-5$ is a factor of $p(x)$.
- $x-2$ is a factor of $p(x)$.
- $x+2$ is a factor of $p(x)$.
- The remainder when $p(x)$ is divided by $x-3$ is -2 .

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: The surface area of a cube is $6\left(\frac{a}{4}\right)^{2}$, where $a$ is a positive constant. Which of the following gives the perimeter of one face of the cube?

Choices:
- $\frac{a}{4}$
- $a$
- $4 a$
- $6 a$


Question: Which of the following is an equivalent form of $(1.5 x-2.4)^{2}-\left(5.2 x^{2}-6.4\right) ?$

Choices:
- $-2.2 x^{2}+1.6$
- $-2.2 x^{2}+11.2$
- $-2.95 x^{2}-7.2 x+12.16$
- $-2.95 x^{2}-7.2 x+0.64$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: A circle in the $x y$-plane has center $(5,7)$ and radius 2. Which of the following is an equation of the circle?

Choices:
- $(x-5)^{2}+(y-7)^{2}=4$
- $(x+5)^{2}+(y+7)^{2}=4$
- $(x-5)^{2}+(y-7)^{2}=2$
- $(x+5)^{2}+(y+7)^{2}=2$

High--Intermediate Geometry--Equations of Lines and Circles


Question: Of the following four types of savings account plans, which option would yield exponential growth of the money in the account?

Choices:
- Each successive year, $2 \%$ of the initial savings is added to the value of the account.
- Each successive year, $1.5 \%$ of the initial savings and $\$ 100$ is added to the value of the account.
- Each successive year, $1 \%$ of the current value is added to the value of the account.
- Each successive year, $\$ 100$ is added to the value of the account.

High--Intermediate Numbers and Algebra--Sequences


Question: Dreams Recalled during One Week\begin{center}\begin{tabular}{|l|c|c|c|c|}\hline& None & 1 to 4 & 5 or more & Total \\\hline\hlineGroup X & 15 & 28 & 57 & 100 \\\hlineGroup Y & 21 & 11 & 68 & 100 \\\hlineTotal & 36 & 39 & 125 & 200 \\\hline\end{tabular}\end{center}The data in the table above were produced by a sleep researcher studying the number of dreams people recall when asked to record their dreams for one week. Group X consisted of 100 people who observed early bedtimes, and Group Y consisted of 100 people who observed later bedtimes. If a person is chosen at random from those who recalled at least 1 dream, what is the probability that the person belonged to Group Y ?

Choices:
- $\frac{68}{100}$
- $\frac{79}{100}$
- $\frac{79}{164}$
- $\frac{164}{200}$

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: If the 180 food calories in a granola bar come entirely from $p$ grams of protein, $f$ grams of fat, and $c$ grams of carbohydrate, which of the following expresses $f$ in terms of $p$ and $c$ ?

Choices:
- $f=20+\frac{4}{9}(p+c)$
- $f=20-\frac{4}{9}(p+c)$
- $f=20-\frac{4}{9}(p-c)$
- $f=20+\frac{9}{4}(p+c)$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: The density $d$ of an object is found by dividing the mass $m$ of the object by its volume $V$. Which of the following equations gives the mass $m$ in terms of $d$ and $V$ ?

Choices:
- $m=d V$
- $m=\frac{d}{V}$
- $m=\frac{V}{d}$
- $m=V+d$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $$\begin{aligned}& 2 x-3 y=-14 \\& 3 x-2 y=-6\end{aligned}$$If $(x, y)$ is a solution to the system of equations above, what is the value of $x-y$ ?

Choices:
- -20
- $\quad-8$
- -4
- 8

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: A researcher surveyed a random sample of students from a large university about how often they see movies. Using the sample data, the researcher estimated that $23 \%$ of the students in the population saw a movie at least once per month. The margin of error for this estimation is $4 \%$. Which of the following is the most appropriate conclusion about all students at the university, based on the given estimate and margin of error?

Choices:
- It is unlikely that less than $23 \%$ of the students see a movie at least once per month.
- At least 23\%, but no more than $25 \%$, of the students see a movie at least once per month.
- The researcher is between $19 \%$ and $27 \%$ sure that most students see a movie at least once per month.
- It is plausible that the percentage of students who see a movie at least once per month is between $19 \%$ and $27 \%$.

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: $$x+1=\frac{2}{x+1}$$In the equation above, which of the following is a possible value of $x+1$ ?

Choices:
- $1-\sqrt{2}$
- $\sqrt{2}$
- 2
- 4

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: The world's population has grown at an average rate of 1.9 percent per year since 1945. There were approximately 4 billion people in the world in 1975. Which of the following functions represents the world's population $P$, in billions of people, $t$ years since $1975 ?(1$ billion $=1,000,000,000)$

Choices:
- $P(t)=4(1.019)^{t}$
- $P(t)=4(1.9)^{t}$
- $P(t)=1.19 t+4$
- $P(t)=1.019 t+4$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $$\begin{aligned}& 3 x+b=5 x-7 \\& 3 y+c=5 y-7\end{aligned}$$In the equations above, $b$ and $c$ are constants.If $b$ is $c$ minus $\frac{1}{2}$, which of the following is true?

Choices:
- $x$ is $y$ minus $\frac{1}{4}$.
- $x$ is $y$ minus $\frac{1}{2}$.
- $x$ is $y$ minus 1 .
- $x$ is $y$ plus $\frac{1}{2}$.

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question:
Given are two sets: $A=\{a ; b ; e ; g\}$ and $B=\{a ; b ; c ; d ; f\}$.
By listing its elements, give the set $B \backslash A$.

Choices:
- $\{c; d; f\}$
- $\{a; b; c\}$
- $\{a; c; d; f\}$
- $\{a; b; e; f; g\}$

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Bori, Kristóf and Marci are playing a role-playing card game. At the beginning of the game they each select one out of 10 role cards, without replacement. In how many different arrangements of the roles can the game begin?

Choices:
- 90
- 720
- 800
- 72

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: Zita's salary has been raised from $275000$ Ft to $308000$ Ft. By what percentage has Zita's salary been raised?

Choices:
- 122\%
- 112\%
- 12\%
- 1.2\%

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: In triangle $A B C \overrightarrow{A B}=\mathbf{b}, \overrightarrow{A C}=\mathbf{c}$. The midpoint of side $A B$ is point $F$, the midpoint of side $A C$ is $G$. Express vector $\overrightarrow{F G}$ in terms of vectors $\mathbf{b}$ and $\mathbf{c}$. Explain your answer.

Choices:
- $\frac{1}{2} \mathbf{b} + \frac{1}{2} \mathbf{c}$
- $\frac{1}{2} \mathbf{b} - \frac{1}{2} \mathbf{c}$
- $\frac{1}{2} \mathbf{c} + \frac{1}{2} \mathbf{b}$
- $\frac{1}{2} \mathbf{c} - \frac{1}{2} \mathbf{b}$

High--Intermediate Numbers and Algebra--Plane Vectors


Question: Give five positive numbers such that their median is 3 and their range is 7 .

Choices:
- 1, 2, 3, 7, 8
- 1, 2, 3, 7, 9
- 2, 3, 4, 5, 6
- -1, 1, 3, 4, 6

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: Determine the decimal (base 10) value of the binary (base 2) number 101011.

Choices:
- 45
- 43
- 41
- 53

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: It is known that $\log_{2}(x)=5$. Give the value of $\log_{2}(2x)$. Explain your answer.

Choices:
- 36
- 7
- 6
- 12

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: List all integer values of $x$ for which both of the inequalities $-6 \leq x \leq 2$ and $-4<x<10$ are true.

Choices:
- -6; -5; -4; -3; -2; -1; 0; 1; 2
- -4; -3; -2; -1; 0; 1; 2
- -4; -3; -2; -1; 0; 1
- -3; -2; -1; 0; 1; 2

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question: Sixteen teams have signed up for the school indoor football championship. In how many different ways can the two teams playing the opening game be selected?

Choices:
- 120
- 140
- 240
- 17

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: The sides of the right triangle $A B C$ are $a=7, b=24, c=25$ units long. Calculate the length of the height that belongs to the hypotenuse. Explain your answer.

Choices:
- 7
- 6.72
- 6.5
- 8

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: Given is line $e: 5 x-y=7$. Give one normal vector of line $e$.

Choices:
- $(5 ; -1)$
- $(-5 ; -1)$
- $(5 ; 1)$
- $(-5 ; 1)$

High--Intermediate Geometry--Equations of Lines and Circles


Question: Given is line $e: 5 x-y=7$. Give the equation of the line that passes through the point $P(3 ; 2)$ and is parallel to line $e$.

Choices:
- $x - 5 y = 7$
- $x - 5 y = 13$
- $5 x - y = 13$
- $5 x + y = 13$

High--Intermediate Geometry--Equations of Lines and Circles


Question:
Function $f$ below is defined over the set of non-negative real numbers. Functions $g$ and $h$ are defined over the set of real numbers.
$$f(x)=\sqrt{x}-2 \quad g(x)=(x-2)^{2}-3 \quad h(x)=2 \sin x$$
For which function is the following statement true: The minimum value of the function is (-2).

Choices:
- $g$
- $f, h$
- $f, g, h$
- $g, h$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Function $f$ below is defined over the set of non-negative real numbers. Functions $g$ and $h$ are defined over the set of real numbers. $$f(x)=\sqrt{x}-2 \quad g(x)=(x-2)^{2}-3 \quad h(x)=2 \sin x$$ For which function is the following statement true: The function has at least two zeros.

Choices:
- $g$
- $f, h$
- $f, g, h$
- $g, h$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question:
Consider the following two events:

A: Roll a fair gambling dice once, the number shown is a divisor of 24.

B: Roll a fair gambling dice twice, none of the numbers shown is a 6 .

Which of the two events has a higher probability?

Choices:
- B event
- A event
- C event
- D event

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question:
In a physics class, students were experimenting in pairs, measuring the acceleration of an object sliding down a slope. Each of the six pairs measured the acceleration four times. Here are the results obtained by Emma and Norbi:

Trial 1: $1.9 m/s^2$
Trial 2: $2.0 m/s^2$
Trial 3: $1.8 m/s^2$
Trial 4: $2.3 m/s^2$

Calculate the standard deviation of the four results Emma and Norbi obtained.

Choices:
- 0.287$\:({\mathrm{m}}/{\mathrm{s}}^{2})$
- 0.187$\:({\mathrm{m}}/{\mathrm{s}}^{2})$
- 0.387 $\:({\mathrm{m}}/{\mathrm{s}}^{2})$
- 0.487 $\:({\mathrm{m}}/{\mathrm{s}}^{2})$

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question:
In a physics class, students were experimenting in pairs, measuring the acceleration of an object sliding down a slope. Each of the six pairs measured the acceleration four times. Here are the results obtained by Emma and Norbi:

Trial 1: $1.9 m/s^2$
Trial 2: $2.0 m/s^2$
Trial 3: $1.8 m/s^2$
Trial 4: $2.3 m/s^2$

The average of the 20 values obtained by the other five pairs is exactly $1.9 m/s^2$.

What is the average of the 24 results of the six pairs? Round your answer to two decimal places.

Choices:
- 2.92$\:({\mathrm{m}}/{\mathrm{s}}^{2})$
- 1.92$\:({\mathrm{m}}/{\mathrm{s}}^{2})$
- 3.92$\:({\mathrm{m}}/{\mathrm{s}}^{2})$
- 4.92$\:({\mathrm{m}}/{\mathrm{s}}^{2})$

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question:
In another experiment, students examined the motion of a ball that had been shot vertically upwards from the ground and then fell back. According to their results, the distance $h$ of the ball from the ground follows the rule $h(t)=6 t-5 t^{2}$, where $t$ is the time passed since the ball has been shot upwards. (Time values are measured in seconds, distances in metres.)

According to the formula above, how high above the ground is the ball 0.5 seconds after it has been shot upwards?

Choices:
- 0.75 (metres).
- 1.75 (metres).
- 2.75 (metres).
- 3.75 (metres).

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question:
In another experiment, students examined the motion of a ball that had been shot vertically upwards from the ground and then fell back. According to their results, the distance $h$ of the ball from the ground follows the rule $h(t)=6 t-5 t^{2}$, where $t$ is the time passed since the ball has been shot upwards. (Time values are measured in seconds, distances in metres.)

How many seconds after it has been shot upwards will the ball reach a height of 1m above the ground?

Choices:
- t=2,t=1
- t=0.2,t=1
- t=0.2,t=0.1
- t=2,t=0.1

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question:
The side of a square is 4 cm. Two line segments are drawn into this square such that they trisect one of the right angles of the square, dividing the square into two right triangles and a quadrilateral. One of the right triangles is shaded gray.

Calculate the area of the triangle shaded grey in the diagram.

Choices:
- 3.62$cm^2$
- 4.62$cm^2$
- 5.62$cm^2$
- 6.62$cm^2$

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question:
Select one interior point of a square of the same size and connect it to the four vertices, as seen in the diagram. The triangles obtained will be coloured in either blue, green or yellow. All three colours must be used and each triangle is coloured in a single colour. No adjacent triangles (sharing a common side) may be coloured the same.

In how many different ways can the square be coloured if the above rules are followed?

Choices:
- 11
- 12
- 13
- 14

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question:
Solve the following equations in the set of real numbers.

$$2 \cdot \sqrt{3 - x} = x + 5$$

Choices:
- x = 0
- x = –1
- x = 1
- x = 2

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question:
Solve the following equations in the set of real numbers.

$$\frac{x}{x+1} + \frac{x^2}{x^2 - 1} = 2$$

Choices:
- x = 1
- x = 2
- x = 3
- x = 4

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question:
The first term of an arithmetic sequence is 18. The sum of the first six terms of the sequence is equal to the sum of the first seven terms.

Show that the sum of the first 13 terms of the sequence is 0 and calculate the 13th term of the sequence.

Choices:
- − 20
- − 18
- − 16
- − 14

High--Intermediate Numbers and Algebra--Sequences


Question:
In the year 2018 factory A produced goods worth 500 million forints, while factory B produced goods worth 400 million forints. According to their long-term plans, factory A will increase production by 5% each year, while factory B will increase production by 6% per year.

Calculate the total worth, in million forints, of the goods produced in factory A in the next 20 years (from 2019 to 2038) altogether.

Choices:
- 16 360 (million forints)
- 17 360 (million forints)
- 18 360 (million forints)
- 19 360 (million forints)

High--Intermediate Numbers and Algebra--Sequences


Question:
In the year 2018 factory A produced goods worth 500 million forints, while factory B produced goods worth 400 million forints. According to their long-term plans, factory A will increase production by 5% each year, while factory B will increase production by 6% per year.

The above plans were published on an economy website. A debate started in the comment section of the article: one user claimed that the difference between the volumes of production between the two factories would get smaller and smaller in the next couple of years.

Calculate the missing values of the table below.

| Year | Production volume of factory A (million Ft) | Production volume of factory B (million Ft) |
|------|--------------------------------------------|---------------------------------------------|
| 2018 | 500 | 400 |
| 2019 | | |
| 2020 | | |
| 2021 | | |

Choices:
-
| Year | Production volume of factory A (million Ft) | Production volume of factory B (million Ft) |
|------|--------------------------------------------|---------------------------------------------|
| 2018 | 500 | 400 |
| 2019 | 525 | 424|
| 2020 | 551.3 | 449.4 |
| 2021 | 578.8 | 476.4 |
-
| Year | Production volume of factory A (million Ft) | Production volume of factory B (million Ft) |
|------|--------------------------------------------|---------------------------------------------|
| 2018 | 500 | 400 |
| 2019 | 535 | 424|
| 2020 | 551.3 | 449.4 |
| 2021 | 578.8 | 476.4 |
-
| Year | Production volume of factory A (million Ft) | Production volume of factory B (million Ft) |
|------|--------------------------------------------|---------------------------------------------|
| 2018 | 500 | 400 |
| 2019 | 525 | 444|
| 2020 | 551.3 | 449.4 |
| 2021 | 578.8 | 476.4 |
-
| Year | Production volume of factory A (million Ft) | Production volume of factory B (million Ft) |
|------|--------------------------------------------|---------------------------------------------|
| 2018 | 500 | 400 |
| 2019 | 525 | 444|
| 2020 | 551.3 | 449.4 |
| 2021 | 578.8 | 466.4 |

High--Intermediate Numbers and Algebra--Sequences


Question:
In the year 2018 factory A produced goods worth 500 million forints, while factory B produced goods worth 400 million forints. According to their long-term plans, factory A will increase production by 5% each year, while factory B will increase production by 6% per year.

Calculate the year in which the production volume of factory B will reach the production volume of factory A. (Assuming the increase in production will, in fact, follow the plans.)

Choices:
- 2041
- 2042
- 2043
- 2044

High--Intermediate Numbers and Algebra--Sequences


Question:
Magic Spheres dessert is sold in boxes that have a shape of a regular hexagon-based prism. Each edge of the regular hexagonal base is 5 cm long and the height of the box is 3 cm. The box contains 6 chocolate balls, each has a diameter of 2.8 cm.

What percentage of the volume of the box is the volume of the chocolate?

Choices:
- 34.4 percent
- 35.4 percent
- 36.4 percent
- 37.4 percent


Question:
Chocolate balls in Magic Spheres dessert are individually wrapped in either gold or red wrapping foil. The sorting machine randomly drops the chocolate balls into the boxes, the probability of a red one is $\frac{1}{3}$, the probability of a gold one is $\frac{2}{3}$, a total six chocolate balls in each box.

Calculate the probability that at least 5 of the 6 chocolate balls in a box will be wrapped in gold foil.

Choices:
- 0.341
- 0.351
- 0.361
- 0.371

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question:
Each side of the regular hexagon $ABCDEF$ is 5 cm long. The hexagon is rotated around the perpendicular bisector of side $AB$.

Calculate the volume of the resulting solid.

Choices:
- 394.8 $cm^3$
- 396.8 $cm^3$
- 398.8 $cm^3$
- 400.8 cm^3


Question: A data set consists of  $6$ (not distinct) positive integers:  $1$ ,  $7$ ,  $5$ ,  $2$ ,  $5$ , and  $X$ . The average (arithmetic mean) of the  $6$ numbers equals a value in the data set. What is the sum of all possible values of  $X$ ?

Choices:
- 10
- 32
- 26
- 36

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question:
Define  $x\diamond y$ to be  $|x-y|$ for all real numbers  $x$ and  $y.$ What is the value of  $(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?$

 $

Choices:
- 1
- {-}2
- 2
- {-}1

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question:
Let  $f(n) = \left( \frac{-1+i\sqrt{3}}{2} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n$ , where  $i = \sqrt{-1}$ . What is  $f(2022)$ ?

 $

Choices:
- -1
- 2
- -2
- \sqrt{3}

High--Intermediate Numbers and Algebra--Complex Numbers


Question:
Kayla rolls four fair  $6$ -sided dice. What is the probability that at least one of the numbers Kayla rolls is greater than  $4$ and at least two of the numbers she rolls are greater than  $2$ ?

 $

Choices:
- \frac{61}{81}
- \frac{19}{27}
- \frac{7}{9}
- \frac{59}{81}

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: One of the following numbers is not divisible by any prime number less than  $10.$ Which is it?

Choices:
- 2^{606}-1
- 2^{607}-1
- 2^{606}+1
- 2^{607}+1

High--Intermediate Numbers and Algebra--Sequences


Question:
How many  $4 \times 4$ arrays whose entries are  $0$ s and  $1$ s are there such that the row sums (the sum of the entries in each row) are  $1, 2, 3,$ and  $4,$ in some order, and the column sums (the sum of the entries in each column) are also  $1, 2, 3,$ and  $4,$ in some order? For example, the array
 $\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ \end{array} \right]$
satisfies the condition.

Choices:
- 336
- 576
- 144
- 240

High--Intermediate Numbers and Algebra--Sequences


Question: In  $\triangle{ABC}$ medians  $\overline{AD}$ and  $\overline{BE}$ intersect at  $G$ and  $\triangle{AGE}$ is equilateral. Then  $\cos(C)$ can be written as  $\frac{m\sqrt p}n$ , where  $m$ and  $n$ are relatively prime positive integers and  $p$ is a positive integer not divisible by the square of any prime. What is  $m+n+p?$

Choices:
- 48
- 44
- 56
- 60

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question:
Let  $P(x)$ be a polynomial with rational coefficients such that when  $P(x)$ is divided by the polynomial
 $x^2 + x + 1$ , the remainder is  $x+2$ , and when  $P(x)$ is divided by the polynomial  $x^2+1$ , the remainder
is  $2x+1$ . There is a unique polynomial of least degree with these two properties. What is the sum of
the squares of the coefficients of that polynomial?

 $

Choices:
- 23
- 10
- 20
- 13

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question:
Let  $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each  $x_k$ is either  $0$ or  $1$ . For each positive integer  $n$ , define
 $S_n = \sum_{k=0}^{n-1} x_k 2^k$
Suppose  $7S_n \equiv 1 \pmod{2^n}$ for all  $n \geq 1$ . What is the value of the sum
 $x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?$

Choices:
- 6
- 7
- 15
- 12

High--Intermediate Numbers and Algebra--Sequences


Question: How many of the first ten numbers of the sequence  $121, 11211, 1112111, \ldots$ are prime numbers?

Choices:
- 1
- 4
- 2
- 0

High--Intermediate Numbers and Algebra--Sequences


Question:
For how many values of the constant  $k$ will the polynomial  $x^{2}+kx+36$ have two distinct integer roots?

 $

Choices:
- 8
- 6
- 14
- 16

High--Intermediate Numbers and Algebra--Sequences


Question:
The point  $(-1, -2)$ is rotated  $270^{\circ}$ counterclockwise about the point  $(3, 1)$ . What are the coordinates of its new position?

 $

Choices:
- (0,5)
- (-3, -4)
- (6,-3)
- (4,3)

High--Intermediate Geometry--Equations of Lines and Circles


Question:
Consider the following  $100$ sets of  $10$ elements each:

$$
\begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21,22,23,\ldots,30\},\\ &\vdots\\ &\{991,992,993,\ldots,1000\}. \end{align*}
$$

How many of these sets contain exactly two multiples of  $7$ ?

 $

Choices:
- 43
- 50
- 40
- 42

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question:
Camila writes down five positive integers. The unique mode of these integers is  $2$ greater than their median, and the median is  $2$ greater than their arithmetic mean. What is the least possible value for the mode?

 $

Choices:
- 5
- 11
- 9
- 13

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: What is the graph of  $y^4+1=x^4+2y^2$ in the coordinate plane?

Choices:
- \text{a circle and two parabolas}
- text{two intersecting parabolas}
- \text{a circle and a hyperbola}
- \text{two nonintersecting parabolas}

High--Intermediate Geometry--Equations of Conic Sections


Question:
The sequence  $a_0,a_1,a_2,\cdots$ is a strictly increasing arithmetic sequence of positive integers such that  $2^{a_7}=2^{27} \cdot a_7.$ What is the minimum possible value of  $a_2$ ?

 $

Choices:
- 12
- 17
- 22
- 8

High--Intermediate Numbers and Algebra--Sequences


Question: Cities  $A$ and  $B$ are  $45$ miles apart. Alicia lives in  $A$ and Beth lives in  $B$ . Alicia bikes towards  $B$ at 18 miles per hour. Leaving at the same time, Beth bikes toward  $A$ at 12 miles per hour. How many miles from City  $A$ will they be when they meet?

Choices:
- 27
- 20
- 25
- 24

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question: Positive real numbers  $x$ and  $y$ satisfy  $y^3=x^2$ and  $(y-x)^2=4y^2$ . What is  $x+y$ ?

Choices:
- 24
- 36
- 18
- 42

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: What is the degree measure of the acute angle formed by lines with slopes  $2$ and  $\frac{1}{3}$ ?

Choices:
- ~37.5
- ~45
- ~60
- ~30

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question:
What is the value of
 $2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?$

Choices:
- 3235
- 3159
- 2023
- 2941

High--Intermediate Numbers and Algebra--Sequences


Question: In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was  $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?

Choices:
- 48
- 15
- 36
- 66

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: How many complex numbers satisfy the equation  $z^5=\overline{z}$ , where  $\overline{z}$ is the conjugate of the complex number  $z$ ?

Choices:
- ~5
- ~3
- ~6
- ~2

High--Intermediate Numbers and Algebra--Complex Numbers


Question:
Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance  $m$ with probability  $\frac{1}{2^m}$ .

What is the probability that Flora will eventually land at 10?

Choices:
- ~\frac{45}{1024}
- ~\frac{511}{1024}
- ~\frac{1}{2}
- ~\frac{5}{512}

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question:
What is the product of all solutions to the equation
 $\log_{7x}2023\cdot \log_{289x}2023=\log_{2023x}2023$

Choices:
- ~(\log_{2023}7\cdot \log_{2023}289)^2
- ~(\log_7 2023\cdot\log_{289} 2023)^2
- ~\log_{7}2023\cdot \log_{289}2023
- ~1

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: The weight of  $\frac{1}{3}$ of a large pizza together with  $3 \frac{1}{2}$ cups of orange slices is the same as the weight of  $\frac{3}{4}$ of a large pizza together with  $\frac{1}{2}$ cup of orange slices. A cup of orange slices weighs  $\frac{1}{4}$ of a pound. What is the weight, in pounds, of a large pizza?

Choices:
- 1\frac{4}{5}
- 3\frac{3}{5}
- 2\frac{2}{5}
- 3

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question: If  $A$ and  $B$ are vertices of a polyhedron, define the distance  $d(A,B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect  $A$ and  $B$ . For example, if  $\overline{AB}$ is an edge of the polyhedron, then  $d(A, B) = 1$ , but if  $\overline{AC}$ and  $\overline{CB}$ are edges and  $\overline{AB}$ is not an edge, then  $d(A, B) = 2$ . Let  $Q$ ,  $R$ , and  $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that  $d(Q, R) > d(R, S)$ ?

Choices:
- \frac{1}{3}
- \frac{5}{12}
- frac{7}{22}
- \frac{1}{2}


Question: Let  $f$ be the unique function defined on the positive integers such that  $\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1$ for all positive integers  $n$ . What is  $f(2023)$ ?

Choices:
- ~96
- ~116
- ~108
- ~-1536

High--Intermediate Numbers and Algebra--Sequences


Question:
How many ordered pairs of positive real numbers  $(a,b)$ satisfy the equation
 $(1+2a)(2+2b)(2a+b) = 32ab?$

Choices:
- 1
- 3
- 0
- \text{an infinite number}

High--Intermediate Numbers and Algebra--Sequences


Question: Let  $K$ be the number of sequences  $A_1$ ,  $A_2$ ,  $\dots$ ,  $A_n$ such that  $n$ is a positive integer less than or equal to  $10$ , each  $A_i$ is a subset of  $\{1, 2, 3, \dots, 10\}$ , and  $A_{i-1}$ is a subset of  $A_i$ for each  $i$ between  $2$ and  $n$ , inclusive. For example,  $\{\}$ ,  $\{5, 7\}$ ,  $\{2, 5, 7\}$ ,  $\{2, 5, 7\}$ ,  $\{2, 5, 6, 7, 9\}$ is one such sequence, with  $n = 5$ .What is the remainder when  $K$ is divided by  $10$ ?

Choices:
- 1
- 9
- 3
- 5

High--Intermediate Numbers and Algebra--Sequences


Question:
There is a unique sequence of integers  $a_1, a_2, \cdots a_{2023}$ such that
 $\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}$ whenever  $\tan 2023x$ is defined. What is  $a_{2023}?$

Choices:
- -1
- 1
- 2023
- -2022

High--Intermediate Numbers and Algebra--Sequences


Question: How many positive perfect squares less than  $2023$ are divisible by  $5$ ?

Choices:
- 11
- 12
- 10
- 8

High--Intermediate Numbers and Algebra--Sequences


Question: How many digits are in the base-ten representation of  $8^5 \cdot 5^{10} \cdot 15^5$ ?

Choices:
- ~16
- ~18
- ~14
- ~17

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Janet rolls a standard  $6$ -sided die  $4$ times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal  $3$ ?

Choices:
- \frac{49}{216}
- \frac{13}{54}
- \frac{17}{72}
- \frac{2}{9}

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: Points  $A$ and  $B$ lie on the graph of  $y=\log_{2}x$ . The midpoint of  $\overline{AB}$ is  $(6, 2)$ . What is the positive difference between the  $x$ -coordinates of  $A$ and  $B$ ?

Choices:
- ~9
- ~4\sqrt{5}
- ~8
- ~4\sqrt{3}

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: A digital display shows the current date as an  $8$ -digit integer consisting of a  $4$ -digit year, followed by a  $2$ -digit month, followed by a  $2$ -digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in  $2023$ will each digit appear an even number of times in the 8-digital display for that date?

Choices:
- ~5
- ~7
- ~8
- ~9

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an  $11$ on the next quiz, her mean will increase by  $1$ . If she scores an  $11$ on each of the next three quizzes, her mean will increase by  $2$ . What is the mean of her quiz scores currently?

Choices:
- 4
- 8
- 6
- 7

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only  $\frac{1}{3}$ full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?

Choices:
- \frac{2}{9}
- \frac{1}{6}
- frac{1}{12}
- \frac{1}{8}

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question:
In the  $xy$ -plane, a circle of radius  $4$ with center on the positive  $x$ -axis is tangent to the  $y$ -axis at the origin, and a circle with radius  $10$ with center on the positive  $y$ -axis is tangent to the  $x$ -axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?

 $

Choices:
- \dfrac{1}{\sqrt{29}}
- \dfrac{2}{7}
- \dfrac{2}{5}
- \dfrac{3}{7}

High--Intermediate Geometry--Equations of Lines and Circles


Question:
For complex number  $u = a+bi$ and  $v = c+di$ (where  $i=\sqrt{-1}$ ), define the binary operation

 $u \otimes v = ac + bdi$

Suppose  $z$ is a complex number such that  $z\otimes z = z^{2}+40$ . What is  $|z|$ ?

Choices:
- 2
- 5\sqrt{2}
- \sqrt{10}
- \sqrt{5}

High--Intermediate Numbers and Algebra--Complex Numbers


Question:
For how many ordered pairs  $(a,b)$ of integers does the polynomial  $x^3+ax^2+bx+6$ have  $3$ distinct integer roots?

 $

Choices:
- 6
- 4
- 8
- 5

High--Intermediate Numbers and Algebra--Sequences


Question:
Suppose  $a$ ,  $b$ , and  $c$ are positive integers such that $\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.$ Which of the following statements are necessarily true?

I. If  $\gcd(a,14)=1$ or  $\gcd(b,15)=1$ or both, then  $\gcd(c,210)=1$ .

II. If  $\gcd(c,210)=1$ , then  $\gcd(a,14)=1$ or  $\gcd(b,15)=1$ or both.

III.  $\gcd(c,210)=1$ if and only if  $\gcd(a,14)=\gcd(b,15)=1$ .

Choices:
- ~\text{I only}
- ~\text{III only}
- ~\text{II and III only}
- ~\text{I, II, and III}

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: In the state of Coinland, coins have values  $6,10,$ and  $15$ cents. Suppose  $x$ is the value in cents of the most expensive item in Coinland that cannot be purchased using these coins with exact change. What is the sum of the digits of  $x?$

Choices:
- 11
- 10
- 8
- 7

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question: Triangle  $ABC$ has side lengths in arithmetic progression, and the smallest side has length  $6.$ If the triangle has an angle of  $120^\circ,$ what is the area of  $ABC$ ?

Choices:
- 15\sqrt{3}
- 14\sqrt{2}
- 20\sqrt{2}
- 12\sqrt{3}

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda's average on all the quizzes she took during the first semester was  $3$ points higher than Zelda's average on all the quizzes she took during the first semester. Yolanda's average on all the quizzes she took during the second semester was  $18$ points higher than her average for the first semester and was again  $3$ points higher than Zelda's average on all the quizzes Zelda took during her second semester. Which one of the following statements cannot possibly be true?

Choices:
- If Zelda had scored  3 points higher on each quiz she took, then she would have had the same average for the academic year as Yolanda.
- Zelda's quiz average for the academic year equaled Yolanda's.
- Yolanda's quiz average for the academic year was  3 points higher than Zelda's.
- Yolanda's quiz average for the academic year was  22 points higher than Zelda's.

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: Each of  $2023$ balls is randomly placed into one of  $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?

Choices:
- \frac{1}{2}
- \frac{1}{4}
- \frac{1}{3}
- frac{2}{3}

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by  $20\%$ on every pair of shoes. Carlos also knew that he had to pay a  $7.5\%$ sales tax on the discounted price. He had  $\$43$ dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?

Choices:
- 47
- 48
- 50
- 46

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question: Cyrus the frog jumps  $2$ units in a direction, then  $2$ more in another direction. What is the probability that he lands less than  $1$ unit away from his starting position?

Choices:
- ~\frac{2\arcsin \frac{1}{4}}{\pi}
- ~\frac{\sqrt{3}}{8}
- ~\frac{\arctan \frac{1}{2}}{\pi}
- ~\frac{1}{5}

High--Intermediate Geometry--Equations of Lines and Circles


Question: A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is  $3\sqrt3$ inches, its top diameter is  $6$ inches, and its bottom diameter is  $12$ inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey?

Choices:
- 6 + 6\pi
- 6\sqrt5
- 6\sqrt3 + \pi
- 6 + 3\pi


Question: A real-valued function  $f$ has the property that for all real numbers  $a$ and  $b,$  $f(a + b) + f(a - b) = 2f(a) f(b).$ Which one of the following cannot be the value of  $f(1)?$

Choices:
- 2
- 1
- -2
- -1

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: When  $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of  $936$ possible values. What is  $n$ ?

Choices:
- ~10
- ~11
- ~9
- ~6

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question:
Suppose that  $a$ ,  $b$ ,  $c$ and  $d$ are positive integers satisfying all of the following relations.

 $abcd=2^6\cdot 3^9\cdot 5^7$
 $\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3$
 $\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3$
 $\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3$
 $\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2$
 $\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2$
 $\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2$

What is  $\text{gcd}(a,b,c,d)$ ?

Choices:
- ~45
- ~3
- ~30
- ~15

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: A  $3-4-5$ right triangle is inscribed in circle  $A$ , and a  $5-12-13$ right triangle is inscribed in circle  $B$ . What is the ratio of the area of circle  $A$ to the area of circle  $B$ ?

Choices:
- \frac{4}{25}
- \frac{25}{169}
- \frac{1}{5}
- \frac{9}{25}

High--Intermediate Geometry--Equations of Lines and Circles


Question: Jackson's paintbrush makes a narrow strip with a width of  $6.5$ millimeters. Jackson has enough paint to make a strip  $25$ meters long. How many square centimeters of paper could Jackson cover with paint?

Choices:
- 1,625
- 162.5
- 162,500
- 16,250


Question:
When the roots of the polynomial

 $P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}$

are removed from the number line, what remains is the union of 11 disjoint open intervals. On how many of these intervals is  $P(x)$ positive?

Choices:
- ~6
- ~4
- ~7
- ~5

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question: For how many integers  $n$ does the expression $\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}}$ represent a real number, where log denotes the base  $10$ logarithm?

Choices:
- 900
- 2
- 901
- 2

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: How many nonempty subsets  $B$ of  ${0, 1, 2, 3, \cdots, 12}$ have the property that the number of elements in  $B$ is equal to the least element of  $B$ ? For example,  $B = {4, 6, 8, 11}$ satisfies the condition.

Choices:
- 156
- 256
- 108
- 144

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question:
What is the area of the region in the coordinate plane defined by

 $| | x | - 1 | + | | y | - 1 | \le 1$ ?

Choices:
- 8
- 15
- 12
- 2

High--Intermediate Geometry--Equations of Lines and Circles


Question: If  $x$ and  $y$ are positive integers such that their product is  $64,$ what is the sum of all distinct values for  $xy+x+y?$

Choices:
- 393
- 392
- 394
- 391

High--Intermediate Numbers and Algebra--Sequences


Question: There are  $\binom{8}{4}=70$ distinct quadrilaterals that can be formed from the vertices of a regular octagon. Which of these statements must hold true for all those quadrilaterals?

Choices:
- \text{All of the 70 quadrilaterals are kites.}
- \text{None of the above statements are true.}
- \text{All of the 70 quadrilaterals are cyclic.}
- \text{All of the 70 quadrilaterals are rectangles.}

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: (fidgetboss_4000) Mark rolled two standard dice. Given that he rolled two distinct values, find the probability that he rolled two primes.

Choices:
- \frac{2}{5}
- \frac{1}{5}
- \frac{1}{12}
- \frac{1}{7}

High--Intermediate Probability and Statistics--Foundations of Probability Theory