2022 College Scholastic Ability Test

Mathematics

Multiple Choice Questions
What is the value of \(\big(2^\sqrt{3}\times 4\big)^{\sqrt{3}\;\!-\;\!2}\)? [2 points]
  1. \(\dfrac{1}{4}\)
  2. \(\dfrac{1}{2}\)
  3. \(1\)
  4. \(2\)
  5. \(4\)
Let \(f(x)=x^3+3x^2+x-1\). What is the value of \(f'(1)\)? [2 points]
  1. \(6\)
  2. \(7\)
  3. \(8\)
  4. \(9\)
  5. \(10\)
An arithmetic progression \(\{a_n\}\) satisfies
\(a_2=6\:\) and \(\:a_4+a_6=36\).
What is the value of \(a_{10}\)? [3 points]
  1. \(30\)
  2. \(32\)
  3. \(34\)
  4. \(36\)
  5. \(38\)
The graph of the function \(y=f(x)\) is shown below.
What is the value of \(\displaystyle\lim_{x\;\!\to\;\!-1-}\!f(x)+\lim_{x\;\!\to\;\!2}f(x)\)? [3 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)

Mathematics

Let \(\{a_n\}\) be a sequence with an initial term of \(1\) such that
\(a_{n+1}=\begin{cases} 2a_n &\; (a_n<7)\\\\ a_n-7 &\; (a_n\geq7) \end{cases}\)
for all positive integers \(n\).
What is the value of \(\displaystyle\sum_{k\;\!=\;\!1}^8 a_k\)? [3 points]
  1. \(30\)
  2. \(32\)
  3. \(34\)
  4. \(36\)
  5. \(38\)
What is the number of integers \(k\) such that the equation \(2x^3-3x^2-12x+k=0\) has three distinct real solutions? [3 points]
  1. \(20\)
  2. \(23\)
  3. \(26\)
  4. \(29\)
  5. \(32\)
Suppose \(\pi<\theta<\dfrac{3}{2}\pi\). Given that \(\tan\!\theta-\dfrac{6}{\tan\!\theta}=1\), what is the value of \(\sin\theta+\cos\theta\)? [3 points]
  1. \(-\dfrac{2\sqrt{10}}{5}\)
  2. \(-\dfrac{\sqrt{10}}{5}\)
  3. \(0\)
  4. \(\dfrac{\sqrt{10}}{5}\)
  5. \(\dfrac{2\sqrt{10}}{5}\)

Mathematics

Suppose the line \(x=k\) bisects the region enclosed by the curve \(y=x^2-5x\) and the line \(y=x\). What is the value of the constant \(k\)? [3 points]
  1. \(3\)
  2. \(\dfrac{13}{4}\)
  3. \(\dfrac{7}{2}\)
  4. \(\dfrac{15}{4}\)
  5. \(4\)
Let \(\mathrm{P}\) and \(\mathrm{Q}\) be points where the line \(y=2x+k\) meets the graphs of functions
\(y=\left(\!\dfrac{2}{3}\!\right)^{\!x+3}\!\!+1\:\) and \(\:y=\left(\!\dfrac{2}{3}\!\right)^{\!x+1}\!\!+\dfrac{8}{3}\)
respectively. Given that \(\overline{\mathrm{PQ}}=\sqrt{5}\), what is the value of the constant \(k\)? [4 points]
  1. \(\dfrac{31}{6}\)
  2. \(\dfrac{16}{3}\)
  3. \(\dfrac{11}{2}\)
  4. \(\dfrac{17}{3}\)
  5. \(\dfrac{35}{6}\)
Let \(f(x)\) be a cubic function such that points \((0,0)\) and \((1,2)\) are on the curve \(y=f(x)\), and the tangent line to the curve \(y=f(x)\) at the two points are the same line. What is the value of \(f'(2)\)? [4 points]
  1. \(-18\)
  2. \(-17\)
  3. \(-16\)
  4. \(-15\)
  5. \(-14\)

Mathematics

For positive numbers \(a\), Let \(f(x)\) be a function defined on the set \(\left\{x\middle|-\dfrac{a}{2}<x\leq a, x\ne\dfrac{a}{2}\right\}\) such that
\(f(x)=\tan\dfrac{\pi x}{a}\).
As the figure shows, a line passes through points \(\mathrm{O, A}\) and \(\mathrm{B}\) on the graph of \(y=f(x)\).
Let \(\mathrm{C}\) be the point where the line passing through point \(\mathrm{A}\) parallel to the \(x\)-axis, meets the graph of \(y=f(x)\) again \((\mathrm{C}\ne \mathrm{A})\). Given that triangle \(\mathrm{ABC}\) is equilateral, what is the area of triangle \(\mathrm{ABC}\)?
(※ \(\mathrm{O}\) is the origin.) [4 points]
  1. \(\dfrac{3\sqrt{3}}{2}\)
  2. \(\dfrac{17\sqrt{3}}{12}\)
  3. \(\dfrac{4\sqrt{3}}{3}\)
  4. \(\dfrac{5\sqrt{3}}{4}\)
  5. \(\dfrac{7\sqrt{3}}{6}\)
Let \(f(x)\) be a function that is continuous on the set of all real numbers, such that
\(\{f(x)\}^3-\{f(x)\}^2-x^2f(x)+x^2=0\)
for all real numbers \(x\). Given that the function \(f(x)\) has a global maximum value of \(1\) and a global minimum value of \(0\), what is the value of \(f\!\left(\!-\dfrac{4}{3}\!\right)+f(0)+f\!\left(\!\dfrac{1}{2}\!\right)\)? [4 points]
  1. \(\dfrac{1}{2}\)
  2. \(1\)
  3. \(\dfrac{3}{2}\)
  4. \(2\)
  5. \(\dfrac{5}{2}\)

Mathematics

For some constants \(a\) and \(b\) \((1<a<b)\), consider a line that passes through points \((a,\log_2 a)\) and \((b,\log_2 b)\), and a line that passes through points \((a,\log_4 a)\) and \((b,\log_4 b)\). Suppose the \(y\)-intercepts of the two lines are equal.
Let \(f(x)=a^{bx}+b^{ax}\).
If \(f(1)=40\), what is the value of \(f(2)\)? [4 points]
  1. \(760\)
  2. \(800\)
  3. \(840\)
  4. \(880\)
  5. \(920\)
Suppose a point \(\mathrm{P}\) is moving on the number line, and its position \(x(t)\) at time \(t\) is
\(x(t)=t(t-1)(at+b)\)
for some constants \(a\) and \(b\). Let \(v(t)\) be the velocity of point \(\mathrm{P}\) at time \(t\). If \(v(t)\) satisfies \(\displaystyle\int_0^1|v(t)|dt=2\), what is the list of correct statements in the <List>? [4 points]
  1. \(\displaystyle\int_0^1v(t)dt=0\)
  2. There exists some \(t_1\) in the open interval \((0,1)\) such that \(|x(t_1)|>1\).
  3. If \(|x(t)|<1\) for all \(t\) satisfying \(0\leq t\leq 1\), then there exists some \(t_2\) in the open interval \((0,1)\) such that \(x(t_2)=0\).
  1. a
  2. a, b
  3. a, c
  4. b, c
  5. a, b, c

Mathematics

Consider circles \(C_1\) and \(C_2\) with centers \(\mathrm{O_1}\) and \(\mathrm{O_2}\), both with a radius of \(\overline{\mathrm{O_1O_2}}\). Figure shows three distinct points \(\mathrm{A, B}\) and \(\mathrm{C}\) on circle \(C_1\), and a point \(\mathrm{D}\) on circle \(C_2\). Points \(\mathrm{A, O_1}\) and \(\mathrm{O_2}\) are on a line, and points \(\mathrm{C, O_2}\) and \(\mathrm{D}\) are on a line.
Let \(\angle\mathrm{BO_1A}\!=\!\theta_1, \angle\mathrm{O_2O_1C}\!=\!\theta_2\) and \(\angle\mathrm{O_1O_2D}\!=\!\theta_3\).
Given that \(\overline{\mathrm{AB}}:\overline{\mathrm{O_1D}}=1:2\sqrt{2}\:\) and \(\:\theta_3=\theta_1+\theta_2\),
The following process computes the ratio of the lengths of line segments \(\mathrm{AB}\) and \(\mathrm{CD}\).
\(\angle\mathrm{CO_2O_1}+\angle\mathrm{O_1O_2D}=\pi\), thus \(\theta_3=\dfrac{\pi}{2}+\dfrac{\theta_2}{2}\).
\(\theta_3=\theta_1+\theta_2\), thus \(2\theta_1+\theta_2\!=\!\pi\) and \(\angle\mathrm{CO_1B}=\theta_1\).
Since \(\angle\mathrm{O_2O_1B}=\theta_1+\theta_2=\theta_3\),
triangles \(\mathrm{O_1O_2B}\) and \(\mathrm{O_2O_1D}\) are congruent.
Let \(\overline{\mathrm{AB}}=k\).
\(\overline{\mathrm{BO_2}}=\overline{\mathrm{O_1D}}=2\sqrt{2}k\), thus \(\overline{\mathrm{AO_2}}=\fbox{\(\;(\alpha)\;\)}\).
\(\angle\mathrm{BO_2A}=\dfrac{\theta_1}{2}\), thus \(\cos\!\dfrac{\theta_1}{2}=\fbox{\(\;(\beta)\;\)}\).
In triangle \(\mathrm{O_2BC}\), we have
\(\overline{\mathrm{BC}}=k, \overline{\mathrm{BO_2}}=2\sqrt{2}k\), and \(\angle\mathrm{CO_2B}=\dfrac{\theta_1}{2}\),
thus \(\overline{\mathrm{O_2C}}=\fbox{\(\;(\gamma)\;\)}\) by the law of cosines.
\(\overline{\mathrm{CD}}=\overline{\mathrm{O_2D}}+\overline{\mathrm{O_2C}} =\overline{\mathrm{O_1O_2}}+\overline{\mathrm{O_2C}}\), therefore
\(\overline{\mathrm{AB}}:\overline{\mathrm{CD}}= k:\Big(\dfrac{\fbox{\(\;(\alpha)\;\)}}{2}+\fbox{\(\;(\gamma)\;\)}\,\Big)\).
Let \(f(t)\) and \(g(t)\) be the correct expressions for \((\alpha)\) and \((\gamma)\) respectively, and let \(p\) be the correct number for \((\beta)\). What is the value of \(f(p)\times g(p)\)? [4 points]
  1. \(\dfrac{169}{27}\)
  2. \(\dfrac{56}{9}\)
  3. \(\dfrac{167}{27}\)
  4. \(\dfrac{166}{27}\)
  5. \(\dfrac{55}{9}\)
Short Answer Questions
Compute \(\log_2 120-\dfrac{1}{\log_{15}2}\). [3 points]
Let \(f(x)\) be a function such that \(f'(x)=3x^2+2x\) and \(f(0)=2\). Compute \(f(1)\). [3 points]

Mathematics

Let \(\{a_n\}\) be a sequence such that
\(\displaystyle\sum_{k\;\!=\;\!1}^{10}a_k - \sum_{k\;\!=\;\!1}^7 \dfrac{a_k}{2} = 56\)
and \(\:\displaystyle\sum_{k\;\!=\;\!1}^{10}2a_k - \sum_{k\;\!=\;\!1}^8 a_k = 100\).
Compute \(a_8\). [3 points]
Given that the function \(f(x)=x^3+ax^2-(a^2-8a)x+3\) increases on the set of all real numbers, compute the maximum value of the real number \(a\). [3 points]
A function \(f(x)\) is differentiable on the set of all real numbers and satisfies the following.
  1. \(f(x)=x\) over the closed interval \([0,1]\).
  2. For some constants \(a\) and \(b\), \(f(x+1)-xf(x)=ax+b\) over the interval \([0,\infty)\).
Compute \(\displaystyle 60\times\!\int_1^2 \!f(x)dx\). [4 points]

Mathematics

A sequence \(\{a_n\}\) satisfies the following.
  1. \(\big|\,a_1\big|=2\)
  2. \(\big|\,a_{n+1}\big| = 2\big|\,a_n\,\big|\) for all positive integers \(n\).
  3. \(\displaystyle\sum_{n\;\!=\;\!1}^{10}a_n=-14\)
Compute \(a_1+a_3+a_5+a_7+a_9\). [4 points]
Consider a cubic function \(f(x)\) with a leading coefficient of \(\dfrac{1}{2}\). For all real numbers \(t\), let \(g(t)\) be the number of real solutions to the equation \(f'(x)=0\) in the closed interval \([t,t+2]\). \(g(x)\) satisfies the following.
  1. \(\displaystyle\lim_{t\;\!\to\;\!\,a+}\!g(t)\!+\!\lim_{t\;\!\to\;\!\,a-}\!g(t)\leq 2\) for all real numbers \(a\).
  2. \(g(f(1))=g(f(4))=2\:\) and \(\:g(f(0))=1\).
Compute \(f(5)\). [4 points]

2022 College Scholastic Ability Test

Mathematics (Prob. & Stat.)

Multiple Choice Questions
In the expansion of the polynomial \((x+2)^7\), what is the coefficient of \(x^5\)? [2 points]
  1. \(42\)
  2. \(56\)
  3. \(70\)
  4. \(84\)
  5. \(98\)
A random variable \(X\) follows the binomial distribution \(\mathrm{B}\!\left(\!n,\dfrac{1}{3}\!\right)\).
If \(\mathrm{V}(2X)=40\), what is the value of \(n\)? [3 points]
  1. \(30\)
  2. \(35\)
  3. \(40\)
  4. \(45\)
  5. \(50\)

Mathematics (Prob. & Stat.)

What is the number of \(5\)-tuples \((a,b,c,d,e)\) where \(a,b,c,d\) and \(e\) are positive integers that satisfy the following? [3 points]
  1. \(a+b+c+d+e=12\)
  2. \(\big|\,a^2-b^2\big|=5\)
  1. \(30\)
  2. \(32\)
  3. \(34\)
  4. \(36\)
  5. \(38\)
There is a sack containing \(10\) cards marked with integers from \(1\) to \(10\) respectively. Suppose \(3\) cards are randomly taken out from this sack at the same time. Let \(a\) be the smallest number among the three integers on those cards. What is the probability that either \(a \leq 4\) or \(a \geq 7\)? [3 points]
  1. \(\dfrac{4}{5}\)
  2. \(\dfrac{5}{6}\)
  3. \(\dfrac{13}{15}\)
  4. \(\dfrac{9}{10}\)
  5. \(\dfrac{14}{15}\)
† Added the variable \(a\) for better readability.

Mathematics (Prob. & Stat.)

The range(distance covered by a single charge) of electric cars produced in some company follows a normal distribution with a mean of \(m\) and a standard deviation of \(\sigma\).
Suppose \(100\) electric cars produced in this company were randomly sampled, and the sample mean of their range was \(\overline{x_1}\). A \(95\%\) confidence interval for \(m\) computed with this sample is \(a\leq m\leq b\).
Suppose \(400\) electric cars produced in this company were randomly sampled, and the sample mean of their range was \(\overline{x_2}\). A \(99\%\) confidence interval for \(m\) computed with this sample is \(c\leq m\leq d\).
Given that \(\overline{x_1}-\overline{x_2}=1.34\) and \(a=c\), what is the value of \(b-a\)?
(※ The unit of range is \(\text{km}\). For a random variable \(Z\) that follows the standard normal distribution, suppose \(\mathrm{P}(|Z| \leq 1.96) = 0.95\) and \(\mathrm{P}(|Z| \leq 2.58) = 0.99\).) [3 points]
  1. \(5.88\)
  2. \(7.84\)
  3. \(9.80\)
  4. \(11.76\)
  5. \(13.72\)
For sets \(X=\{1,2,3,4,5\}\) and \(Y=\{1,2,3,4\}\), what is the number of functions \(f\) from \(X\) to \(Y\) that satisfy the following? [4 points]
  1. \(f(x)\geq \sqrt{x}\) for all elements \(x\) in set \(X\).
  2. The image of the function \(f\) has \(3\) elements.
  1. \(128\)
  2. \(138\)
  3. \(148\)
  4. \(158\)
  5. \(168\)

Mathematics (Prob. & Stat.)

Short Answer Questions
Two absolutely continuous random variables \(X\) and \(Y\) take the value of \(0\leq X\leq6\) and \(0\leq Y\leq6\). The probability density function of \(X\) and \(Y\) are \(f(x)\) and \(g(x)\) respectively. The graph of \(f(x)\) is shown below.
Suppose that
\(f(x)+g(x)=k\) (\(k\) is a constant)
for all \(x\) in \(0\leq x\leq 6\).
Given that \(\mathrm{P}(6k\leq Y\leq 15k)=\dfrac{q}{p}\), Compute \(p+q\).
(※ \(p\) and \(q\) are positive integers that are coprime.) [4 points]
There is a basket containing at least \(10\) white balls and black balls each, and there is an empty sack. Let us perform the following trial with a die.
Throw the die once.
If it lands on a number that is \(5\) or more,
move \(2\) white balls from the basket to the sack.
If it lands on a number that is \(4\) or less,
move \(1\) black ball from the basket to the sack.
Let us repeat this trial \(5\) times. Let \(a_n\) and \(b_n\) be the number of white balls and black balls in the sack after the \(n\)th trial \((1\leq n\leq5)\), respectively.
Given that \(a_5+b_5\geq 7\), the probability that there exists an integer \(k\) \((1\leq k\leq5)\) such that \(a_k=b_k\),
is equal to \(\dfrac{q}{p}\). Compute \(p+q\). (※ \(p\) and \(q\) are positive integers that are coprime.) [4 points]

2022 College Scholastic Ability Test

Mathematics (Calculus)

Multiple Choice Questions
What is the value of \( \displaystyle\lim_{n\;\!\to\;\!\infty} \dfrac{ \dfrac{5}{n}+\dfrac{3}{n^2} }{ \dfrac{1}{n}-\dfrac{2}{n^3} } \)? [2 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
Let \(f(x)\) be a function that is differentiable on the set of all real numbers, such that
\(f\big(x^3+x\big)=e^x\)
for all real numbers \(x\). What is the value of \(f'(2)\)? [3 points]
  1. \(e\)
  2. \(\dfrac{e}{2}\)
  3. \(\dfrac{e}{3}\)
  4. \(\dfrac{e}{4}\)
  5. \(\dfrac{e}{5}\)

Mathematics (Calculus)

Let \(\{a_n\}\) be a geometric progression such that
\(\displaystyle\sum_{n\;\!=\;\!1}^\infty (a_{2n-1}-a_{2n})=3\:\) and \(\:\displaystyle\sum_{n\;\!=\;\!1}^\infty {a_n}^2=6\).
What is the value of \(\displaystyle\sum_{n\;\!=\;\!1}^\infty a_n\)? [3 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
What is the value of \( \displaystyle\lim_{n\;\!\to\;\!\infty}\sum_{k\;\!=\;\!1}^n \dfrac{k^2+2kn}{k^3+3k^2n+n^3} \)? [3 points]
  1. \(\ln5\)
  2. \(\dfrac{\ln5}{2}\)
  3. \(\dfrac{\ln5}{3}\)
  4. \(\dfrac{\ln5}{4}\)
  5. \(\dfrac{\ln5}{5}\)

Mathematics (Calculus)

Suppose a point \(\mathrm{P}\) is moving on the \(xy\)-plane, and its position at time \(t \, (t > 0)\) is the midpoint of the two intersections of the curve \(y=x^2\) and the line \(y=t^2x-\dfrac{\ln t}{8}\). What is the distance that point \(\mathrm{P}\) travels from time \(t=1\) to \(t=e\)? [3 points]
  1. \(\dfrac{e^4}{2}-\dfrac{3}{8}\)
  2. \(\dfrac{e^4}{2}-\dfrac{5}{16}\)
  3. \(\dfrac{e^4}{2}-\dfrac{1}{4}\)
  4. \(\dfrac{e^4}{2}-\dfrac{3}{16}\)
  5. \(\dfrac{e^4}{2}-\dfrac{1}{8}\)
Let \(f(x)=6\pi(x-1)^2\), and let the function \(g(x)\) be
\(g(x)=3f(x)+4\cos f(x)\).
What is the number of values \(x\) in \(0<x<2\) where the function \(g(x)\) has a local minimum? [4 points]
  1. \(6\)
  2. \(7\)
  3. \(8\)
  4. \(9\)
  5. \(10\)

Mathematics (Calculus)

Short Answer Questions
As the figure shows, consider a semicircle whose diameter is the line segment \(\mathrm{AB}\) with a length of \(2\). Let \(\mathrm{P}\) and \(\mathrm{Q}\) be points on arc \(\mathrm{AB}\) such that \(\angle\mathrm{PAB}=\theta\) and \(\angle\mathrm{QBA}=2\theta\). Let \(\mathrm{R}\) be the intersection of lines \(\mathrm{AP}\) and \(\mathrm{BQ}\). Let \(\mathrm{S, T}\) and \(\mathrm{U}\) be points on line segments \(\mathrm{AB, BR}\) and \(\mathrm{AR}\) respectively, such that line \(\mathrm{UT}\) is parallel to line \(\mathrm{AB}\), and triangle \(\mathrm{STU}\) is equilateral. Let \(f(\theta)\) be the area of the region enclosed by the arc \(\mathrm{AQ}\) and line segments \(\mathrm{AR}\) and \(\mathrm{QR}\). Let \(g(\theta)\) be the area of triangle \(\mathrm{STU}\).
Given that \(\displaystyle\lim_{\theta\;\!\to\;\!0+}\!\dfrac{g(\theta)}{\theta\times\!f(\theta)}=\dfrac{q}{p}\sqrt{3}\), compute \(p+q\).
(※ \(0<\theta<\dfrac{\pi}{6}\), and \(p\) and \(q\) are positive integers that are coprime.) [4 points]
A strictly increasing function \(f(x)\) is differentiable on the set of all real numbers, and satisfies the following.
  1. \(f(1)=1\:\) and \(\:\displaystyle\int_1^2f(x)dx=\dfrac{5}{4}\).
  2. Let \(g(x)\) be the inverse of the function \(f(x)\). Then \(g(2x)=2f(x)\) for all \(x\geq1\).
Given that \(\displaystyle\int_1^8xf'(x)dx=\dfrac{q}{p}\), compute \(p+q\).
(※ \(p\) and \(q\) are positive integers that are coprime.) [4 points]

2022 College Scholastic Ability Test

Mathematics (Geometry)

Multiple Choice Questions
In \(3\)-dimensional space, let point \(\mathrm{P}\) be the reflection of point \(\mathrm{A}(2,1,3)\) about the \(xy\)-plane, and let point \(\mathrm{Q}\) be the reflection of point \(\mathrm{A}\) about the
\(yz\)-plane. What is the length of the line segment \(\mathrm{PQ}\)? [2 points]
  1. \(5\sqrt{2}\)
  2. \(2\sqrt{13}\)
  3. \(3\sqrt{6}\)
  4. \(2\sqrt{14}\)
  5. \(2\sqrt{15}\)
A hyperbola \(\dfrac{x^2}{a^2}-\dfrac{y^2}{6}=1\) has a focus with coordinates \((3\sqrt{2}, 0)\). What is the length of the major axis of this hyperbola? (※ \(a\) is a positive number.) [3 points]
  1. \(3\sqrt{3}\)
  2. \(\dfrac{7\sqrt{3}}{2}\)
  3. \(4\sqrt{3}\)
  4. \(\dfrac{9\sqrt{3}}{2}\)
  5. \(5\sqrt{3}\)

Mathematics (Geometry)

Let \(\theta\) be the acute angle between the lines
\(\dfrac{x+1}{2}=y-3\:\) and \(\:x-2=\dfrac{y-5}{3}\)
on the \(xy\)-plane. What is the value of \(\cos\theta\)? [3 points]
  1. \(\dfrac{1}{2}\)
  2. \(\dfrac{\sqrt{5}}{4}\)
  3. \(\dfrac{\sqrt{6}}{4}\)
  4. \(\dfrac{\sqrt{7}}{4}\)
  5. \(\dfrac{\sqrt{2}}{2}\)
Consider the ellipse \(\dfrac{x^2}{64}+\dfrac{y^2}{16}=1\) with two foci \(\mathrm{F}\) and \(\mathrm{F'}\), and a point \(\mathrm{A}\) on the ellipse in the \(1\)st quadrant. Let \(C\) be a circle tangent to the lines \(\mathrm{AF}\) and \(\mathrm{AF'}\) at the same time, whose center is on the \(y\)-axis with a negative \(y\)-coordinate. Let \(\mathrm{B}\) be the center of circle \(C\). Given that the quadrilateral \(\mathrm{AFBF'}\) has an area of \(72\), what is the radius of circle \(C\,\)? [3 points]
  1. \(\dfrac{17}{2}\)
  2. \(9\)
  3. \(\dfrac{19}{2}\)
  4. \(10\)
  5. \(\dfrac{21}{2}\)

Mathematics (Geometry)

Figure shows a cube \(\mathrm{ABCD-EFGH}\) with side lengths of \(4\). Let \(\mathrm{M}\) be the midpoint of the line segment \(\mathrm{AD}\). What is the area of the triangle \(\mathrm{MEG}\)? [3 points]
  1. \(\dfrac{21}{2}\)
  2. \(11\)
  3. \(\dfrac{23}{2}\)
  4. \(12\)
  5. \(\dfrac{25}{2}\)
Let \(\mathrm{F_1}\) be the focus of the parabola \((y-a)^2=4px\), and let \(\mathrm{F_2}\) be the focus of the parabola \(y^2=-4x\), where \(a\) and \(p\) are positive numbers. Let \(\mathrm{P}\) and \(\mathrm{Q}\) be points where the line segment \(\mathrm{F_1F_2}\) meets the two parabolas respectively. Given that \(\overline{\mathrm{F_1F_2}}=3\) and \(\overline{\mathrm{PQ}}=1\), what is the value of \(a^2+p^2\)? [4 points]
  1. \(6\)
  2. \(\dfrac{25}{4}\)
  3. \(\dfrac{13}{2}\)
  4. \(\dfrac{27}{4}\)
  5. \(7\)

Mathematics (Geometry)

Short Answer Questions
Consider a parallelogram \(\mathrm{ABCD}\) on the \(xy\)-plane with \(\overline{\mathrm{OA}}=\sqrt{2}, \overline{\mathrm{OB}}=2\sqrt{2}\) and \(\cos(\angle\mathrm{AOB})=\dfrac{1}{4}\).
A point \(\mathrm{P}\) on the \(xy\)-plane satisfies the following.
  1. \(\overrightarrow{\mathrm{OP}}= s\,\overrightarrow{\mathrm{OA}}+t\,\overrightarrow{\mathrm{OB}} \; (0\leq s\leq1, 0\leq t\leq1)\)
  2. \(\overrightarrow{\mathrm{OP}}\cdot\overrightarrow{\mathrm{OB}}+ \overrightarrow{\mathrm{BP}}\cdot\overrightarrow{\mathrm{BC}}=2\)
Consider a circle with center \(\mathrm{O}\) that passes through point \(\mathrm{A}\). For a point \(\mathrm{X}\) moving on this circle, Let \(M\) and \(m\) be the maximum value and minimum value of \(\big|3\overrightarrow{\mathrm{OP}}-\overrightarrow{\mathrm{OX}}\big|\) respectively.
Given that \(M\times m=a\sqrt{6}+b\), compute \(a^2+b^2\).
(※ \(a\) and \(b\) are rational numbers.) [4 points]
In \(3\)-dimensional space, consider the sphere
\(S:\,(x-2)^2+(y-\sqrt{5})^2+(z-5)^2=25\)
with center \(\mathrm{C}(2,\sqrt{5},5)\) that passes through point \(\mathrm{P}(0,0,1)\). Consider a circle which is the intersection of sphere \(S\) and plane \(\mathrm{OPC}\). Suppose a point \(\mathrm{Q}\) is moving on this circle, and a point \(\mathrm{R}\) is moving on the sphere \(S\). Let \(\mathrm{Q_1}\) and \(\mathrm{R_1}\) be the projections of points \(\mathrm{Q}\) and \(\mathrm{R}\) onto the \(xy\)-plane respectively.
Let us fix points \(\mathrm{Q}\) and \(\mathrm{R}\) such that the area of triangle \(\mathrm{OQ_1R_1}\) is maximized. Then, the projection of triangle \(\mathrm{OQ_1R_1}\) onto plane \(\mathrm{PQR}\) has an area of \(\dfrac{q}{p}\sqrt{6}\). Compute \(p+q\).
(※ \(\mathrm{O}\) is the origin. Points \(\mathrm{O, Q_1}\) and \(\mathrm{R_1}\) are not on a line. \(p\) and \(q\) are positive integers that are coprime.) [4 points]