2021년 12월 18일 토요일

서울대 수시합격자 10명 중 4명 특목고·영재교·자사고 출신…일반고 줄어

 

서울 고교 출신 31.4%…여학생 44.3%로 예년과 비슷


서울대 정문 전경. /뉴스1 © News1 황덕현 기자


올해 서울대 수시 합격생 10명 중 4명 이상이 특수목적고(과학고·외국어고·국제고·예술고·체육고)나 영재학교, 자사고 출신인 것으로 나타났다.

서울대가 17일 발표한 '2022학년도 수시모집 선발 결과'에 따르면 서울대 수시 합격자(2391명) 가운데 특목고·영재학교·자사고 출신이 42.5%(1017명)에 달하는 것으로 조사됐다. 

올해 서울대 수시 합격생 10명 중 4명 이상이 특수목적고(과학고·외국어고·국제고·예술고·체육고)나 영재학교, 자사고 출신인 것으로 나타났다.

서울대가 17일 발표한 '2022학년도 수시모집 선발 결과'에 따르면 서울대 수시 합격자(2391명) 가운데 특목고·영재학교·자사고 출신이 42.5%(1017명)에 달하는 것으로 조사됐다. 

합격생의 출신 학교 유형별로는 일반고 합격자 비중이 줄고 특목고 및 영재학교, 자사고 출신이 늘었다. 

일반고 출신 비중은 지난해 48.3%에서 1.6%포인트 줄었다. 그 다음으로 영재고가 13.1%였고 자사고 12.1%, 외국어고 9.2%, 예술·체육고 6.3%, 과학고 6.1%, 자율형공립고 3.7%, 국제고 2.0%, 검정고시 0.4%, 특성화고 0.3% 순이었다. 

영재고 출신은 지난해 12.0%보다 1.1%포인트 증가했고 외국어고도 지난해(8.6%)보다 0.6%포인트 늘었다. 

합격생을 1명 이상 배출한 고교는 809곳으로 2020학년도(875곳)에 비해 크게 줄었다. 다만 최근 3년 동안 합격생이 없었던 81개 일반고도 합격생을 배출했다고 서울대는 설명했다. 

올해 서울대 수시 합격생의 31.4%는 서울에 있는 고교를 나왔다. 중소도시 소재 고교를 나온 입학생은 37.1%, 광역시 고교 출신은 26.7%였다. 

올해 수시 합격생 가운데 여학생은 44.3%로 지난해 44.3%, 2020년 44.6% 등과 비슷했다.

2022학년도 수시모집 합격생의 등록 기간은 18일부터 21일 오후 4시까지다. 미등록 인원이 생기면 두 차례에 걸쳐 충원 합격자를 발표할 예정이다. 

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2021년 12월 13일 월요일

2021 Fall AMC 12B 문제

 

Problem 1

What is the value of $1234+2341+3412+4123?$

$(\textbf{A})\: 10{,}000\qquad(\textbf{B}) \: 10{,}010\qquad(\textbf{C}) \: 10{,}110\qquad(\textbf{D}) \: 11{,}000\qquad(\textbf{E}) \: 11{,}110$

Solution

Problem 2

What is the area of the shaded figure shown below?[asy] size(200); defaultpen(linewidth(0.4)+fontsize(12)); pen s = linewidth(0.8)+fontsize(8); pair O,X,Y; O = origin; X = (6,0); Y = (0,5); fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2)); for (int i=1; i<7; ++i) { draw((i,0)--(i,5), gray+dashed); label("${"+string(i)+"}$", (i,0), 2*S); if (i<6) { draw((0,i)--(6,i), gray+dashed); label("${"+string(i)+"}$", (0,i), 2*W); } } label("$0$", O, 2*SW); draw(O--X+(0.15,0), EndArrow); draw(O--Y+(0,0.15), EndArrow); draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5); [/asy]

$(\textbf{A})\: 4\qquad(\textbf{B}) \: 6\qquad(\textbf{C}) \: 8\qquad(\textbf{D}) \: 10\qquad(\textbf{E}) \: 12$

Solution

Problem 3

At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$

$(\textbf{A})\: 10\qquad(\textbf{B}) \: 30\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 100\qquad(\textbf{E}) \: 120$

Solution

Problem 4

Let $n=8^{2022}$. Which of the following is equal to $\frac{n}{4}?$

$(\textbf{A})\: 4^{1010}\qquad(\textbf{B}) \: 2^{2022}\qquad(\textbf{C}) \: 8^{2018}\qquad(\textbf{D}) \: 4^{3031}\qquad(\textbf{E}) \: 4^{3032}$

Solution

Problem 5

Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?

$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

Solution

Problem 6

The largest prime factor of $16384$ is $2$ because $16384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16383$?

$\textbf{(A)} \: 3\qquad\textbf{(B)} \: 7\qquad\textbf{(C)} \: 10\qquad\textbf{(D)} \: 16\qquad\textbf{(E)} \: 22$

Solution

Problem 7

Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation\[x(x-y)+y(y-z)+z(z-x) = 1?\]

$\textbf{(A)} \: x>y$ and $y=z$

$\textbf{(B)} \: x=y-1$ and $y=z-1$

$\textbf{(C)} \: x=z+1$ and $y=x+1$

$\textbf{(D)} \: x=z$ and $y-1=x$

$\textbf{(E)} \: x+y+z=1$

Solution

Problem 8

The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle?

$\textbf{(A)} \: 105 \qquad\textbf{(B)} \: 120 \qquad\textbf{(C)} \: 135 \qquad\textbf{(D)} \: 150 \qquad\textbf{(E)} \: 165$

Solution

Problem 9

Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$?

$\textbf{(A)} \: 9\pi \qquad\textbf{(B)} \: 12\pi \qquad\textbf{(C)} \: 18\pi \qquad\textbf{(D)} \: 24\pi \qquad\textbf{(E)} \: 27\pi$

Solution

Problem 10

What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\cos(40^\circ),\sin(40^\circ))$, $(\cos(60^\circ),\sin(60^\circ))$, and $(\cos(t^\circ),\sin(t^\circ))$ is isosceles?

$\textbf{(A)} \: 100 \qquad\textbf{(B)} \: 150 \qquad\textbf{(C)} \: 330 \qquad\textbf{(D)} \: 360 \qquad\textbf{(E)} \: 380$

Solution

Problem 11

Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6{ }$ numbers obtained. What is the probability that the product is divisible by $4?$

$\textbf{(A)}\: \frac34\qquad\textbf{(B)} \: \frac{57}{64}\qquad\textbf{(C)} \: \frac{59}{64}\qquad\textbf{(D)} \: \frac{187}{192}\qquad\textbf{(E)} \: \frac{63}{64}$

Solution

Problem 12

For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of n is divided by n. For example,\[f(14)=(1+2+7+14)\div 14=\frac{12}{7}\]What is $f(768)-f(384)?$

$\textbf{(A)}\ \frac{1}{768} \qquad\textbf{(B)}\ \frac{1}{192} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \frac{4}{3} \qquad\textbf{(E)}\ \frac{8}{3}$

Solution

Problem 13

Let $c = \frac{2\pi}{11}.$ What is the value of\[\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?\]

$\textbf{(A)}\ -1 \qquad\textbf{(B)}\ \frac{\sqrt{-11}}{5} \qquad\textbf{(C)}\ \frac{\sqrt{11}}{5} \qquad\textbf{(D)}\ \frac{10}{11} \qquad\textbf{(E)}\ 1$

Solution

Problem 14

Suppose that $P(z), Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z)=R(z)$. What is the minimum possible value of $N$?

$\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 5$

Solution

Problem 15

Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c$?

IMAGE

$(\textbf{A})\: 75\qquad(\textbf{B}) \: 93\qquad(\textbf{C}) \: 96\qquad(\textbf{D}) \: 129\qquad(\textbf{E}) \: 147$

Solution

Problem 16

Suppose $a$, $b$, $c$ are positive integers such that\[a+b+c=23\]and\[\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.\]What is the sum of all possible distinct values of $a^2+b^2+c^2$?

$(\textbf{A})\: 259\qquad(\textbf{B}) \: 438\qquad(\textbf{C}) \: 516\qquad(\textbf{D}) \: 625\qquad(\textbf{E}) \: 687$

Solution

Problem 17

A bug starts at a vertex of a grid made of equilateral triangles of side length $1$. At each step the bug moves in one of the $6$ possible directions along the grid lines randomly and independently with equal probability. What is the probability that after $5$ moves the bug never will have been more than $1$ unit away from the starting position?

$\textbf{(A)}\ \frac{13}{108} \qquad\textbf{(B)}\ \frac{7}{54} \qquad\textbf{(C)}\ \frac{29}{216} \qquad\textbf{(D)}\ \frac{4}{27} \qquad\textbf{(E)}\ \frac{1}{16}$

Solution

Problem 18

Set $u_0 = \frac{1}{4}$, and for $k \ge 0$ let $u_{k+1}$ be determined by the recurrence\[u_{k+1} = 2u_k - 2u_k^2.\]

This sequence tends to a limit; call it $L$. What is the least value of $k$ such that\[|u_k-L| \le \frac{1}{2^{1000}}?\]

$(\textbf{A})\: 10\qquad(\textbf{B}) \: 87\qquad(\textbf{C}) \: 123\qquad(\textbf{D}) \: 329\qquad(\textbf{E}) \: 401$

Solution

Problem 19

Regular polygons with $5$, $6$, $7$, and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?

$(\textbf{A})\: 52\qquad(\textbf{B}) \: 56\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 64\qquad(\textbf{E}) \: 68$

Solution

Problem 20

A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)

$(\textbf{A})\: 7\qquad(\textbf{B}) \: 8\qquad(\textbf{C}) \: 9\qquad(\textbf{D}) \: 10\qquad(\textbf{E}) \: 11$

Solution

Problem 21

For real numbers $x$, let\[P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)\]where $i = \sqrt{-1}$. For how many values of $x$ with $0\leq x<2\pi$ does\[P(x)=0?\]

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

Solution

Problem 22

Right triangle $ABC$ has side lengths $BC=6$, $AC=8$, and $AB=10$.

A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. What is $OP$?

$\textbf{(A)}\ \frac{23}{8} \qquad\textbf{(B)}\ \frac{29}{10} \qquad\textbf{(C)}\ \frac{35}{12} \qquad\textbf{(D)}\ \frac{73}{25} \qquad\textbf{(E)}\ 3$

Solution

Problem 23

What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{ 1, 2, 3, …, 30\}$? (For example the set $\{1, 17, 18, 19, 30\}$ has $2$ pairs of consecutive integers.)

$\textbf{(A)}\ \frac{2}{3} \qquad\textbf{(B)}\ \frac{29}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\ \frac{29}{30} \qquad\textbf{(E)}\ 1$

Solution

Problem 24

Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$, and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$. The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $B$ and $F \ne B$. What is $CF$?

$\textbf{(A) } 28 \qquad \textbf{(B) } 20\sqrt{2} \qquad \textbf{(C) } 30 \qquad \textbf{(D) } 32 \qquad \textbf{(E) } 20\sqrt{3}$

Solution

Problem 25

For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

Solution


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