SEAMO 2027 - Paper D - SET 09 - Self-study

Question 1

SEAMO self-study edition 2027; not an official 2027 paper. Original source years are retained. 25 questions, 90 minutes; Q1–10: 3 points, Q11–20: 4 points, Q21–25: 6 points, total 100. Original response types and all subparts are preserved. Verified is an editorial convention; source defects, necessary assumptions and unresolved mathematical uncertainty are disclosed.

How many consecutive zeros are at the end of the decimal representation of $70!$?

Question 2

Let $x,y$ be real numbers satisfying $$y=\sqrt{\frac{2008x+2009}{2010x-2011}}+\sqrt{\frac{2008x+2009}{2011-2010x}}+2010.$$ Find $y$.

2010
2011
2012
2025
None of the above

Question 3

In triangle $ABC$, $E$ and $D$ lie in the interiors of sides $AC$ and $BC$, respectively. In the figure, $AF$ bisects $\angle CAD$ and $BF$ bisects $\angle CBE$. Prove $$\angle AEB+\angle ADB=2\angle AFB.$$

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Question 4

Let $m,n$ be integers with $m\gt n$, $m+n=20$ and $m^2+n^2=328$. Find $m^2-n^2$.

Question 5

Evaluate $$\frac12+\frac24+\frac38+\frac4{16}+\cdots+\frac{10}{2^{10}}.$$

$\frac{509}{256}$
$\frac{513}{256}$
$\frac{356}{512}$
$\frac{418}{512}$
$\frac{47}{512}$

Question 6

A bag contains 7 black, 5 white and 4 orange balls. Four balls are drawn without replacement. Find the probability that at least 3 are white.

$\frac{23}{362}$
$\frac1{18}$
$\frac{23}{363}$
$\frac1{19}$
None of the above

Question 7

For rational $m$, find the smallest possible value of $$|m-1|+|m-3|+|m+5|+|m+6|.$$

Question 8

$a_1,a_2,a_3,\ldots$ is an arithmetic sequence with common difference 1. Given $a_1+a_2+\cdots+a_{98}=137$, find $a_2+a_4+\cdots+a_{98}$.

89
91
93
97
None of the above

Question 9

15 countries took part in the World Cup. Each team played exactly 2 matches. A win earns 3 points, a loss 0 points, and a draw 1 point for each team. At most how many more points did the champion score than the runner-up? Original choices: (A) 48; (B) 51; (C) 53; (D) 56; (E) 59.

Source inconsistency: the printed two-match rule and the supplied choices are incompatible. Answer the printed question in the response box; the original choices are retained for reference.

Question 10

Square $ABCD$ has side length $2\sqrt{15}$. Points $E,F$ are the midpoints of $AB,BC$, respectively. Line $AF$ meets $DE$ at $M$ and $DB$ at $N$. Find the area of $\triangle DMN$.

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8
9
10
11
None of the above

Question 11

In the cryptarithm shown, different letters represent different digits. Find the digit represented by $B$.

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Question 12

In triangle $ABC$, $D$ lies on $BA$ with $BD:DA=1:2$, and $E$ lies on $CB$ with $CE:EB=1:4$. Segments $DC$ and $AE$ meet at $F$. Express $CF:FD$ using two relatively prime positive integers.

$1:2$
$3:8$
$5:8$
$2:3$
None of the above

Question 13

Evaluate $$\frac{2023^2+2123^2-2(2023)(2123)}{25}.$$

Question 14

Solve $\log_x(2x^2-7x+12)=2$.

$0,1$
$1$
$-2,3$
$2,3$
$3,4$

Question 15

Evaluate $$\frac{(4\times7+2)(6\times9+2)(8\times11+2)\cdots(100\times103+2)}{(5\times8+2)(7\times10+2)(9\times12+2)\cdots(99\times102+2)}.$$

Question 16

Given $x/2=y/3=z/5$, where $x,y,z$ are nonzero, find $$\frac{xy-2yz+xz}{x^2-2y^2+z^2}.$$

$-\frac{11}{13}$
$-\frac{14}{11}$
$0$
$\frac{11}{14}$
$\frac{14}{25}$

Question 17

The quadrants $OAD$ and $OBC$ have radii 5 cm and 10 cm, respectively, and $\angle COD=30^\circ$. Find the area of the shaded region $ABCD$ in $\text{cm}^2$.

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$8\pi$
$\frac{35\pi}4$
$\frac{55\pi}4$
$15\pi$
$\frac{75\pi}4$

Question 18

If $|x-10|+x-10=0$, what is the range of $x$?

$x<10$
$x>10$
$x\le10$
$x\ge10$
None of the above

Question 19

How many six-digit positive integers are divisible by 36 and have their last two digits equal?

Question 20

How many squares are in the following figure, in which every cell is a square?

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28
29
30
31
None of the above