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SMO 2012 Junior Round 1

Question 1

Let $\alpha,\beta$ be the roots of $x^2+2bx+b=1$. The smallest possible value of $(\alpha-\beta)^2$ is:
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

Question 2

It is known that $n^{2012}+n^{2010}$ is divisible by $10$ for some positive integer $n$. Which number is not a possible value of $n$?
  1. 2
  2. 13
  3. 35
  4. 47
  5. 59

Question 3

Using the vertices of a cube as vertices, how many triangular pyramids can you form?
  1. 54
  2. 58
  3. 60
  4. 64
  5. 70

Question 4

$AB$ is a chord of a circle with centre $O$. The diameter $CD$ is perpendicular to $AB$, with $AB$ closer to $C$ than to $D$. Given $\angle AOB=90^\circ$, find the quotient $[ABC]/[AOD]$.
  1. $\sqrt2-1$
  2. $2-\sqrt2$
  3. $\sqrt2/2$
  4. $(1+\sqrt2)/2$
  5. $1/2$

Question 5

In the diagram, $ABCD$ is a parallelogram, and $AE,BE$ are straight lines. Let $F,G$ be the intersections of $BE$ with $CD,AC$, respectively. Given $BG=EF$, find $DE/AE$.

Question illustration
  1. $(3-\sqrt5)/2$
  2. $(3-\sqrt6)/2$
  3. $(3-\sqrt6)/4$
  4. $(3-\sqrt7)/2$
  5. $(3-\sqrt7)/4$

Question 6

Four circles each of radius $x$ and a square are arranged within a circle of radius $8$ as shown. What is the range of $x$?

Question illustration
  1. $0\lt x\lt 4$
  2. $0\lt x\lt 8(\sqrt2+1)$
  3. $4-2\sqrt2\lt x\lt 4$
  4. $4-2\sqrt2\lt x\lt 8(\sqrt2-1)$
  5. $4-\sqrt2\lt x\lt 4(\sqrt2+1)$

Question 7

Adam has a triangular field $ABC$ with $AB=5$, $BC=8$, $CA=11$. He plans a straight fence from $A$ to a point $D$ on $BC$ so that $AD$ bisects $\angle BAC$. Find the area of the part $ABD$.
  1. $4\sqrt{21}/11$
  2. $4\sqrt{21}/5$
  3. $5\sqrt{21}/11$
  4. $5\sqrt{21}/4$
  5. None of the above.

Question 8

For a real number $x$, let $\lfloor x\rfloor$ be the largest integer at most $x$, and let $\{x\}=x-\lfloor x\rfloor$. Real numbers $a,b$, with $b\ne0$, satisfy
$$a=b\left\lfloor\frac ab\right\rfloor-b\left\{\frac ab\right\}.$$
Which statement is incorrect?
  1. If b is an integer, then a is an integer.
  2. If a is a nonzero integer, then b is an integer.
  3. If b is rational, then a is rational.
  4. If a is a nonzero rational number, then b is rational.
  5. If b is an even number, then a is an even number.

Question 9

Given that $y$, defined by the expression below, is an integer, which statement is incorrect?

Question illustration

Editorial transcription note: the numerator's special symbols are corrupted in the supplied PDF and cannot be transcribed reliably. The original formula is retained here for review; no definite replacement is assumed.
  1. x can be any nonzero integer.
  2. x can be any positive number.
  3. x can be any negative number.
  4. y can take the value 2.
  5. y can take the value −2.

Question 10

Teachers $A,B,C$ work in different schools $X,Y,Z$ and specialize in different subjects: Mathematics, Latin and Music. It is known that:
(i) $A$ does not teach Mathematics and $B$ does not work in $Z$;
(ii) the teacher in $Z$ teaches Music;
(iii) the teacher in $X$ does not teach Latin;
(iv) $B$ does not teach Mathematics.
Which statement is correct?
  1. B works in X and C works in Y.
  2. A teaches Latin and works in Z.
  3. B teaches Latin and works in Y.
  4. A teaches Music and C teaches Latin.
  5. None of the above.

Question 11

Let $a,b$ be real numbers such that $a>b$, $2^a+2^b=75$ and $2^{-a}+2^{-b}=12^{-1}$. Find $2^{a-b}$.

Question 12

Find the sum of all positive integers $x$ for which $\frac{x^3-x+120}{(x-1)(x+1)}$ is an integer.

Question 13

Consider $\sqrt{3x^2-8x+1}+\sqrt{9x^2-24x-8}=3$. The largest root is $-k$ times the smallest root. Find $k$.

Question 14

Find the four-digit number $\overline{abcd}$ satisfying $2\overline{abcd}+1000=\overline{dcba}$. (For example, if $a=1,b=2,c=3,d=4$, then $\overline{abcd}=1234$.)

Question 15

Suppose real numbers $x,y$ satisfy $x^2+y^2-22x-20y+221=0$. Find $xy$.

Question 16

Let $m,n$ be positive integers satisfying $mn^2+876=4mn+217n$. Find the sum of all possible values of $m$.

Question 17

For real $x$, let $\lfloor x\rfloor$ be the largest integer at most $x$. Find $\lfloor x\rfloor$ for the smallest $x$ satisfying $\lfloor x^2\rfloor-\lfloor x\rfloor^2=100$.

Question 18

Suppose real numbers $x_1,x_2,\ldots,x_{49}$ satisfy $x_1^2+2x_2^2+\cdots+49x_{49}^2=1$. Find the maximum of $x_1+2x_2+\cdots+49x_{49}$.

Question 19

Find the minimum value of
$$\sqrt{x^2+(20-y)^2}+\sqrt{y^2+(21-z)^2}+\sqrt{z^2+(20-w)^2}+\sqrt{w^2+(21-x)^2}.$$

Question 20

Let $A$ be a four-digit integer. Increase its first (leftmost) and third digits by $n$, and decrease its second and fourth digits by $n$. The resulting number is $nA$. Find $A$.