SEAMO X 2022 — Paper E — Grade 9–10

Given $\binom n{r-1}=715$, $\binom nr=1287$ and $\binom n{r+1}=1716$, find $r$.

An unbiased coin is weighted so that the probability of heads on the $n$th toss is $1/2^n$. In $10$ tosses, let $p$ be the probability of exactly $9$ heads and $1$ tail. Find $2^{55}p$.

A $6$-metre truck travels through a tunnel. If its length is doubled and its speed reduced by $20\%$, the time for the whole truck to pass through the tunnel increases by $50\%$. Find the tunnel length in metres.

In triangle $ABC$, $AB=9$, $BC=8$, $CA=7$. The bisector of $\angle BAC$ meets $BC$ at $D$. A circle through $A$, tangent to $BC$ at $D$, meets $AB$ again at $X$ and $AC$ again at $Y$. Find $XY$.

Let $\alpha,\beta$ be the roots of $x^2-20x-22=0$. If $\dfrac{1-\alpha}{1+\alpha}+\dfrac{1-\beta}{1+\beta}=-m/n$ with coprime positive integers $m,n$, find $m+n$.

Find the minimum value of $y=(x^2+4x+5)(x^2+4x+1)+3x^2+12x+5$ for real $x$.

Find the sum of all reduced fractions with denominator $3$ that lie strictly between $1$ and $20$.

How many fractions $p/q$ in lowest terms satisfy $0\lt p/q\lt 1$ and $pq=20!$?

How many real roots does $x|x-1|-4|x|+3=0$ have?

One number is randomly selected from all multiples of $4000$. What is the probability that it is divisible by $25000$?

Thirteen fractions use the integers $1,2,\ldots,26$ each exactly once as a numerator or denominator. At most how many fractions can simplify to integers?

Triangle $ABC$ is isosceles with $AB=AC$. Point $D$ on $AC$ is such that $BD$ bisects $\angle ABC$. Given $BC=BD+AD$, find $\angle BAC$ in degrees.

Tev chose five numbers from $\{1,2,3,4,5,6,7\}$. Knowing their product would not suffice for Terry to decide whether their sum is even or odd. Find their product.

Primes $p,q$ are such that $x^2-px+q=0$ has distinct positive integer roots. Find $p+q$.

Given $\tan A=12/5$, $\cos B=-3/5$ and that $A,B$ are in the same quadrant, find $\cos(A-B)$.