AMO 2027 Grades 10-12 SET 2 - Self-study

Find the maximum value of the following expression: $$\sqrt{781-X}+\sqrt{X-59}.$$

Enter one whole number, without units.

A quadratic equation of the form $f(x)=x^2+bx+c$ opens upwards and has its vertex 5 units above the $x$-axis and 7 units to the right of the $y$-axis. If a linear equation $g(x)=mx+10$ were added to the original equation to give $h(x)=f(x)+g(x)$, find the $x$-coordinate of the vertex if the graph of $h(x)$ just touches the positive $x$-axis.

Choose one option.

In a class of 40 students, 20 boys and 20 girls, when given a choice to play either football or chess, 5 girls and 12 boys preferred football, with the rest choosing chess. What is the probability that the winner of the chess tournament held between the students who choose chess is a girl if all students are equally likely to win?

Choose one option.

If the following is true: $$\frac{3+\cot78^\circ\cot18^\circ}{\cot78^\circ+\cot18^\circ}=\frac{\tan n+\sin m}{\cos m},$$ where $n$ and $m$ are in degrees and are acute angles, find the product $mn$.

**Editorial clarification.** Determine whether the data force a unique product. Enter that product if unique; otherwise enter NOT UNIQUE. No unstated relation between the two angles is assumed.

Use the response format specified above.

10 lines are drawn on a plane. Find the maximum number of unique line segments that can be identified from these 10 lines on the plane. Note: the line segment identified must be between 2 points that already lie on one of the 10 lines and $AB$ is equivalent to $BA$.

**Editorial clarification.** Count segments whose two distinct endpoints are intersections of the drawn lines and whose entire segment lies on one of those lines. The ten lines are distinct; their arrangement may be chosen to maximize the count.

Enter one whole number, without units.

A person invests 100 dollars every month in a recurring deposit which compounds monthly, promising a 10% increase in the principal invested every year. Find the total value of the recurring deposit after 4 years, or after the 48th installment is paid. (Given $1.1^{1/12}=1.00797$.)

**Editorial clarification.** Use an effective annual growth factor of 1.1 and equal monthly intervals. Value the account immediately after the 48th payment, with no interest yet on that payment. Choose the closest listed monetary amount; the options need not contain the correctly rounded exact total. The given decimal is rounded.

Choose one option.

Find the product of all integers $n\ge1$ such that $\frac{n^3+9}{n^2+13}$ is an integer.

Choose one option.

In the adjoining figure, if the value of the larger 2 circles is 72 and 18, find the radius of the smallest circle. All circles are tangent to each other and to the common ray $B$.



**Editorial clarification.** Interpret the unspecified values 72 and 18 as the radii of the two larger circles. Use the externally tangent configuration on one side of the ray shown in the figure, with the smallest circle in the intervening gap.

Choose one option.

Find $abc$ if $a,b,c$ are distinct prime numbers and $$a+b+c=92\quad\text{and}\quad ab+bc+ca=2201.$$

Enter one whole number, without units.

In the adjoining figure, segment $DE$ is perpendicular to diameter $BC$ of the circle and intersects $BC$ at $P$. Find the length of the line segment if the radius is 10 and $PB=2$.



**Editorial clarification.** The requested segment is the whole chord DE; P is the point on diameter BC shown inside the circle.

Enter one whole number, without units.

If $a>b>1$ and $\frac1{\log_a b}+\frac1{\log_b a}=\sqrt{293}$, find the value of $$\frac1{\log_{ab}b}-\frac1{\log_{ab}a}.$$

Choose one option.

Find the value of $$X=\sum_{n=1}^{7}\tan^2\frac{n\pi}{16}.$$

Enter one whole number, without units.

If $a$ and $b$ are relatively prime positive integers, find $a+b$ if $$\frac{a^3-b^3}{(a-b)^3}=\frac{133}{3}.$$

Enter one whole number, without units.

If $a,b,c\in\mathbb R$, $a+b+c=36$ and $\frac1a+\frac1b+\frac1c=39$, find $$\frac ab+\frac bc+\frac ca+\frac ac+\frac cb+\frac ba.$$

Enter one whole number, without units.

If $N=5!\,8!\,9!$, then find the number of factors of $N$ which are square integers.

Choose one option.