AMO 2027 Grades 10-12 SET 1 - Self-study

Find the ordered set of 4-tuples, of positive integers, $[w,x,y,z]$ such that: $$w+x+y+z=21.$$

**Editorial clarification.** The source wording asks for the set, while its key gives a count. For the answer box, enter the number of such ordered tuples. The solution also describes the full set.

Enter one whole number, without units.

If $x+\frac1x=\sqrt3$, then find the value of $x^3+\frac1{x^3}$.

**Editorial clarification.** Interpret $x$ as a nonzero complex number; the original source does not specify a domain.

Choose one option.

In the adjoining figure, 4 semi-circles are constructed and the area of overlap between any 2 semi-circles is shaded purple. Find the area of the shaded region if the side of the square is 10.



Choose one option.

In the adjoining figure, the areas of the regions are mentioned in the figure (not to scale). Find the value of $m+n$, if $\frac mn$ is the unknown area $x$ of the quadrilateral, in lowest terms.



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.

Find the sum of all the prime factors of $N$ if: $$\log_3\!\left(\log_5\!\left(\frac N2\right)\right)=6.$$

**Editorial clarification.** Count each distinct prime divisor once, as intended by the source options.

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.

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.

Find the value of $$X=\cos(57^\circ)\cos(27^\circ)+\sin(57^\circ)\sin(27^\circ),$$ if $\cos(57^\circ)=0.5446$, $\cos(27^\circ)=0.8910$, and all values inside sine and cosine are in degrees.

**Editorial clarification.** Treat the printed decimal cosine values as rounded approximations. Use the stated angles exactly.

Choose one option.

**Authorized replacement.** The original Grades 10–12 source has no Question 9 stem. This question is supplied from Grades 9–11 SET 1 Q13 of the 2027 self-study edition (original booklet SET 1 Q15, statement PDF page 10, worked solution page 35). It is a replacement, not a reconstruction. The orphan source Q9 key A does not apply to it.

Given $f:\mathbb{R}\to\mathbb{R}$ is a quadratic polynomial with $$f(1)=1,\qquad f(2)=\frac12,\qquad f(3)=\frac13,$$ find $f(4)$.



Choose one option.

Find the following sum if $i=\sqrt{-1}$: $$S=i+2i^2+3i^3+\cdots+2020i^{2020}+2021i^{2021}.$$

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.

If $\frac1{x+1}+\frac{12}{y+12}+\frac{144}{z+144}=1$, then find the value of $$\frac{x^2}{x^2+x}+\frac{y^2}{y^2+12y}+\frac{z^2}{z^2+144z}.$$

**Editorial clarification.** Assume every displayed fraction is defined. The six quantities $x$, $x+1$, $y$, $y+12$, $z$, $z+144$ must all be nonzero.

Choose one option.

A tower 40 m tall has an elevation of 45 degrees from a point $B$ on the ground, and 53 degrees from a point $C$ on the ground. If point $A$ lies on the base of the tower 40 m tall, what is the distance between $B$ and $C$ on the ground if $\angle BAC=90^\circ$? (Given $\sin37^\circ=\frac34$.)

**Editorial clarification.** For this self-study item, use $\tan37^\circ\approx3/4$ (equivalently $\cot53^\circ\approx3/4$) instead of the source’s printed $\sin37^\circ=3/4$, and choose the distance under this approximation. Assume a vertical tower and level ground.

Choose one option.

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.