AMO 2027 Grades 9-11 SET 4 - Self-study

Find $MN-M-N$ if the following identity holds for all real $x$ for which the denominators are nonzero: $$\frac{52x+73}{14x^2+83x+33}=\frac{M}{7x+3}+\frac{N}{2x+11}.$$ Here $M,N$ are constants.

Choose one option.

If $h(x)=2+\sqrt{x}$ and $f(h(x))=7+5\sqrt{x}+x$ for $x\ge0$, find $f(3)$.

Choose one option.

The positive whole numbers are arranged five at a time in each row as shown, with the direction alternating from row to row. What is the sum of the row number and the column number containing 2019? Rows and columns are numbered starting at 1.



Choose one option.

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

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

Enter one positive integer, without units.

How many positive four-digit integers have exactly three positive factors?

Enter one positive integer, without units.

A cricket climbs a staircase by leaping either 1 step or 2 steps upward with each bound. In how many different ways can it climb a 9-step staircase? For example, a 3-step staircase has exactly three ways: $1+1+1$, $1+2$, and $2+1$.

Enter one positive integer, without units.

Samuel cycles at constant speed along a straight street from $A$ to $B$. When he is $\frac6{19}$ of the way from $A$ to $B$, he hears a car approaching from behind $A$ at $418$ km/h. If he immediately turns back, he meets the car at $A$; if he continues toward $B$, the car overtakes him at $B$. How fast is Samuel cycling, in km/h?

Use the idealized source model: the car keeps its speed, and Samuel keeps the same speed in either direction, with no turning delay.

Enter one positive integer, without units.

In triangle $ABC$, $D$ lies on $AB$ with $BD=3AD$, and $E$ lies on $AC$ with $DE\parallel BC$. Point $F$ lies on the line $DE$ as shown, with $AB\parallel CF$. Find $EF$ if $BC=16$.



Choose one option.

Rearrange the digits of 2018 to form all four-digit integers, using each digit exactly once. (0128 is only a three-digit number.) Find the first three digits of the sum of all these integers.

Enter one positive integer, without units.

Seven distinct awards are to be handed to seven students, one award per student, with one intended recipient for each award. Assume all $7!$ assignments are equally likely. The probability that every award goes to an incorrect recipient is $\frac ab$ in lowest terms. Find $a+b$. The corresponding probabilities with five and six awards are given as $\frac{11}{30}$ and $\frac{53}{144}$, respectively.

Enter one positive integer, without units.

Study the figures below. Let $M$ be the total number of dots in figures 1 through 37, inclusive. Find $\frac{M}{37}-37$.

Continue the illustrated diamond-shaped pattern by adding one complete outer diamond layer at each stage, with the same dot spacing. This specifies the intended continuation of the finite picture.



Enter one positive integer, without units.

An ant is at $(7,10)$ and its anthill is at $(19,25)$. What is the shortest distance the ant must travel to reach the anthill if it must touch the $x$-axis, along which sugar is spread? It may touch the axis at any point.

Enter one positive integer, without units.

A rectangular box has volume $70\text{ cm}^3$. Its length, width and height, measured in centimetres, are three different prime numbers. What is its total surface area, in $\text{cm}^2$?

Choose one option.

A sequence starts with 457 as its first term. Each later term is the sum of the cubes of the decimal digits of the preceding term. Find its 2021st term.

Choose one option.

Jessica lies on Wednesdays, Thursdays and Fridays, and tells the truth on every other day. Tim lies on Sundays, Mondays and Tuesdays, and tells the truth on every other day. On what day can both say, “Tomorrow, I will lie”?

Enter the weekday code: Monday = 001, Tuesday = 002, Wednesday = 003, Thursday = 004, Friday = 005, Saturday = 006, Sunday = 007. The original paper asks for shaded digits; here type the code (leading zeros are optional).

Enter one positive integer, without units.