AMO 2027 Grades 9-11 SET 2 - Self-study

Find the greatest positive integer that divides 298, 348 and 873 with the same remainder in each case.



Enter one positive integer, without units.

Let $S$ be the smallest positive integer such that $7S$ has 2019 digits in base ten. Find the tens digit of $S$.

Enter one positive integer, without units.

Simplify: $$\sqrt[5]{9^{10}}+\sqrt{(-2)^6}.$$ Use the nonnegative square root.



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What is the greatest number of regions into which 7 straight lines can divide the inside of a circle?

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How many unique necklaces can be made with 7 equally spaced beads of 7 different colours, using each colour once? As in the source method, necklaces that differ only by rotation or by turning the necklace over are considered the same.



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Rationalise the denominator of
$$\frac1{\sqrt6-\sqrt3+\sqrt2+1}.$$
The printed source offers the following expressions, retained unchanged for comparison:
- A. $\dfrac{\sqrt6+\sqrt3-\sqrt2+1}{2}$
- B. $\dfrac{\sqrt6-\sqrt3+\sqrt2+1}{2}$
- C. $\sqrt6-\sqrt3+\sqrt2+1$
- D. $\sqrt6+\sqrt3-\sqrt2+1$
- E. $\sqrt6-\sqrt3-\sqrt2+1$

For this edition, submit your rationalised expression rather than an option letter. Write the result as $(a+b\sqrt2+c\sqrt3+d\sqrt6)/e$, where the five coefficients are integers with no common factor greater than 1 and $e>0$. Enter a;b;c;d;e in that order, separated by semicolons, with no spaces. Keep every sign, including negative coefficients.

The first five terms of a sequence are $5,9,16,28,47$. Find the sum of the eighth and seventh terms.
Use the continuation intended in the source: the third differences remain constant. A first difference is the next term minus the preceding term; repeat this operation to obtain second and third differences.

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The points $(0,1)$ and $(1,2)$ are vertices of a square. Find the sum of the $y$-coordinates of all possible vertices diagonally opposite to $(0,1)$. The two given vertices are not required to be adjacent; include every possible square.

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What is the greatest number of consecutive integers whose sum is $-100$?

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If $f(x)=2^x+86$ and $g(x)=3x^2+x-4$, find $$g\!\left[f^{-1}(g(14))\right].$$ Here $f^{-1}$ denotes the inverse function, not the reciprocal of $f$.



Enter one positive integer, without units.

The spinner shown has five equal sectors labelled 1, 2, 3, 4 and 5. Spin it once. If it lands on a perfect square, flip a fair coin once. If it lands on a prime number, flip the coin twice. What is the probability of seeing at least one head? A fair coin has probability $1/2$ of heads on each flip.
Assume independent flips, independent of the spinner. Write the probability as $p/q$ in lowest terms with $q>0$, then enter $p+q$.



Enter one positive integer, without units.

Find the positive difference between the sum of all possible integer $x$ values and the sum of all possible integer $y$ values satisfying $x^y=256$.
Use the source's convention: sum the $x$-coordinates and the $y$-coordinates over all distinct ordered solution pairs $(x,y)$, counting a repeated coordinate once for each pair in which it appears.

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Consider negative integers as Cold Cubes and positive integers as Hot Cubes. Adding 5 Hot Cubes raises a liquid's temperature by five degrees, and adding 5 Cold Cubes lowers it by five degrees. How many Cold Cubes must be removed from a liquid at $-97$ degrees to reach $-42$ degrees? Use this integer model, in which each cube changes the temperature by one degree.



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Given $$x^4+x^3+x^2+x+1=0,$$ find the value of $x^{2021}$.
Treat $x$ as an algebraic or complex number satisfying the equation; a real solution is not assumed. Preserve the printed target exactly. For this edition, enter the simplest expression in terms of $x$, using a lowercase letter, rather than forcing a numerical value.

How many perfect square numbers are strictly between $7^6$ and $4^8$? Do not include either endpoint.



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