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AMO 2023 – Junior College (Grade 11–12)

Question 1

Evaluate $\sqrt{100000^2+99999+100002}$.

Question illustration
  1. 99999
  2. 100000
  3. 100001
  4. 100002

Question 2

The equation $x^2+x+1=0$ has roots $a,b$. Which equation has roots $a^2,b^2$?

Question illustration
  1. $x^2-x+1=0$
  2. $x^2+x+1=0$
  3. $x^2+2x+1=0$
  4. $x^2+3x+1=0$

Question 3

The vector $\vec a=\begin{pmatrix}2\\4\end{pmatrix}$ is transformed by $M=\begin{pmatrix}0&3\\3&0\end{pmatrix}$. Find $M\vec a$.

Question illustration
  1. $\begin{pmatrix}-12\\4\end{pmatrix}$
  2. $\begin{pmatrix}12\\6\end{pmatrix}$
  3. $\begin{pmatrix}23\\34\end{pmatrix}$
  4. $\begin{pmatrix}-16\\8\end{pmatrix}$

Question 4

Find $\frac1{\sqrt9+\sqrt{11}}+\frac1{\sqrt{11}+\sqrt{13}}+\frac1{\sqrt{13}+\sqrt{15}}+\cdots+\frac1{\sqrt{959}+\sqrt{961}}$.

Question illustration
  1. 10
  2. 12
  3. 14
  4. 16

Question 5

A polynomial $f(x)$ leaves remainder 16 on division by $x-3$ and remainder 2 on division by $x+4$. Find its remainder on division by $x^2+x-12$.

Question illustration
  1. $x-12$
  2. $5x+6$
  3. $2x+10$
  4. $3x-18$

Question 6

For positive integers, define $\operatorname{Euclid}(a,b)$ as follows: if $a\bmod b=0$, return $b$; otherwise return $\operatorname{Euclid}(b,a\bmod b)$. Here $a\bmod b$ is the remainder when $a$ is divided by $b$. What is $\operatorname{Euclid}(36,50)$?

Question illustration

Code notation: def defines a function; if/else selects a branch; return gives its result; print displays it; % means remainder and == tests equality.
  1. 9
  2. 6
  3. 3
  4. 2

Question 7

Write every subset of the set of the eight smallest primes and calculate each subset sum. What is the sum of all these sums?

Question illustration
  1. 10368
  2. 9856
  3. 6898
  4. 5624

Question 8

Rectangle $ABCD$ is divided by perpendicular segments $EF$ and $GH$, meeting at $O$, into four rectangles. The areas of $BEOG,ECHO,GOFA$ are respectively $6,9,8$ square units. Find the area of $OHDF$.

Question illustration
  1. 24 square units
  2. 18 square units
  3. 16 square units
  4. 12 square units

Question 9

Let $f(x)=3x^2-2x+5$, $A=(3,f(3))$, and $B=(3+\delta,f(3+\delta))$ for small nonzero $\delta$. As $\delta\to0$, what value does $R=\frac{f(3+\delta)-f(3)}{\delta}$ approach?

Question illustration
  1. 12
  2. 14
  3. 16
  4. 18