AIME

Conducting Body: Mathematical Association of America (MAA)
Eligibility: By invitation based on AMC 10/12 scores
Syllabus: Advanced algebra, geometry, trigonometry, number theory, combinatorics
Question Type: 15 integer-answer questions (0-999)
Duration: 3 hours
Marking: 1 point per correct answer, no negative marking
Frequency: Annually, with two testing dates
Qualifies For: USA Mathematical Olympiad (USAMO/USAJMO)

The American Invitational Mathematics Examination (AIME) is the second stage in the series of contests used to select the United States team for the International Mathematical Olympiad. It is an invitational exam for high school students who achieve a top score on the AMC 10 or AMC 12 competitions. A student's AIME score is combined with their AMC score to determine qualification for the USA Mathematical Olympiad (USAMO) and USA Junior Mathematical Olympiad (USAJMO).

The paper consists of 15 questions to be answered in three hours. Unlike the AMC, the AIME is not a multiple-choice test. Every answer is an integer from 0 to 999, and students must derive this number themselves and write it on the answer sheet. There is no partial credit awarded for showing work.

The AIME is substantially more difficult than the AMC. The problems require a deeper understanding of topics like number theory, geometry, and combinatorics, and often need creative, multi-step solutions. The integer-answer format makes the exam particularly challenging, as a small arithmetic error leads to a completely wrong answer. Managing time is also difficult, with an average of only twelve minutes available for each complex problem.

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From a mathematics paper rather than AIME, worked the way every question on this site is. AIME questions are being added.

JEE Advanced 2009

Let (x, y, z) be points with integer coordinates satisfying the system of homogeneous equations $3x - y - z = 0, -3x + z = 0, -3x + 2y + z = 0.$ Then the number of such points for which $x^{2} + y^{2} + z^{2} \le 100$ is ______.

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$3x - y - z = 0,\quad -3x + z = 0,\quad -3x + 2y + z = 0$

$x, y, z \in \mathbb{Z}$

$\text{To find: number of points with } x^2 + y^2 + z^2 \le 100$

$\text{Step 1: solve the system} \Rightarrow \text{all integer solutions } (x, y, z)$

$\text{Step 2: keep those with } x^2 + y^2 + z^2 \le 100$

$-3x + z = 0 \Rightarrow z = 3x$

$3x - y - z = 0 \Rightarrow 3x - y - 3x = 0$

$\Rightarrow y = 0$

$-3x + 2y + z = -3x + 0 + 3x = 0\ \checkmark$

$(x, y, z) = (t, 0, 3t),\quad t \in \mathbb{Z}$

$\text{line through the origin, direction } (1, 0, 3)$

$x^2 + y^2 + z^2 \le 100$

$t^2 + 0 + (3t)^2 = 10t^2 \le 100$

$t^2 \le 10 \Rightarrow |t| \le 3$

$t \in \{-3, -2, -1, 0, 1, 2, 3\}$

$\text{Number of values of } t = 7$

$(t, 0, 3t):\ t = -3,\dots,3 \Rightarrow 7 \text{ points}$

$\boxed{7}$

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