Ze Committee ZeMC

Ze Committee Ze Math Competition

1st Annual

ZIME
Tuesday, January 16th, 2024
INSTRUCTIONS
1. DO NOT OPEN THIS BOOKLET UNTIL YOU TELL YOU TO BEGIN.
2. This is a 15 question, 3 hour examination. All answers are integers ranging from 000 to 999, inclusive. Your
score will be the number of correct answers. There is neither partial credit nor penalties for wrong answers.
3. Despite the name, Ze Invitational Mathematics Examination is open for anybody to take. There will not
be a 2024 ZeJMO/ZeMO.
4. Submit your answers by PMing them through AoPS to “ihatemath123“, or DMing them through Discord
to ”bennywang“. If you use Discord, please specify your AoPS username.
You may format your answers in any way, as long as it is clear which problem each answer corresponds to.
If you wish to remain anonymous on the leaderboard, or wish to remain anonymous if your score is below a
certain threshold, make sure to specify this in your message.
DO NOT edit your message; you may be considered for cheating.
5. You should receive a response with your score and distribution within 24 hours, in addition to a link with
access to a private discussion forum.
6. Only blank scratch paper, rulers, protractors, and erasers are allowed as aids. Calculators, Dotted Cac-
ulators, grid paper and lined paper are NOT allowed. No problems on the contest require the use of a
calculator.
7. Figures are not necessarily drawn to scale.

The Ze Committee ZIME Office reserves the right to disqualify scores from an individual if it determines that the rules or the
nonexistent required security procedures were not followed.

The publication, reproduction, or communication of the problems or solutions of this exam during the period when students are
eligible to participate seriously jeopardizes the integrity of the results. Dissemination via phone, email, or digital media of any
type during this period is a violation of the competition rules.
1. If two positive palindromes (numbers which read the same forwards and backwards) sum to 2024, find the
sum of all possible values for the smaller palindrome.

2. Find the number of ways that the set {1, 2, 3, . . . , 18} can be split into two indistinguishable sets of nine
elements, such that one set has a median of 6 and the other set has a median of 12.

3. Evan thinks of two positive integers. Their quotient, which leaves no remainder, divides their sum, and their
product is 784,000. Find the remainder when the absolute difference between Evan’s two numbers is divided
by 1000.

4. Julie picks positive reals b and x with b ̸= 1 and writes down the logarithm logb x. If she were to erase b
and replace it with 2b , the value of the logarithm would increase by 12. Instead, if she were to erase b and
replace it with 2b, the value of the logarithm would decrease by 9. Find log2 (x).

5. Find the number of ways to split an eight by eight square into five rectangles with integer side lengths, no
two of which share more than one vertex in common. (Rotations and reflections are considered distinct.)

6. Let m and b be real numbers. Distinct points A, B , C and D lie on the line y = mx + b in that order,
equally spaced. Given that A and C lie on the parabola y = x2 + 9x + 19 and B and D lie on the parabola
y = x2 + x + 15, find mb.

7. In △ABC , points D, E and F lie on segments BC , AC and AB such that BD = DC = 5, DE√

bisects
∠ADC and DF bisects ∠ADB . If DE = 6 and DF = 2, the length DA can be expressed as c , where
a+ b

a, b and c are positive integers with gcd(a, c) = 1. Find a + b + c.

8. If a, b and c are complex numbers such that



|a| + b + c = 7i
a + |b| + c = 9i


a + b + |c| = 10i,
find |a + b + c|2 .

9. Parallelogram ABCD has an area of 350 and satisfies AB = 35. Let F and G be points in the interior of
the parallelogram such that F G = 24 and F G ∥ AB . If there exists an ellipse with foci F and G tangent
to all four sides of the parallelogram, find BC 2 .
10. Alex and Oron are playing a game. They take turns spinning a fair spinner with ten sectors of equal size,
numbered 1, 2, . . . , 10. Alex goes first. After the first spin, if a player spins a number less than or equal
to the number previously spun, the game ends and the other player wins. The probability that Oron loses
the game can be expressed as m n
, where m and n are coprime integers. Find the remainder when m + n is
divided by 1000.

11. Let n and k be positive integers such that the sum of the n smallest perfect powers of k (including 1) is a
multiple of 1001. Find the number of possible values of n less than 1001.

12. Points D and E lie on sides AB and AC of △ABC , respectively, such that the circumcircles ω1 and ω2
of △ABE and △ADC , respectively, meet on side BC . Line DE meets ω1 and ω2 at points X and Y ,
respectively,
√
such that BC = 14, XY = 19, BD = 6 and CE = 9. If the length DE can be expressed as
a− b
c
, where a, b and c are positive integers with gcd(a, c) = 1, find a + b + c.

13. Let {an } and {bn } be two sequences of real numbers such that a1 = 20, b1 = 23 and
 √

ai+1 = |ai · bi | + i 2 i
a +b


b √
i+1 = |ai · bi | − ai +bi
2

for all positive integers i. Find the smallest integer k for which ak = bk .

14. Alexandre forms a piece of cookie dough in the shape of a regular hexagon with a side length of 8 cm, and
places it at the center of a square baking pan with a side length of 50 cm, as shown in the diagram below.
He then drops a circular cookie cutter with a radius of 7 cm randomly and uniformly inside the baking pan,
such that the entire cutter lies within the pan. The expected number of pieces that the cookie dough gets
cut into can be expressed as m n
for coprime positive integers m and n. Find m + n.

(For example, in the diagram below, the cookie is cut into two pieces. If the cookie cutter does not touch
the dough, the cookie dough is in 1 piece.)

15. Let △ABC be an acute triangle with circumcenter O. Let OB and OC be the circumcenters of △BOA and
△COA, respectively, and let P be the circumcenter of △OOB OC . If the circumradii of △ABC , △OOB OC
and △P BC are 9, 15 and 11, respectively, find AP 2 .
 Ze Committee ZeMC

ZIME
DO NOT OPEN UNTIL Tuesday, January 16th, 2024

**Administration on an earlier date will probably occur.**

• None of the information needed to administer this competition is contained in the ZIME
Teacher’s Manual. PLEASE DO NOT READ THE MANUAL AS IT DOES NOT EXIST.
• Answer sheets must be returned to the Ze Committee ZeMC office within 2.9 seconds of the
competition administration. Use an overnight or 2-day shipping service, with a tracking
number, to guarantee the timely arrival of these answer sheets. If you wish for all of the
answer sheets to get thrown in an incinerator, USPS overnight is strongly recommended.
• The publication, reproduction or communication of the problems or solutions of this com-
petition during the period when students are eligible to participate seriously jeopardizes
the integrity of the results. Dissemination via phone, email, friends (if you have them),
or digital media of any type during this period is a violation of competition rules.

The ZeMC competition series is made possible by the contributions of the following
problem-writers and test-solvers:

asbodke, ayush_agarwal, bissue, Geometry285, ihatemath123, kante314, OronSH,

peace09, P_Groudon, Significant and Turtwig113.
Thank you for taking our mock AIME!