Algebra
Geometry
Combinatorics
/Probability
/Probability
Number Theory
Mystery Box
100
There are integers 
and 
each greater than 
such that
for all 
. What is 
?
B. 3
100
Mary divides a circle into 1212 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
A. 5 B.6 C. 8 D. 10 E. 12
12
100
Using the letter A,M,O,S, and U we can form 120 five-letter "words". If these words are arranged in alphabetical order, then the word USAMO occupires postions
A. 112
B.113
C.114
D.115
E.116
D.115
100
How many ordered pairs of integers (m,n) satisfy (n^2−49)^1/2=m
A. 1
B. 2
C. 3
D. 4
E. infinitely many
D. 4
100
100
proof the Ceva’s Theorem
200
Which of the following is equivalent to
3^128 - 2^128
200
Let points 
and 
. Quadrilateral 
is cut into equal area pieces by a line passing through 
. This line intersects 
at point 
, where these fractions are in lowest terms. What is 
?
58
200
The numbers from 1 to 8 are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?
18
200
The digits 1, 2, 3, 4, and 5 are each used once to write a five-digit number P QRST. The three-digit number P QR is divisible by 4, the three-digit number QRS is divisible by 5, and the three-digit number RST is divisible by 3. What is P?
1
200
The tetradecagon is a polygon with how many sides?
14
300
A rectangular floor measures 
by feet, where and are positive integers with 
. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width 
foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair 
?
2
300
Six spheres of radius are positioned so that their centers are at the vertices of a regular hexagon of side length 
. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?
3/2
300
How many different integers can be expressed as the sum of three distinct members of the set 
?
13
300
In year 
, the 
day of the year is a Tuesday. In year 
, the 
day is also a Tuesday. On what day of the week did the 
th day of year 
occur?
Thursday
300
A number that is not algebraic, meaning, more or less, not the root of a finite polynomial with rational coefficients, is known by what term?
Transcendental Number
400
Find 
if 
and 
are integers such that 
.
588
400
In 
, 
, and 
. A circle with center and radius 
intersects 
at points 
and 
. Moreover 
and 
have integer lengths. What is 
?
61
400
For some particular value of , when 
is expanded and like terms are combined, the resulting expression contains exactly 
terms that include all four variables 
and 
, each to some positive power. What is ?
14
400
Let S be a subset of {1, 2, 3, ..., 50} such that no pair of distinct elements in S has a sum divisible by 7. What is the maximum number of elements in S?
23
400
Lagrange points L1, L2, and L3 are points of gravitational equilibrium between the Earth and the Moon. They are named after the Italian Joseph Lagrange, but he is actually the 2nd person to discover these points, with _____ already discovering them around a decade prior in 1750.
Euler
500
Let
Which of the following polynomials is a factor of 
?
x^6 + x³+1
500
is a square of side length 
. Point 
is on 
such that 
. The square region bounded by is rotated 
counterclockwise with center , sweeping out a region whose area is 
, where , , and 
are positive integers and 
. What is 
?
19
500
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
47/256
500
(1959 IMO #1) Prove that the fraction 21n+4 14n+3 is irreducible for every natural number n.
Euclidean Algo ahh proof
500
If you get this one, you’ll be all that and a…can of chips? What geometric shape is a Pringle?
Hyperbolic Paraboloid