Ford has an unfair random number generator that generates positive integers. When the generator is run, each positive integer n comes up with probability n(n+1)1. For example, the probability of generating an 8 is 721. Ford calls a positive integer “interdimensional” if it is one less than a triangular number. (The first few interdimensional numbers are 2,5,9,14,….) What is the probability that Ford’s random number generator shows an interdimensional number when run? Express your answer as a common fraction.
How many distinct arrangements of the letters in the word PROBABILITY contain the sub-arrangement PRO, whose letters must appear consecutively and in that order anywhere in the arrangement?
For every positive integer n, let Q(n) be the sum of the greatest integers less than or equal to the square roots of the integers 1,2,3,…,n. For example, Q(3)=1+1+1=3 and Q(10)=1+1+1+2+2+2+2+2+3+3=19. What is the least positive integer n for which Q(1)+Q(2)+Q(3)+⋯+Q(n)≥2026?
A dartboard is the region B in the coordinate plane consisting of points (x,y) such that ∣x∣+∣y∣≤8. A target T is the region where (x2+y2−25)2≤49. A dart is thrown and lands at a random point in B. The probability that the dart lands in T can be expressed as nm⋅π, where m and n are relatively prime positive integers. What is m+n?
Points P and Q are chosen uniformly and independently at random on sides AB and AC, respectively, of equilateral triangle △ABC. Which of the following intervals contains the probability that the area of △APQ is less than half the area of △ABC?
Integers a and b are randomly chosen without replacement from the set of integers with absolute value not exceeding 10. What is the probability that the polynomial x3+ax2+bx+6 has 3 distinct integer roots?
There are n values of x in the interval 0<x<2π where f(x)=sin(7π⋅sin(5x))=0. For t of these n values of x, the graph of y=f(x) is tangent to the x-axis. Find n+t.
There are 8!=40320 eight-digit positive integers that use each of the digits 1,2,3,4,5,6,7,8 exactly once. Let N be the number of these integers that are divisible by 22. Find the difference between N and 2025.
An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is 3, and the area of the trapezoid is 72. Let the parallel sides of the trapezoid have lengths r and s, with r=s. Find r2+s2.
In the drawing below, equilateral triangles are erected on the outside of a square, with shared edges of the square and each triangle coinciding exactly. The total area of the shaded region shown is 1003 units2. What is the total unshaded area inside ABCD?
If the positive integer c has positive integer divisors a and b with c=ab, then a and b are said to be complementary divisors of c. Suppose that N is a positive integer that has one complementary pair of divisors that differ by 20 and another pair of complementary divisors that differ by 23. What is the sum of the digits of N?
Circle C1 and C2 each have radius 1, and the distance between their centers is 21. Circle C3 is the largest circle internally tangent to both C1 and C2. Circle C4 is internally tangent to both C1 and C2 and externally tangent to C3. What is the radius of C4?
Let K be the number of sequences A1,A2,…,An such that n is a positive integer less than or equal to 10, each Ai is a subset of {1,2,3,…,10}, and Ai−1 is a subset of Ai for each i between 2 and n, inclusive. For example, {},{5,7},{2,5,7},{2,5,7},{2,5,6,7,9} is one such sequence, with n=5. What is the remainder when K is divided by 10?
Rows 1,2,3,4, and 5 of a triangular array of integers are shown below: 1117151311151711 Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digit of the sum of the 2023 numbers in the 2023rd row?
Let f be the unique function defined on the positive integers such that ∑d∣nd⋅f(dn)=1 for all positive integers n, where the sum is taken over all positive divisors of n. What is f(2023)?
Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance m with probability 2m1. What is the probability that Flora will eventually land at 10?
A regular pentagon with area 1+5 is printed on paper and cut out. All five vertices are folded to the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
Isosceles trapezoid ABCD has parallel sides AD and BC, with BC<AD and AB=CD. There is a point P in the plane such that PA=1, PB=2, PC=3, and PD=4. What is ADBC?
A triangular number is a positive integer that can be expressed in the form tn=1+2+3+⋯+n, for some positive integer n. The three smallest triangular numbers that are also perfect squares are t1=1=12, t8=36=62, and t49=1225=352. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?
The MATHCOUNTS Question Writing Committee is meeting to finalize the problems for competition year 2026. They are seated around a circular table where the seat of the committee chair, Greg, has already been determined in advance. The remaining six members must sit such that exactly one of the following pairs of writers must be seated next to each other: Thinula and Matthew, Ryoko and Michelle, or Rachel and Liam. In how many ways can the members be seated around the table?
A sequence is said to be non-decreasing when each term is greater than or equal to the term before it. For example, 4,4,4,5,6 is non-decreasing, but 4,4,4,3,5,6 is not because 3 is less than 4, and 3 comes after 4 in the sequence. How many non-decreasing sequences of positive integers with length 2026 terms start with 1 and end with 3?
Xing has an unfair coin with probability 32 of getting heads. What is the expected number of flips he will need to get a head followed by two tails in three consecutive flips? Express your answer as a common fraction.
Chandra writes down all of the nonempty subsets of {1,2,3,…,2025}, and for each subset she then erases everything except the largest element. What is the mean value of all of these largest elements? Express your answer to the nearest integer.
Five of the six edges of a tetrahedron each have length 3 inches, and the sixth edge has length 4 inches. What is the volume of the tetrahedron, in cubic inches? Express your answer in simplest radical form.
Makayla finds all the possible ways to draw a path in a 5×5 diamond-shaped grid. Each path starts at the bottom of the grid and ends at the top, always moving one unit northeast or northwest. She computes the area of the region between each path and the right side of the grid. Two examples are shown in the figures below. What is the sum of the areas determined by all possible paths?
In trapezoid ABCD, angles B and C measure 60∘ and AB=DC. The side lengths are all positive integers, and the perimeter of ABCD is 30 units. How many non-congruent trapezoids satisfy all of these conditions?
Lakshmi has 5 round coins of diameter 4 centimeters. She arranges the coins in 2 rows on a table top, as shown below, and wraps an elastic band tightly around them. In centimeters, what will be the length of the band?
Rodrigo has a very large sheet of graph paper. First he draws a line segment connecting point (0,4) to point (2,0) and colors the 4 cells whose interiors intersect the segment, as shown below. Next Rodrigo draws a line segment connecting point (2000,3000) to point (5000,8000). How many cells will he color this time?
