MathDB

2020 AIME Problems

Part of AIME Problems

Subcontests

(15)
11
2

Ordered triples

For integers aa, bb, cc, and dd, let f(x)=x2+ax+bf(x) = x^2 + ax + b and g(x)=x2+cx+dg(x) = x^2 + cx + d. Find the number of ordered triples (a,b,c)(a,b,c) of integers with absolute values not exceeding 1010 for which there is an integer dd such that g(f(2))=g(f(4))=0g(f(2)) = g(f(4)) = 0.

Polynomials.....again

Let P(x)=x2−3x−7P(x) = x^2 - 3x - 7, and let Q(x)Q(x) and R(x)R(x) be two quadratic polynomials also with the coefficient of x2x^2 equal to 11. David computes each of the three sums P+QP + Q, P+RP + R, and Q+RQ + R and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If Q(0)=2Q(0) = 2, then R(0)=mnR(0) = \dfrac mn, where mm and nn are relatively prime positive integers. Find m+nm+n.
6
2
4
2

More factoring

Let SS be the set of positive integers NN with the property that the last four digits of NN are 20202020, and when the last four digits are removed, the result is a divisor of NN. For example, 42,02042,020 is in SS because 44 is a divisor of 42,02042,020. Find the sum of all the digits of all the numbers in SS. For example, the number 42,02042,020 contributes 4+2+0+2+0=84+2+0+2+0=8 to this total.

Triangle Rotation

Triangles △ABC\triangle ABC and △A′B′C′\triangle A'B'C' lie in the coordinate plane with vertices A(0,0)A(0,0), B(0,12)B(0,12), C(16,0)C(16,0), A′(24,18)A'(24,18), B′(36,18)B'(36,18), and C′(24,2)C'(24,2). A rotation of mm degrees clockwise around the point (x,y)(x,y), where 0<m<1800<m<180, will transform △ABC\triangle ABC to △A′B′C′\triangle A'B'C'. Find m+x+ym+x+y.
3
2
7
2

What is Vandermonde

A club consisting of 1111 men and 1212 women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as 11 member or as many as 2323 members. Let NN be the number of such committees that can be formed. Find the sum of the prime numbers that divide NN.

Sphere in Cone Intersection

Two congruent right circular cones each with base radius 33 and height 88 have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance 33 from the base of each cone. A sphere with radius rr lies inside both cones. The maximum possible value for r2r^2 is mn\frac mn, where mm and nn are relatively prime positive integers. Find m+nm+n.
9
2

Dividing Divisors

Let SS be the set of positive integer divisors of 209.20^9. Three numbers are chosen independently and at random from the set SS and labeled a1,a2,a_1,a_2, and a3a_3 in the order they are chosen. The probability that both a1a_1 divides a2a_2 and a2a_2 divides a3a_3 is mn,\frac mn, where mm and nn are relatively prime positive integers. Find m.m.

Did Anybody Not Bash

While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.
10
2
2
2

I could have used these three minutes on #10...

There is a unique positive real number xx such that the three numbers log⁡8(2x),log⁡4x,\log_8(2x),\log_4x, and log⁡2x,\log_2x, in that order, form a geometric progression with positive common ratio. The number xx can be written as mn,\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.

Slopes...

Let PP be a point chosen uniformly at random in the interior of the unit square with vertices at (0,0),(1,0),(1,1)(0,0), (1,0), (1,1), and (0,1)(0,1). The probability that the slope of the line determined by PP and the point (58,38)\left(\frac58, \frac38 \right) is greater than 12\frac12 can be written as mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
5
2

Ordered Cards

Six cards numbered 1 through 6 are to be lined up in a row. Find the number of arrangements of these six cards where one of the cards can be removed leaving the remaining five cards in either ascending or descending order.

Sums of digits in multiple bases

For each positive integer nn, let f(n)f(n) be the sum of the digits in the base-four representation of nn and let g(n)g(n) be the sum of the digits in the base-eight representation of f(n)f(n). For example, f(2020)=f(133210four)=10=12eightf(2020) = f(133210_\text{four}) = 10 = 12_\text{eight}, and g(2020)=the digit sum of 12eight=3g(2020) = \text{the digit sum of } 12_\text{eight} = 3. Let NN be the least value of nn such that the base-sixteen representation of g(n)g(n) cannot be expressed using only the digits 00 through 9.9. Find the remainder when NN is divided by 1000.1000.
14
2

Quadratic iteration

Let P(x)P(x) be a quadratic polynomial with complex coefficients whose x2x^2 coefficient is 11. Suppose the equation P(P(x))=0P(P(x))=0 has four distinct solutions, x=3,4,a,bx=3,4,a,b. Find the sum of all possible values of (a+b)2(a+b)^2.

Sad Fractional Parts

For real number xx let ⌊x⌋\lfloor x\rfloor be the greatest integer less than or equal to xx, and define {x}=x−⌊x⌋\{x\}=x-\lfloor x\rfloor to be the fractional part of xx. For example, {3}=0\{3\}=0 and {4.56}=0.56\{4.56\}=0.56. Define f(x)=x{x}f(x)=x\{x\}, and let NN be the number of real-valued solutions to the equation f(f(f(x)))=17f(f(f(x)))=17 for 0≤x≤20200\leq x\leq 2020. Find the remainder when NN is divided by 10001000.
15
2

Orthocenter tangent

Let ABCABC be an acute triangle with circumcircle ω\omega and orthocenter HH. Suppose the tangent to the circumcircle of △HBC\triangle HBC at HH intersects ω\omega at points XX and YY with HA=3HA=3, HX=2HX=2, HY=6HY=6. The area of △ABC\triangle ABC can be written as mnm\sqrt n, where mm and nn are positive integers, and nn is not divisible by the square of any prime. Find m+nm+n.

Projections and Tangents

Let △ABC\triangle ABC be an acute scalene triangle with circumcircle ω\omega. The tangents to ω\omega at BB and CC intersect at TT. Let XX and YY be the projections of TT onto lines ABAB and ACAC, respectively. Suppose BT=CT=16BT=CT=16, BC=22BC=22, and TX2+TY2+XY2=1143TX^2+TY^2+XY^2=1143. Find XY2XY^2.
13
2

Angle bisector

Point DD lies on side BCBC of △ABC\triangle ABC so that AD‾\overline{AD} bisects ∠BAC\angle BAC. The perpendicular bisector of AD‾\overline{AD} intersects the bisectors of ∠ABC\angle ABC and ∠ACB\angle ACB in points EE and FF, respectively. Given that AB=4AB=4, BC=5BC=5, CA=6CA=6, the area of △AEF\triangle AEF can be written as mnp\tfrac{m\sqrt n}p, where mm and pp are relatively prime positive integers, and nn is a positive integer not divisible by the square of any prime. Find m+n+pm+n+p.

Makes me think of Brianchon's

Convex pentagon ABCDEABCDE has side lengths AB=5AB=5, BC=CD=DE=6BC=CD=DE=6, and EA=7EA=7. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of ABCDEABCDE.
1
2
12
2

LTE or Binomial Theorem

Let nn be the least positive integer for which 149n−2n149^n - 2^n is divisible by 33⋅55⋅773^3 \cdot 5^5 \cdot 7^7. Find the number of positive divisors of nn.

Large grid

Let mm and nn be odd integers greater than 1.1. An m×nm\times n rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers 11 through nn, those in the second row are numbered left to right with the integers n+1n + 1 through 2n2n, and so on. Square 200200 is in the top row, and square 20002000 is in the bottom row. Find the number of ordered pairs (m,n)(m,n) of odd integers greater than 11 with the property that, in the m×nm\times n rectangle, the line through the centers of squares 200200 and 20002000 intersects the interior of square 1099.1099.
8
2

A Bug's Life

A bug walks all day and sleeps all night. On the first day, it starts at point OO, faces east, and walks a distance of 5 units due east. Each night the bug rotates 60∘60 ^\circ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to point PP. Then OP2=mnOP^2 = \frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm + n.

Scary recursive function

Define a sequence of functions recursively by f1(x)=∣x−1∣f_1(x) = |x-1| and fn(x)=fn−1(∣x−n∣)f_n(x)=f_{n-1}(|x-n|) for integers n>1n > 1. Find the least value of nn such that the sum of the zeros of fnf_n exceeds 500,000500{,}000.