Weekly Contest #92

(Ended)

Standings

Q1.

Find the number of ordered pairs of integers $(a, b)$ such that the sequence\[3, 4, 5, a, b, 30, 40, 50\]is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.

Answer must be a floating-point or integer value and precision error less than 10^-6 is allowed.


Editorial
Tags: Combinatorics Number Theory

Q2.

Find the three-digit positive integer $\underline{a}\,\underline{b}\,\underline{c}$ whose representation in base nine is $\underline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},$ where $a,$ $b,$ and $c$ are (not necessarily distinct) digits.

Answer must be a floating-point or integer value and precision error less than 10^-6 is allowed.


Editorial
Tags: Number Theory

Q3.

Quadratic polynomials $P(x)$ and $Q(x)$ have leading coefficients $2$ and $-2,$ respectively. The graphs of both polynomials pass through the two points $(16,54)$ and $(20,53).$ Find $P(0) + Q(0).$

Answer must be a floating-point or integer value and precision error less than 10^-6 is allowed.


Editorial
Tags: Algebra

Q4.

Let $\triangle ABC$ have side lengths $AB=5$, $BC=9$, $CA=10$. The tangents to circumcircle of $\triangle ABC$ at $B$ and $C$ intersect at point $D$, and $\overline{AD}$ intersects the circumcircle at $P \neq A$. The length of $AP$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$.

Diagram

[asy] import olympiad; unitsize(15); pair A, B, C, D, E, F, P, O; C = origin; A = (10,0); B = (7.8, 4.4899); draw(A--B--C--cycle); draw(A..B..C..cycle, red+dotted); O = circumcenter(A, B, C); E = rotate(90,B) * (O); F = rotate(90,C) * (O); D = IP(B..E + (B-E)*4, C..F + (C-F)*-3); draw(B--D--C--D--A); P = IP(D..A, A..B..C); dot(A); dot(B); dot(C); dot(D); dot(P); label("$A$", A, dir(335)); label("$B$", B, dir(65)); label("$C$", C, dir(200)); label("$D$", D, dir(135)); label("$P$", P, dir(235)); [/asy]

Answer must be a floating-point or integer value and precision error less than 10^-6 is allowed.


Editorial
Tags: Geometry