## Problem of the Week Description

## Problems curated by Problem Czar, **Andy Tang**

Solve fun and interesting AMC-level problems to develop your skills! We will go over the weekly problem at the beginning of the following meeting.

Submit answers through online form on website and earn points for both correct answers and elegant solutions.

Current Problem of the Week

**11/30:**

Let N be the coefficient of x^2 in (1 + x + x^2)(1 + 2x + 2x^2) (1 + 3x + 3x^2)...(1+20x+20x^2). What is the remainder when N is divided by 1000?

## Past Problems of the Week

**9/14**:

Compute the number of distinct 11 letter palindromic permutations of the letters of MISSISSIPPI.

**9/21**:

In acute-angled triangle

*ABC*,

*m<A*= (x+15)°,

*m<B*= (2x-6)°, and the exterior angle at C has measure (3x+9)°. Compute the number of possible integral values of

*x*.

**9/28: Relay!:**

Set #2 P1. Square ABCD is inscribed in a circle. Square EFGH has vertices E and F on CD and vertices G and H on the circle. Find the ratio of the area of square ABCD to the area of square EFGH.

Set #2 P2. Let T = TNYWR. Find the number of 0s at the right end of the decimal representation of 1!2!3!4!...(T-1)!T!

Set #2 P3. Let T = TNYWR. The function f has the property such that for all real numbers, f(x) + f(x-1) + f(x-2) = x^2. Given that, f(T) = 125, find f(125).

*Provide answers/solutions to all the questions that you solve*

**in one form submission**

**10/5:**Compute the number of ordered triples (x,y,z), 1729 < x, y, z < 1999 which satisfy

x^2 + xy + y^2 = y^3 - x^3 and yz + 1 = y^2 + z

**10/12:**How many integers n with 10 ≤ n ≤ 500 have the property that the hundreds digit of 17n and 17n+ 17 are different?

**10/19:**In isosceles triangle ABC with base BC of length 23 cm, points P and Q are chosen on side BC with BP = QC = 9 cm. If segments AP and AQ trisect angle BAC, what is the perimeter of triangle ABC?

**10/26:**

Define an increasing sequence a1, a2, a3, ..., ak of integers to be n-true if it satisfies the following conditions for positive integers n and k:

i) The difference between any two consecutive terms is less than n.

ii) The sequence must start with 0 and end with 10. How many 5-true sequences exist?

**11/2:**

Find the sum of all integers y on the interval 0 < y < 100 such that for some positive integer x, (x+sqrt(y)) + 1/(x+sqrt(y)) is an integer.

**11/9:**

In rectangle ABCD, AB = 2016 and BC = 1. Points E and F lie on segment AB such that BE = 2BF. The midpoints of EC and FD are M and N, respectively. If MN = 1000, what is the area of quadrilateral ENMB?

**11/16:**

How many ways can we pick four 3-element subsets of {1, 2, ..., 6} so that each pair of subsets share exactly one element?