[ Decimal number system ] [ Divisibility of numbers. General properties ]

Find the last two digits in the decimal representation of the number $1!+2!+\ldots+2001!+2002!$.
From some point, factorials start to be divisible by 100.

## Solution

Notice that 10! is divisible by 100, and \( n! \) for \( n > 10 \) will also be divisible by 100. Therefore, the last two digits of the number \( 1! + 2! + \ldots + 2001! + 2002! \) are the same as the last two digits of the number \( 1! + 2! + \ldots + 9! \). The last two digits of this number can be found directly (the calculations are simplified by the fact that, starting from \( n = 5 \), \( n! \) is divisible by 10).

## Answer

13.

## Angles between angle bisectors ] Problem 53388 Topics: [Sum of angles in a triangle. Theorem about the exterior angle.] [ Arithmetic progression $]$

The angles at vertices \( A, B, C \) of triangle \( ABC \) form an arithmetic progression with a common difference of \( \pi / 7 \). The angle bisectors of this triangle intersect at point \( D \). Points \( A_1, B_1, C_1 \) are located on the extensions of segments \( DA, DB, DC \) beyond points \( A, B, C \) respectively, at the same distance from point \( D \). Prove that the angles \( A_1, B_1, C_1 \) also form an arithmetic progression. Find its common difference.

## Hint

\(\angle BDC = \pi / 2 + 1 / 2 \angle A\).

## Solution

Let \(\angle A = \alpha\) be the middle angle of triangle \( ABC \), \(\angle C = \alpha - \pi / 7\), \(\angle B = \alpha + \pi / 7\). \(\angle B_1DC_1 = \pi / 2 + \alpha / 2\) (see problem \(\underline{55448}\)),

\(\angle DB_1C_1 = \angle DC_1B_1 = 1 / 2 (\pi - \pi / 2 - \alpha / 2) = \pi / 4 - \alpha / 4\). Similarly, \(\angle DA_1C_1 = \angle DC_1A_1 = \pi / 4 - \alpha / 4 - \pi / 28\), \(\angle DA_1B_1 = \angle DB_1A_1 = \pi / 4 - \alpha / 4 + \pi / 28\). Therefore,

\(\angle A_1 = \angle DA_1C_1 + \angle DA_1B_1 = \pi / 2 - \alpha / 2\), \(\angle B_1 = \pi / 2 - \alpha / 2 + \pi / 28\), \(\angle C_1 = \pi / 2 - \alpha / 2 - \pi / 28\).

![](https://cdn.mathpix.com/cropped/2024_05_06_5008c33ea747074d348fg-08.jpg?height=469&width=692&top_left_y=1043&top_left_x=680)

## Answer

\(\pi / 28\).