2010 xIMC keystage2-EMIC individual

A billboard reads IMC 2010 INCHEON. Every minute the first character of each word moves to its end: after one minute it reads MCI 0102 NCHEONI. After how many minutes do the original three words first reappear?

What is the sum of the digits of $10^{2010}-2010$?

The notation $d_n$ means $n$ repetitions of digit $d$, with adjacent blocks concatenated. For example $5_3=555$, $4_3 9_5 8_1 3_6=444999998333333$. If $2_w3_x5_y+3_y5_w2_x=5_3 7_2 8_z5_1 7_3$, find $w+x+y+z$.

A man weighs $60$ kg plus one-quarter of his weight. His wife weighs $64$ kg plus one-fifth of hers. Find their absolute weight difference in kg.

Quadrilateral $ABCD$ has $AB=6$, $AD=4$, $BC=7$, $CD=15$ cm. If diagonal $AC$ is an integer number of cm, find it.

The current is $1$ km/h. A boat rows at a constant still-water speed upstream for $3$ hours, then downstream for $2$ hours back to its start. How far away was the turning point?

In $ABCD$, $AB\parallel DC$ and $AD=BC$. Eight identical copies form the hollow regular octagon shown. Find $\angle BAD$ in degrees.

Let $\overline{abc}$ and $\overline{def}$ be different three-digit numbers. If $\overline{abcdef}-\overline{defabc}$ is divisible by $2010$, find the largest possible sum of the two numbers.

Find the average of all distinct nine-digit numbers containing five digits $5$ and four digits $4$.

Rectangle $ABCD$ has $AB=4$, $BC=6$ cm. Points $E,F,G,H$ lie on $AB,BC,CD,DA$, with $AE=CG=3$, $BF=DH=4$ cm. Interior point $P$ satisfies $[AEPH]=5$ cm$^2$. Find $[PFCG]$.

Rice papers $6\times8$ cm are rolled into cylinders, overlapping by $1$ cm. Compare the volume of an $8$-cm-long narrow spring roll with that of a $6$-cm-long wide spring roll, made by rolling the paper the other way.

The largest of $23$ consecutive odd numbers is five times the smallest. Find their average.

Place $1,2,\ldots,9$ once each in the shown nine squares. Every row and column containing three squares sums to $13$; $9$ and $5$ are already entered. Find the starred square.

Each of five students scores $w,x,y$ or $z$ in four subjects, as tabulated. Find the missing Biology class total.

| Subject / Môn | Anna | Gail | Mary | Patty | Susie | Total / Tổng |
|---|---|---|---|---|---|---|
| Algebra / Đại số | w | z | w | z | y | 416 |
| Biology / Sinh học | w | x | y | y | z | ? |
| Chemistry / Hóa học | x | y | y | w | x | 428 |
| Dictation / Chính tả | y | w | z | z | x | 401 |
| Individual total / Tổng cá nhân | 349 | 330 | 349 | 326 | 315 | |

Find the largest positive integer $n$ containing no digit $0$, with digit sum $15$, such that the digit sum of $2n$ is less than $20$.