2012 xIMC keystage2-EMIC team

The nine circles contain distinct consecutive positive integers. Each of the seven drawn lines has total $23$. The fixed left-middle number is $6$; top-right is less than bottom-right. Restore the numbers.

The first column of a tablet is $0,1,2,\ldots$. Along row one, increments are $1$; row two $2$; row three $3$; and so on. The surviving fragments are the horizontal triple $(?,2012,2023)$ and vertical triple $(2012,2683,?)$. Find the sum of missing entries.

A row starts with $2012$, then one $1$, then $2012$, then two $1$s, then $2012$, then three $1$s, and so on. Find the sum of its first $2012$ entries.

Andrea answers one-third of a test incorrectly, Barbara answers seven incorrectly, and one-fifth of all questions are answered incorrectly by both. What is the maximum number answered correctly by both?

Five distinct positive integers give ten pairwise products. The smallest is $28$, largest $240$, and $128$ is also a product. Find their sum.

Square $MNPQ$ lies inside rectangle $ABCD$, with parallel corresponding sides and $AB-BC=7$ cm. The total area of trapezoids $ABNM,CDQP$ is $123$ cm$^2$, while $ADQM,BCPN$ total $312$ cm$^2$. Find the square area.

Two companies initially have equal employee counts. The first increases its workforce to eleven times its original size, while the second lays off eleven employees. The first new count is a multiple of the second. Find the maximum original count.

In square $ABCD$, points $K,L,M,N$ divide $BC$ into five equal segments. $E$ is on $AD$ with $AE=BK$. Find $\angle AKE+\angle ALE+\angle AME+\angle ANE+\angle ACE$.

Fill the seven empty squares of the $3\times3$ table with $2,3,4,5,6,7,9$, once each. The fixed entries are $1$ top-left and $8$ middle-right. All four $2\times2$ subtable sums must be equal. Find all solutions.

At each month\'s start an adult red ant produces three baby black ants. Each adult black ant eats one baby black, produces three baby reds, then dies; there are enough babies to eat. Babies mature during the month. Christmas counts are $9000000$ red and $1000000$ black. What was the difference of red and black counts one Christmas earlier?