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MATHEMATICAL CIRCLE S (RUSSIA N EXPERIENCE )

Problem 6 . I n a certai n yea r ther e wer e exactl y fou r Priday s an d exactl y fou r

Mondays i n January . O n wha t da y o f th e wee k di d th e 20t h o f Januar y fal l tha t

year?

Problem 7 . Ho w man y boxe s ar e crosse d b y a diagona l i n a rectangula r tabl e

formed b y 199 x 991 small squares ?

Problem 8 . Cros s out 10 digits fro m th e numbe r 123451234512345123451234 5 s o

that th e remainin g numbe r i s as larg e a s possible .

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Problem 9 . Pete r said : "Th e da y befor e yesterda y I was 10, but I will turn 13 in

the nex t year. " I s this possible ?

Problem 10. Pete' s ca t alway s sneeze s befor e i t rains . Sh e sneeze d today . "Thi s

means i t wil l be raming, " Pet e thinks . I s he right ?

Problem 11. A teache r dre w severa l circle s o n a shee t o f paper . The n h e aske d

a studen t "Ho w man y circle s ar e there? " "Seven, " wa s th e answer . "Correct ! So ,

how many circle s are there?" th e teacher aske d anothe r student . "Five, " answere d

the student . "Absolutel y right! " replie d th e teacher . Ho w many circle s were reall y

drawn o n the sheet ?

Problem 12. Th e so n of a professor's fathe r i s talking t o the fathe r o f the profes -

sor's son, and the professor doe s not tak e part i n the conversation. I s this possible ?

Problem 13. Thre e turtles ar e crawling along a straight roa d headin g in the sam e

direction. "Tw o othe r turtle s ar e behin d me, " say s th e firs t turtle . "On e turtl e i s

behind m e and on e other i s ahead," say s the second. "Tw o turtles ar e ahead o f me

and on e other i s behind," say s th e thir d turtle . Ho w ca n thi s b e possible ?

Problem 14. Thre e scholars are riding in a railway car. Th e train passes through a

tunnel for several minutes, an d they are plunged int o darkness. Whe n they emerge ,

each of them sees that th e faces of his colleagues are black with the soot that flew in

through th e ope n window. The y star t laughin g a t eac h other, but , al l of a sudden ,

the smartes t o f them realize s tha t hi s fac e mus t b e soile d too . Ho w does h e arriv e

at thi s conclusion ?

Problem 15. Thre e tablespoons of milk from a glass of milk are poured into a glass

of tea, an d th e liqui d i s thoroughly mixed . The n thre e tablespoon s o f this mixtur e

are poure d bac k int o th e glas s o f milk . Whic h i s greate r now : th e percentag e o f

milk i n the te a o r th e percentag e o f tea i n the milk ?

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Problem 16. For m a magie square with the digit s 1, 2, 3, 4, 5, 6, 7, 8, and 9 ; tha t

is, plac e the m i n th e boxe s o f a 3 x 3 tabl e s o tha t al l th e sum s o f th e number s

along th e rows , columns, an d tw o diagonals ar e equal .

Problem 17. I n a n arithmeti c additio n proble m th e digit s wer e replace d wit h

letters (equa l digit s b y sam e letters , an d differen t digit s b y differen t letters) . Th e

result is : LOVE S + LIV E = THERE . Ho w many "loves " ar e "there" ? Th e answe r

is the maximu m possibl e valu e of the wor d THERE .

Problem 18. Th e secre t servic e o f Th e Federatio n intercepte d a code d messag e

from Th e Dominio n which read: BLASÃ‰+LBS A = BASES . I t i s known that equa l