SASMO 2014 Secondary 4 (Grade 10) Contest

Tom must travel from Town A to Town B and he plans to go at a certain
E speed. If he travels at a speed 5 km/h faster than planned, he will arrive 5
hours earlier and if he travels at a speed 10 km/h faster than planned, he will
arrive 8 hours earlier. What is his planned speed?

: On the island of nobles and liars, 50 people are standing in a queue. Everyone,
= except the first person in the queue, said, that the person before him in the
ES | queue is a liar. The first man in the queue, said, that all people, standing after
him are liars. How many liars are in the queue? (Nobles always speak the
E truth, and liars always tell lies.)

In a football league with 6 teams (A, B, C, D, E and F), each team plays each of
the other teams exactly once. So far, A has played one match, B has played 2
matches, C has played 3 matches, D has played 4 matches, and E has played 5
matches. How many matches has F played so far?
SS ee

How many integers n › 3 are there such that there exists a convex n-gon
whose angles are in ratio 1: 2:3: ...:n?

A sequence of numbers has 6 as its first term, and every term after the first is
defined as follows:

e If aterm Ais an even number, then the following term in the sequence is é 2

e If aterm B is an odd number, then the following term is 3B +1.
If the first four terms in our sequence are 6, 3, 10 and 5, what is the 200" term
of the sequence?

Tom rolls a standard 6-sided dice. Jerry rolls another standard 6-sided dice.
Tom wins if the values shown on their dice differ by 1. What is the probability
that Tom wins?
eee BN Senet 1 SS, eee a
;
o 8 2

1 oS =

ee

How many integers n (1 ‹n ‹ 200) are there such that (5n)5" is a perfect
square?

How many positive integers not exceeding 2013 are multiples of 3 and 4 but
not 5?

‘|

How many right-angled triangles can be formed by joining three vertices of a
given regular 14-gon?
fel

The first element of the sequence a, = 0, and if n › 1 then any = ay +
(-1)" x n. Find the value of k if a, = 2013.

How many 10-digit numbers only composed of 1, 2, and 3 exist, in which any
two neighbouring digits differ by 1?

Four different numbers a, b, c and d are chosen from the list —1,—2, —3, —4,
and —5. What is the largest possible value of a? + c*
ee) SEE Seae eS a S SSO es MM See eee SS Soca haere
i
@ = :

=

FSS] See eee
/

i Evaluate the following sum
ee :
QVT+1v2 3V24+2V3 4V3 + 3V4 2013V2012 + 2012V2013
f
i.

f Ifx?+5=5 and x › 0, find x5 +,
:

Let 5 be the least positive integer such that 10S is a perfect square and 65 is a
{ perfect cube. How many positive factors does the number S have?

i
SSS SS SS Se

Two circles have their centers on the same diagonal of a a
square. They touch each other and the sides of the J
square as shown. The square has a side of length 1 cm.
What is the sum of the lengths of the radii of the circles?

What is the last digit of

15 + 28 4 35 +--+ 20135 + 20148

-
How many 7-digit numbers contain at least one 7?
‘ 2 2

Se SSS eS SSS SSS

Evaluate (in the simplest form)

2013 + 28 [2013 + 28/2013 + 28V...

How many zeros does the following product
1000 x 999 x 998 xX ..X3X2X1

end with?