ESAT Challenge Mathematics 1 Esat-maths1-challenge-2

10 questions10 marks20Updated August 2026

The ESAT Challenge Mathematics 1 Esat-maths1-challenge-2 paper in full: all 10 questions, each with its answer. ESAT is the Engineering and Science Admissions Test. Sit it cold under exam timing, mark it, then work back through anything you missed using the solutions below.

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Question 1

1 mark

How many ordered pairs of positive integers

(a,b)

satisfy

\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{1}{20}

A. $8$
B. $12$
C. $15$
D. $21$
E. $30$

Answer: C

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Question 2

1 mark

What are the last two digits of

7^{2026}

A. $01$
B. $07$
C. $43$
D. $93$
E. $49$

Answer: E

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Question 3

1 mark

A right-angled triangle has sides of integer length, and its area is numerically equal to its perimeter. How many such triangles are there, up to congruence?

A. $1$
B. $2$
C. $3$
D. $4$
E.infinitely many

Answer: B

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Question 4

1 mark

A fair six-sided die is rolled repeatedly until a six first appears. What is the probability that the number of rolls required is even?

A. $\tfrac{5}{11}$
B. $\tfrac12$
C. $\tfrac{5}{12}$
D. $\tfrac{6}{11}$
E. $\tfrac56$

Answer: A

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Question 5

1 mark

In triangle

ABC

, the point

D

lies on

BC

with

BD:DC = 1:2

, and the point

E

lies on

AC

with

AE:EC = 3:1

. The segments

AD

and

BE

meet at

P

. Find

AP:PD

A. $3:1$
B. $4:1$
C. $6:1$
D. $9:1$
E. $12:1$

Answer: D

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Question 6

1 mark

A real number

x

satisfies

x+\dfrac{1}{x}=3

. Find

x^{5}+\dfrac{1}{x^{5}}

A. $117$
B. $123$
C. $108$
D. $126$
E. $141$

Answer: B

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Question 7

1 mark

An egg is to be boiled for exactly

15

minutes. The only clocks available are two sand timers, one that runs for exactly

7

minutes and one that runs for exactly

11

minutes. Either timer may be turned over at any moment, turning one over takes no time, and both begin empty. Measured from the instant the first timer is started, what is the least possible time until the egg is done?

A. $14$ min
B. $15$ min
C. $18$ min
D. $21$ min
E. $22$ min

Answer: B

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Question 8

1 mark

A regular tetrahedron has edge length

6

. Find the distance between the midpoints of two opposite edges.

A. $3\sqrt{2}$
B. $3$
C. $2\sqrt{6}$
D. $3\sqrt{3}$
E. $6$

Answer: A

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Question 9

1 mark

How many four-digit positive integers have digits that are strictly increasing from left to right?

A. $84$
B. $210$
C. $126$
D. $120$
E. $715$

Answer: C

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Question 10

1 mark

Two circles, of radii

3

and

1

, touch each other externally, and both touch the same straight line. Find the distance between their two points of contact with that line.

A. $2$
B. $2\sqrt{2}$
C. $4$
D. $2\sqrt{3}$
E. $3$

Answer: D

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