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ENP Alger Access Contest study materials
ENP Alger Access Contest study resources

ENP Alger Access Contest practice questions

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Question 1Mathematics

Alger's contest rules award 3 points for each correct answer and deduct 1 point for each wrong answer; unanswered questions score 0. A candidate answers 40 questions in total and finishes with 92 points. How many did they answer correctly?

  1. 33 Correct answer.
  2. 27
  3. 30
  4. 36

Explanation

Let c be correct and w be wrong. Then c + w = 40 and 3c − w = 92. Adding the two equations gives 4c = 132, so c = 33. Checking: 33 correct and 7 wrong gives 99 − 7 = 92, which matches.

Question 2Mathematics

A contest score S is computed as S = 5c − 2w, where c is the number of correct answers and w the number of wrong answers. Which expression gives w in terms of S and c?

  1. w = 5c − 2S
  2. w = (S − 5c)/2
  3. w = (5c + S)/2
  4. w = (5c − S)/2 Correct answer.

Explanation

Start from S = 5c − 2w and isolate the term containing w: 2w = 5c − S. Dividing both sides by 2 gives w = (5c − S)/2. Substituting a sample value such as c = 4, w = 1, S = 18 confirms the rearrangement.

Question 3Mathematics

In the Alger Access Contest, each entrant's final mark is the arithmetic mean of four section scores. A candidate scores 72, 85, 68 and 91. What final mark is recorded?

  1. 80
  2. 81
  3. 78
  4. 79 Correct answer.

Explanation

The sum of the four section scores is 72 + 85 + 68 + 91 = 316. Dividing by the number of sections, 316 ÷ 4 = 79. The mean is therefore 79.

Question 4Mathematics

A contest question asks for the value of 2^5 + 5^2. What is the result?

  1. 57 Correct answer.
  2. 100
  3. 45
  4. 35

Explanation

Compute each power separately: 2^5 = 32 and 5^2 = 25. Their sum is 32 + 25 = 57. Exponents apply only to their own base, so the two terms are not combined before evaluation.

Question 5Mathematics

A candidate's raw score R is converted to a scaled contest score by the rule T = 4R + 15. If the candidate's scaled score is 255, what is the raw score?

  1. 55
  2. 60 Correct answer.
  3. 75
  4. 64

Explanation

Solve 4R + 15 = 255 by subtracting 15 from both sides to get 4R = 240. Dividing by 4 gives R = 60. Substituting back gives 4(60) + 15 = 255, which checks.

Question 6Mathematics

In a contest probability round, a fair six-sided die is rolled once. What is the probability that the result is an even number or a number greater than 4?

  1. 1/3
  2. 1/2
  3. 5/6
  4. 2/3 Correct answer.

Explanation

The even results are {2, 4, 6} and the results greater than 4 are {5, 6}. Their union is {2, 4, 5, 6}, which has four outcomes. Since the die is fair with six equally likely outcomes, the probability is 4/6 = 2/3.

Question 7Mathematics

A contest problem states that a number n satisfies 3n − 7 < 11. Which set lists exactly the integer values of n allowed by this inequality?

  1. n ≤ 5 Correct answer.
  2. n < 6
  3. n ≤ 6
  4. n < 5

Explanation

Add 7 to both sides to get 3n < 18, then divide by 3 to get n < 6. The integers satisfying n < 6 are 5, 4, 3 and so on downward, which is exactly the condition n ≤ 5 for integers.

Question 8Mathematics

The contest's marking scheme is proportional: 5 correct answers earn 12 points. At the same rate, how many points do 15 correct answers earn?

  1. 30
  2. 24
  3. 40
  4. 36 Correct answer.

Explanation

The rate is 12 points per 5 correct answers, so each correct answer is worth 12/5 = 2.4 points. For 15 correct answers the score is 15 × 2.4 = 36 points. Equivalently, 15 is three times 5, so the score is three times 12.

Question 9Mathematics

A contestant's four section scores are 64, 78, 82 and x. If the mean of the four scores is 75, what is x?

  1. 80
  2. 72
  3. 76 Correct answer.
  4. 68

Explanation

A mean of 75 over four sections requires a total of 4 × 75 = 300. The three known scores sum to 64 + 78 + 82 = 224. Therefore x = 300 − 224 = 76.

Question 10Mathematics

In the contest's logic round, a statement of the form 'If a contestant scores above 90, then they receive a certificate' is true. A particular contestant did not receive a certificate. What necessarily follows?

  1. The contestant did not score above 90. Correct answer.
  2. The contestant scored exactly 90.
  3. The contestant scored below 60.
  4. Nothing follows about the score.

Explanation

The statement has the form P → Q, and its contrapositive ¬Q → ¬P is logically equivalent. Since the certificate was not received (¬Q), it follows that the score was not above 90 (¬P). This is valid deductive reasoning, unlike denying the antecedent or affirming the consequent.

Question 11Mathematics

A contest question gives the quadratic x² − 7x + 12 = 0. What is the sum of the two roots?

  1. 12
  2. −7
  3. 7 Correct answer.
  4. −12

Explanation

For ax² + bx + c = 0, the sum of the roots equals −b/a. Here a = 1, b = −7 and c = 12, so the sum is 7. Factoring as (x − 3)(x − 4) confirms roots 3 and 4, whose sum is 7.

Question 12Mathematics

A contest score is modelled by S(t) = 20 + 8t − t², where t is the number of practice hours (0 ≤ t ≤ 8). What is the maximum score?

  1. 44
  2. 28
  3. 20
  4. 36 Correct answer.

Explanation

The quadratic −t² + 8t + 20 opens downward, so its vertex is a maximum. The vertex occurs at t = −b/(2a) = −8/(−2) = 4. Substituting gives S(4) = 20 + 32 − 16 = 36.

Question 13Mathematics

A contest question presents the sequence 3, 7, 15, 31, ... where each term after the first is obtained by doubling the previous term and adding 1. What is the next term?

  1. 62
  2. 64
  3. 63 Correct answer.
  4. 47

Explanation

Applying the rule to the fourth term: 2 × 31 + 1 = 63. Checking earlier terms confirms the pattern, since 2 × 15 + 1 = 31. The sequence is t(n) = 2^(n+1) − 1.

Question 14Mathematics

In a contest statistics problem, a set of five scores has a mean of 60. A sixth score of 90 is added. What is the new mean?

  1. 65 Correct answer.
  2. 66
  3. 63
  4. 70

Explanation

The original total is 5 × 60 = 300. Adding 90 gives a new total of 390 over six scores. Dividing, 390 ÷ 6 = 65, which is the new mean.

Question 15Mathematics

A contest problem defines f(x) = 2x + 3 and g(x) = x² − 1. What is f(g(2))?

  1. 15
  2. 21
  3. 9 Correct answer.
  4. 7

Explanation

Evaluate the inner function first: g(2) = 2² − 1 = 3. Then apply f to that result: f(3) = 2(3) + 3 = 9. Composition proceeds from the inside out.

Question 16Mathematics

A contest scoring rule states that a candidate's total T satisfies |T − 70| ≤ 5. Which interval describes all possible totals?

  1. 65 ≤ T ≤ 75 Correct answer.
  2. 70 ≤ T ≤ 75
  3. T ≤ 65
  4. T ≥ 75

Explanation

The inequality |T − 70| ≤ 5 means the distance from T to 70 is at most 5. This is equivalent to −5 ≤ T − 70 ≤ 5. Adding 70 to all parts gives 65 ≤ T ≤ 75.

Question 17Mathematics

A contest problem asks for the number of distinct ways to arrange the letters of the word ALGER. How many arrangements are there?

  1. 60
  2. 120 Correct answer.
  3. 24
  4. 720

Explanation

The word ALGER has five distinct letters, so the number of arrangements is 5! = 5 × 4 × 3 × 2 × 1 = 120. No letters repeat, so no division by repeated factorials is needed.

Question 18Mathematics

In the contest's geometry round, a right triangle has legs of length 9 cm and 12 cm. What is the length of the hypotenuse?

  1. 15 cm Correct answer.
  2. 13 cm
  3. 21 cm
  4. 18 cm

Explanation

By the Pythagorean theorem, the hypotenuse satisfies h² = 9² + 12² = 81 + 144 = 225. Taking the positive square root gives h = 15 cm. This is the 3-4-5 triangle scaled by 3.

Question 19Mathematics

A contest question gives the system 2x + 3y = 12 and 4x − 3y = 6. What is the value of x?

  1. 2
  2. 4
  3. 3 Correct answer.
  4. 1

Explanation

Adding the two equations eliminates y: (2x + 4x) + (3y − 3y) = 12 + 6, so 6x = 18 and x = 3. Substituting back gives 2(3) + 3y = 12, so y = 2, which is consistent.

Question 20Mathematics

A candidate's percentage score P is related to raw marks m by P = (m/80) × 100. A second candidate scores 5 raw marks more and obtains a percentage 6.25 points higher. Which statement correctly explains this result?

  1. Each raw mark is worth 1.25 percentage points, so 5 marks give 6.25 points. Correct answer.
  2. The percentage increase should be 5 points because raw marks and percentages rise equally.
  3. Each raw mark is worth 0.8 percentage points, so 5 marks give 4 points.
  4. The relationship is nonlinear, so the increase cannot be predicted.

Explanation

The formula is linear with slope 100/80 = 1.25 percentage points per raw mark. A gain of 5 raw marks therefore raises the percentage by 5 × 1.25 = 6.25 points. This matches the stated result and confirms proportionality.

Question 21Mathematics

A contest problem defines a sequence where each term is the sum of the two previous terms, starting with 1 and 1. What is the eighth term?

  1. 34
  2. 18
  3. 13
  4. 21 Correct answer.

Explanation

The sequence is 1, 1, 2, 3, 5, 8, 13, 21. Counting positions carefully, the eighth term is 21. Each term after the second is the sum of the two preceding terms.

Question 22Mathematics

In a contest analysis problem, the scores of five candidates are 12, 15, 15, 18 and 40. Which measure best describes a typical score when one extreme value is present?

  1. The mean, 20, because it uses every value in the data set.
  2. The maximum, 40, because it is the highest score achieved.
  3. The range, 28, because it captures the full spread of scores.
  4. The median, 15, because it resists distortion by the outlier 40. Correct answer.

Explanation

The value 40 is far above the other scores and pulls the mean upward to 20, which exceeds four of the five scores. The median is the middle value, 15, and is unaffected by how extreme the outlier is. For skewed data the median is the more representative measure of centre.

Question 23Mathematics

A contest problem states that log₂(x) + log₂(x − 2) = 3. What is the valid solution for x?

  1. 8
  2. 2
  3. 4 Correct answer.
  4. −2

Explanation

Combining the logarithms gives log₂(x(x − 2)) = 3, so x(x − 2) = 8. This yields x² − 2x − 8 = 0, which factors as (x − 4)(x + 2) = 0, giving x = 4 or x = −2. Since logarithms require x > 0 and x − 2 > 0, only x = 4 is valid.

Question 24Mathematics

A contest question asks how many integers from 1 to 300 inclusive are divisible by 3 or by 5. What is the count?

  1. 120
  2. 140 Correct answer.
  3. 160
  4. 150

Explanation

Use inclusion–exclusion. Multiples of 3: 100; multiples of 5: 60; multiples of 15 (counted twice): 20. So the count is 100 + 60 − 20 = 140. Subtracting the overlap avoids double counting.

Question 25Mathematics

A contest score function is f(x) = x³ − 3x² + 4. On the interval 0 ≤ x ≤ 4, at which value of x does f attain its minimum?

  1. x = 2 Correct answer.
  2. x = 1
  3. x = 0
  4. x = 4

Explanation

The derivative is f'(x) = 3x² − 6x = 3x(x − 2), giving critical points at x = 0 and x = 2. Evaluating: f(0) = 4, f(2) = 8 − 12 + 4 = 0, f(4) = 64 − 48 + 4 = 20. The minimum on the interval is 0, attained at x = 2.

Question 26Mathematics

A contest rule awards 3 points for each correct answer and deducts 1 point for each wrong answer. A candidate answers 20 questions and scores 40 points. How many did they answer correctly?

  1. 16
  2. 12
  3. 14
  4. 15 Correct answer.

Explanation

Let c be correct and w be wrong, so c + w = 20 and 3c − w = 40. Substituting w = 20 − c gives 3c − (20 − c) = 40, hence 4c = 60 and c = 15. Checking: 15 correct gives 45 points, 5 wrong removes 5, total 40, which matches.

Question 27Mathematics

In a contest geometry task, a circle has radius 7 cm. What is the area of the sector cut off by a central angle of 90°?

  1. 49π/2 cm²
  2. 7π/2 cm²
  3. 49π/4 cm² Correct answer.
  4. 14π cm²

Explanation

A sector's area equals (θ/360) × πr². Here θ/360 = 90/360 = 1/4 and πr² = 49π, so the area is 49π/4 cm², roughly 38.5 cm².

Question 28Mathematics

A contest problem defines the operation a ★ b = a² − ab. What is the value of 5 ★ 3?

  1. 22
  2. 16
  3. 10 Correct answer.
  4. 4

Explanation

Substituting a = 5 and b = 3 into the definition gives 5² − 5×3 = 25 − 15 = 10. The operation is not symmetric, so the order of the inputs matters.

Question 29Mathematics

A contest puzzle asks for the units digit of 3^2025. Which digit is it?

  1. 9
  2. 7
  3. 3 Correct answer.
  4. 1

Explanation

Powers of 3 cycle through units digits 3, 9, 7, 1 with period 4. Since 2025 = 4×506 + 1, the exponent leaves remainder 1, so the units digit matches 3¹, namely 3.

Question 30Mathematics

A contest score table records that a candidate's marks rose from 45 to 54 between two rounds. What is the percentage increase?

  1. 20% Correct answer.
  2. 9%
  3. 25%
  4. 16.7%

Explanation

Percentage increase equals (new − old)/old × 100. Here (54 − 45)/45 × 100 = 9/45 × 100 = 20%.

Question 31Mathematics

A contest problem gives two similar triangles whose corresponding sides are in the ratio 3 : 5. What is the ratio of their areas?

  1. 27 : 125
  2. 9 : 25 Correct answer.
  3. 3 : 5
  4. 6 : 10

Explanation

For similar figures, the ratio of areas equals the square of the linear scale factor. Squaring 3 : 5 gives 9 : 25.

Question 32Mathematics

A contest question asks for the remainder when 7^100 is divided by 5. What is it?

  1. 4
  2. 1 Correct answer.
  3. 3
  4. 2

Explanation

Modulo 5, powers of 7 ≡ 2 cycle as 2, 4, 3, 1 with period 4. Since 100 is divisible by 4, 7^100 ≡ 1 (mod 5).

Question 33Mathematics

A contest graph problem states that a line passes through (2, 5) and is perpendicular to the line y = 2x + 1. What is the equation of this line?

  1. y = −2x + 9
  2. y = x/2 + 4
  3. y = −x/2 + 6 Correct answer.
  4. y = 2x + 1

Explanation

A perpendicular line has slope equal to the negative reciprocal of 2, namely −1/2. Using point (2, 5): 5 = −1 + b, so b = 6, giving y = −x/2 + 6.

Question 34Mathematics

In a contest combinatorics round, a team of 3 is chosen from 8 candidates. How many different teams are possible?

  1. 336
  2. 512
  3. 56 Correct answer.
  4. 24

Explanation

Order within the team does not matter, so the count is C(8,3) = 8!/(3!5!) = (8×7×6)/(3×2×1) = 56.

Question 35Mathematics

A contest problem asks for the sum of the first 40 positive even integers, 2 + 4 + 6 + … + 80. What is the sum?

  1. 1620
  2. 1560
  3. 3200
  4. 1640 Correct answer.

Explanation

The sum of the first n even numbers is n(n + 1). With n = 40 this gives 40 × 41 = 1640.

Question 36Mathematics

A contest problem states that a quantity doubles every 3 hours. If it starts at 50, which expression gives its value after t hours?

  1. 50 · 2^(3t)
  2. 50 · 3^(t/2)
  3. 50 + 2t/3
  4. 50 · 2^(t/3) Correct answer.

Explanation

Doubling every 3 hours means the exponent counts how many 3-hour periods have passed, namely t/3. The model is therefore 50 · 2^(t/3).

Question 37Mathematics

A contest question gives the polynomial P(x) = x³ − 4x² + x + 6. Given that x = 2 is a root, what are all the roots?

  1. 2, 3, −1 Correct answer.
  2. 2, 2, 3
  3. 2, 3, 1
  4. 2, −3, −1

Explanation

Dividing P(x) by (x − 2) gives x² − 2x − 3, which factors as (x − 3)(x + 1). The three roots are therefore 2, 3 and −1.

Question 38Mathematics

In a contest physics-style problem, a ball is thrown upward with height h(t) = 40t − 5t² metres. At what time does the ball return to the ground?

  1. 4 s
  2. 8 s Correct answer.
  3. 10 s
  4. 5 s

Explanation

Setting h(t) = 0 gives 5t(8 − t) = 0, so t = 0 (launch) or t = 8. The ball returns to the ground after 8 seconds.

Question 39Mathematics

A contest problem asks for the value of sin 30° + cos 60° + tan 45°. What is it?

  1. 2.5
  2. 1.5
  3. 2 Correct answer.
  4. 1

Explanation

The standard exact values are sin 30° = 1/2, cos 60° = 1/2 and tan 45° = 1. Adding them gives 1/2 + 1/2 + 1 = 2.

Question 40Mathematics

A contest problem gives the set A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}. How many elements are in A ∩ B?

  1. 4
  2. 2
  3. 6
  4. 3 Correct answer.

Explanation

The intersection contains elements present in both sets: 2, 4 and 6. That is 3 elements.

Question 41Mathematics

A contest problem defines f(x) = x² − 4x + 7 for real x. What is the minimum value of f?

  1. −3
  2. 7
  3. 4
  4. 3 Correct answer.

Explanation

Completing the square gives f(x) = (x − 2)² + 3. Since the square is never negative, the minimum is 3, attained at x = 2.

Question 42Mathematics

A contest problem asks for the coefficient of x² in the expansion of (2x + 3)⁴. What is it?

  1. 54
  2. 96
  3. 216 Correct answer.
  4. 81

Explanation

By the binomial theorem the x² term is C(4,2)(2x)²(3)² = 6 × 4x² × 9 = 216x². The coefficient is therefore 216.

Question 43Mathematics

A contest problem states that a bag contains 4 red and 6 blue counters. Two counters are drawn without replacement. What is the probability that both are red?

  1. 1/5
  2. 8/45
  3. 4/25
  4. 2/15 Correct answer.

Explanation

The probability is (4/10) × (3/9) = 12/90 = 2/15. Equivalently, C(4,2)/C(10,2) = 6/45 = 2/15.

Question 44Mathematics

A contest question asks how many diagonals a convex polygon with 12 sides has. What is the count?

  1. 54 Correct answer.
  2. 24
  3. 66
  4. 60

Explanation

Each vertex connects to 9 others, giving 12 × 9 = 108 segments, but each diagonal is counted twice, so there are 54. Equivalently, C(12,2) − 12 = 66 − 12 = 54.

Question 45Mathematics

A contest problem gives the arithmetic sequence 5, 9, 13, 17, … What is the 25th term?

  1. 100
  2. 105
  3. 97
  4. 101 Correct answer.

Explanation

The first term is 5 and the common difference is 4. The nth term is 5 + 4(n − 1), so the 25th term is 5 + 4 × 24 = 101.

Question 46Mathematics

A contest problem states that the point (k, 4) lies on the line through (1, 2) and (5, 10). What is k?

  1. 2 Correct answer.
  2. 1
  3. 3
  4. 4

Explanation

The slope is (10 − 2)/(5 − 1) = 2. Using point (1, 2), the line is y = 2x, so y = 4 gives x = 2.

Question 47Mathematics

A contest problem asks for the value of log₃ 81 − log₂ 16. What is it?

  1. 0 Correct answer.
  2. 2
  3. −1
  4. 1

Explanation

Since 3⁴ = 81, log₃ 81 = 4; since 2⁴ = 16, log₂ 16 = 4. Their difference is 0.

Question 48Mathematics

A contest problem states that a right circular cone has base radius 3 cm and height 4 cm. What is its total surface area?

  1. 24π cm² Correct answer.
  2. 15π cm²
  3. 21π cm²
  4. 36π cm²

Explanation

The slant height is √(3² + 4²) = 5. The total surface area is πr² + πrl = 9π + 15π = 24π cm².

Question 49Mathematics

A contest problem states that a function satisfies f(x + 1) = 2f(x) for all x, with f(0) = 3. What is f(3)?

  1. 12
  2. 18
  3. 24 Correct answer.
  4. 6

Explanation

Applying the rule repeatedly: f(1) = 2 × 3 = 6, f(2) = 2 × 6 = 12, f(3) = 2 × 12 = 24. In general f(n) = 3 × 2ⁿ.

Question 50Mathematics

A contest problem gives the inequality x² − 5x + 6 < 0. Which interval contains exactly the real solutions?

  1. (2, 3) Correct answer.
  2. (−∞, 2) ∪ (3, ∞)
  3. (−3, −2)
  4. [2, 3]

Explanation

The quadratic factors as (x − 2)(x − 3), with roots 2 and 3 and an upward parabola. It is negative strictly between the roots, so the solution set is the open interval (2, 3).