Basics of Math 1 Logic and Sets the Statement a Is True, B Is False

Basics of math 1 Logic and sets The statement a is true, b is false.

Both statements are false. 9000086601 (level 1): Let a and b be two sentences in the sense of mathematical logic. It is known that the composite statement 9000086604 (level 1): Let a and b be two sentences in the sense of mathematical ¬(a ∨ b) logic. It is known that the composite statement is true. For each a and b determine whether it is true or false. ¬(a ∧ ¬b)

is false. For each a and b determine whether it is true or false. Both statements are false. The statement a is true, b is false. Both statements are true.

The statement a is true, b is false. Both statements are true.

The statement a is false, b is true. The statement a is false, b is true.

Both statements are false. 9000086602 (level 1): Let a and b be two sentences in the sense of mathematical logic. It is known that the composite statement 9000086605 (level 1): Let a and b be two sentences in the sense of mathematical ¬a ∨ b logic. It is known that the composite statement is false. For each a and b determine whether it is true or false. ¬a =⇒ ¬b

The statement a is true, b is false. is false. For each a and b determine whether it is true or false.

The statement a is false, b is true. Both statements are true.

The statement a is true, b is false. Both statements are false.

Both statements are false. 9000086603 (level 1): Let a and b be two sentences in the sense of mathematical logic. It is known that the composite statement 9000086606 (level 1): Let a and b be two sentences in the sense of mathematical ¬a ∧ b logic. It is known that the composite statement is true. For each a and b determine whether it is true or false. a ⇐⇒ (a ∨ b)

The statement a is false, b is true. is false. For each a and b determine whether it is true or false.

The statement a is false, b is true. Both statements are true.

Both statements are true.

1 The statement a is true, b is false. a ⇐⇒ (b ∨ c) (¬a ∨ b) ∨ c (a ∧ b) ∨ c

Both statements are false. (a ∨ b) =⇒ ¬c

9000086607 (level 1): 9000080901 (level 2): Let a and b be two sentences in the sense of mathematical Find the intersection A ∩ B for A = {−5; 0; 1.5; 2; 6} and logic. It is known that the composite statement B = {x ∈ Z; x ≥ 0}. (¬a ∨ b) ∧ a {0; 2; 6} {0; 1.5; 2; 6} {1.5; 2; 6} Z is true. For each a and b determine whether it is true or false.

Both statements are true. 9000080902 (level 2): Find the intersection A ∩ B for A = {x ∈ Z; x ≥ −2} and The statement a is true, b is false. B = {x ∈ N; x ≤ 5}.

The statement a is false, b is true. {1; 2; 3; 4; 5} {0; 1; 2; 3; 4; 5}

Both statements are false. {0; 1; 2; 3; 4} {−2; −1; 0; 1; 2; 3; 4; 5}

9000086608 (level 1): 9000080903 (level 2): Let a and b be two sentences in the sense of mathematical Find the union A ∪ B for A = {x ∈ Z; x ≥ −3} and logic. It is known that the composite statement B = {x ∈ N; x < 8}. ¬a ⇐⇒ (a ∧ b) {x ∈ Z; x ≥ −3} is true. For each a and b determine whether it is true or false. {1; 2; 3; 4; 5; 6; 7} The statement a is true, b is false. {−3; −2; −1; 0; 1; 2; 3; 4; 5; 6; 7} Both statements are true. Z The statement a is false, b is true.

Both statements are false. 9000080904 (level 2): Find the union A ∪ B for A = N and B = {x ∈ Z; x > 8}.

9000086609 (level 1): ∅ {x ∈ ; x > 8} Let the sentence a be true and the sentences b and c be false. N Z In the following list identify a true statement. Z (a ∨ b) =⇒ ¬c (¬a ∨ b) ∨ c (a ∧ b) ∨ c 9000080905 (level 2): a ⇐⇒ (b ∨ c) Identify the set B which satisﬁes

A ∪ B = C 9000086610 (level 1): Let the sentences a and b be false and c be true. In the if A = {x ∈ N; x < 3} and C = {0; 1; 2}. following list identify a false statement.

2 {0; 1; 2}, {0; 1}, {0; 2}, {0} 9000089001 (level 3): Students from a class have a possibility to work in mathematical and physical hobby groups. no solution exist • There are 31 students in the class. • From the total, 21 students are members of mathematical ∅ group. • Some of the students are members of both groups, but {0; 1; 2}, {0; 1}, {1; 2}, {0; 2} there are 10 which are members of just one group. • There are 3 students which are not members of any of those groups. 9000080906 (level 2): How many students are members of both mathematical and 0 Find BA (the complement to B in A) for A = {x ∈ N; x < 9} physical groups? and B = {4; 5; 6; 7}. 18 16 19 {1; 2; 3; 8} ∅ {4; 5; 6; 7} {0; 1; 2; 3; 8}

9000089002 (level 3): 9000080907 (level 2): Students from a class decided to order books for forthcoming 0 holiday. The book-shop in the neighborhood had two Find BA (the complement to B in A) for A = Z and bestsellers on the stock: a crime novel and a horror stories. B = {x ∈ Z; |x| > 3}. • There were 31 students in the class in total. • From this total, 22 students bought horror stories. {−3; −2; −1; 0; 1; 2; 3} {−2; −1; 0; 1; 2} • Altogether 12 students bought just one of the books. • Two students did not buy any of these books. {0; 1; 2; 3} {1; 2; 3} How many students bought the crime novel?

24 7 5 9000080908 (level 2): Find the set diﬀerence A \ B for A = {−2; −1; 0; 1; 2} and B = {x ∈ Z; x < 2}. 9000089003 (level 3): Students from a class bought a snack in the school lunch-room. {2} {−2; −1; 0; 1; 2} {0; 1} • There were 31 students in the class in total. • Altogether 8 students had snack from their home and they did not buy anything. ∅ • Altogether 12 students bought hamburger and 15 students bought hot-dog. How many students bought both hamburger and hot-dog. 9000080909 (level 2): Find the set diﬀerence B \ A for A = {x ∈ ; x < 2} and Z 4 19 8 B = {x ∈ Z; x < 5}.

{2; 3; 4} {x ∈ Z; x < 2} {3; 4} 9000089004 (level 3): Students in a university are allowed to buy either lunch or ∅ dinner in a student’s canteen. • There are 129 freshmen students in total. • Altogether 116 freshmen students buy the lunch or dinner. 9000080910 (level 2): • From this amount, 62 students buy just one meal. • The number of freshmen students which buy lunch is bigger Find the set diﬀerence A \ B for A = {x ∈ Z; |x| < 3} and by 46 than the number of freshmen students which buy dinner. B = {x ∈ N; x < 5}. How many freshmen students buy only the dinner?

{−2; −1; 0} ∅ {−2; −1} {3; 4} 8 54 62

3 9000089005 (level 3): Complete the following statement: „The number is divisible by There are two types of cheese in a shop. Altogether 153 two if and only if ...” customer were in the shop during a day. • From this total, 65 customers bought the ﬁrst cheese. the last digit of this number is even. • From the same total, 49 customers bought the second cheese. the sum of its digits is divisible by two. • 20% of customers which bought at least one of the types of cheese bought actually both types. the sum of its digits is even. How many customers did not buy any of these two cheeses?

the last digit of this number is 2, 3, 6 or 8. 58 39 19

9000089006 (level 3): 9000115602 (level 1): A questionnaire on popular singers has been ﬁlled by 200 girls Complete the following statement: „The number is divisible by from a high-school. The girls had to mark the singers from three if and only if ...” three most popular (Justin Timberlake, Justin Bieber and Axl Rose) which they like. the sum of its digits is divisible by three. • Justin Timberlake got 78 votes, Justin Bieber got 75 votes and Axl Rose 101 votes. the number constituted from the last two digits is divisi- • There were 28 girls which voted for all three singers. ble by three. • There were 22 girls which voted for two singers. One half of this amount are fans of the pair Bieber and Rose. the sum of its digits is odd. • The number of the girls which like only Justin Bieber is smaller by 7 comparing to the number of girls which like only the last digit of this number is 3, 6 or 9. Justin Timberlake. How many girls did not like any of these three singers? 9000115603 (level 1): 24 32 11 Complete the following statement: „The number is divisible by four if and only if ...”

9000089007 (level 3): the number constituted from the last two digits is divisi- A class in the school has 35 students. On the last holiday the ble by four. students visited Slovakia, Croatia and Bulgaria. From the total amount of 35 the sum of its digits is divisible by four. • 7 students have been in Slovakia, • 7 students have been in Croatia, the last digit of this number is 4. • 5 students have been in Bulgaria, • 21 students have not been abroad, the last digit of this number is even. • one student was in every of these countries, • two students have been in both Bulgaria and Croatia but not in Slovakia, • one student has been in both Bulgaria and Slovakia but not 9000115604 (level 1): in Croatia. Complete the following statement: „The number is divisible by How many students visited either Slovakia or Croatia? ﬁve if and only if ...”

11 7 3 the last digit of this number is 5 or 0.

the sum of its digits is divisible by ﬁve. 2 Elementary arithmetics it is divisible by two and three.

9000115601 (level 1): the last digit of this number is odd.

4 the number constituted from the last two digits is divisible by ﬁve. 9000115605 (level 1): Complete the following statement: „The number is divisible by the last digit of this number is even. six if and only if ...”

it is divisible by both two and three. 9000115609 (level 1): Complete the following statement: „The number is divisible by the sum of its digits is divisible by both two and three. twelve if and only if ...”

the sum of its digits is even and the last digit of this divisible by three and four. number is 3.

the last digit of this number is 6. the sum of its digits is divisible by both two and three.

the sum of the digits is even and the last digit of this number is odd. 9000115606 (level 1): Complete the following statement: „The number is divisible by the sum of the digits is odd and the last digit of this eight if and only if ...” number is even.

the number constituted from the last three digits is divisible by eight. 9000115610 (level 1): Complete the following statement: „The number is divisible by the sum of its digits is divisible by eight. ﬁfteen if and only if ...”

it is divisible by both two and four. divisible by both three and ﬁve.

the number constituted from the last two digits is divisi- the sum of its digits is divisible by both three and ﬁve. ble by eight. the sum of its digits is odd and divisible by ﬁve. 9000115607 (level 1): the last digit of this number is either 5 or 0. Complete the following statement: „The number is divisible by nine if and only if ...” 9000076001 (level 2): the sum of its digits is divisible by nine. In the following list identify a set of a numbers which give remainder 2 after division by 3. i.e. the numbers can be the number constituted from the last two digits is divisi- written in the form 3k + 2, k ∈ . ble by nine. N0 5, 8, 11 5, 10, 15 3, 6, 9 15, 25, 30 the sum of its digits is odd. 4, 5, 6 the last digit of this number is 9.

9000076002 (level 2): 9000115608 (level 1): In the following list identify a set of a numbers which give Complete the following statement: „The number is divisible by remainder 2 after division by 5. i.e. the numbers can be ten if and only if ...” written in the form 5k + 2, k ∈ N0. the last digit of this number is 0. 37, 42, 102 5, 10, 15 17, 27, 100 29, 47, 60

the sum of its digits is divisible by ten. 41, 55, 62

5 is divisible by 5. is divisible by 3. is divisible by 4. 9000076003 (level 2): In the following list identify a set of a numbers which give is divisible by 6. is divisible by 10. remainder 1 after division by 11. i.e. the numbers can be written in the form 11k + 1, k ∈ N0. 9000076009 (level 2): 56, 122, 221 21, 32, 48 18, 88, 115 34, 55, 70 In the following list identify a set of the numbers where all the numbers are primes. 45, 56, 65 3, 7, 89 7, 15, 17 8, 11, 17 2, 7, 91 3, 27, 81

9000076004 (level 2): In the following list identify a set such that each element of 9000076010 (level 2): this set is a divisor of 256. In the following list identify a set of the numbers where any number has exactly three divisors (including the number 1 1, 128, 256 1, 64, 123 4, 8, 104 1, 12, 128 and itself).

16, 30, 64 4, 25, 289 1, 2, 3 25, 36, 49 1, 17, 289

25, 36, 121 9000076005 (level 2): In the following list identify a set such that each element of this set is a divisor of 1 260. 9000084901 (level 2): Which of the following numbers is a prime? 1, 36, 42 4, 8, 630 12, 18, 26 17 1 27 91 289 16, 315, 1 260 1, 17, 256

9000084902 (level 2): 9000076006 (level 2): In the following list ﬁnd the set which does not contain any In the following list identify a set such that each element of prime number. this set is a divisor of 578. 91, 243 13, 100 1, 2, 4 29, 81 101, 211 17, 34, 289 1, 2, 4 13, 15, 17 1, 13, 289

2, 35, 578 9000084903 (level 2): In the following list ﬁnd the set which contains only prime numbers. 9000076007 (level 2): Complete the statement. „The sum of any three consecutive 13, 131 1, 31, 211 289, 291 17, 169 integers ...” 51, 97 is divisible by 3. is not divisible by 6.

is divisible by 6. is not divisible by 3. 9000084904 (level 2): From the following list ﬁnd the number which has just three is divisible by 9. proper divisors (diﬀerent from 1 and itself).

49 21 75 100 250 9000076008 (level 2): Complete the statement. „The sum of any ﬁve consecutive integers ...” 9000084905 (level 2):

6 In the following list find the number such that the prime 9000085602 (level 2): factorization of this number contains just two different primes Evaluate the following numbers and round to the nearest tens. or powers of two different primes. 2 (22)2 100 5 25 120 121

260 510 120 60 9000084906 (level 2): In the following list ﬁnd the number such that the prime factorization of this number contains exactly one cube power. 9000085603 (level 2): Find the sum of the three numbers obtained by rounding the 24 12 63 196 420 number 5 316 to the nearest tens, hundreds and thousands.

15 620 15 610 15 560 15 580 9000084907 (level 2): In the following list ﬁnd the number such that the prime factorization of this number contains more diﬀerent primes 9000085604 (level 2): comparing to the prime factorizations of other numbers from Find the sum of the three numbers obtained by rounding the this list. number 2 013 to the nearest tens, hundreds and thousands.

330 21 100 486 1 024 6 010 6 000 6 020 6 030

9000084908 (level 2): 9000085605 (level 2): Among the numbers in the following list ﬁnd the numbers Find the product of all nonzero one-digit primes and round to which has bigger power in prime factorizations than the other nearest hundreds. numbers from the list. 200 100 300 400 1 024 21 100 330 486

9000085606 (level 2): 9000084909 (level 2): Find the product of all divisors of 12 and round to nearest Among the numbers in the following list find the number such hundreds. that the prime factorization of this number does not contain a power different from 2. 1 700 1 200 600 100 36 24 120 360 512 9000085607 (level 2): Given numbers 8 175 and 3 926, find the difference between 9000084910 (level 2): • the sum of these numbers rounded to the nearest tens, In the following list find the number such that the prime • the sum of these numbers rounded to the nearest hundreds. factorization of this number contains only one prime. 10 20 100 0 125 15 100 250 768

9000085608 (level 2): 9000085601 (level 2): Given numbers 456 138 and 321 814, ﬁnd the diﬀerence Evaluate the following numbers and round to the nearest tens. between 102 + 112 + 122 + 132 + 142 • the sum of these numbers rounded to the nearest tens, • the sum of these numbers rounded to the nearest hundreds.

730 720 740 750 50 100 1 000 0

7 9000085609 (level 2): aa0 a − a0 a f = f = f = a + a0 f = Given the number 45 875, round this number to nearest a + a0 a + a0 a0 thousands, nearest hundreds and subtract the results.

100 200 1 000 0 9000039305 (level 2): Find m1 as a function of the other variables from the following mixing equation. 9000085610 (level 2):

Given the number 82 361, round this number to nearest w1m1 + w2m2 = w3m3 thousands, nearest hundreds and subtract the results.

400 300 200 100 w3m3 − w2m2 w3m3w2m2 m1 = m1 = w1 w1 3 Polynomials and fractions w3m3 + w2m2 w2m2 − w3m3 m1 = m1 = w1 w1

9000039301 (level 2): Given formula for an acceleration of the motion in the form 9000079201 (level 2): v − v Evaluate a = 0 , −x2 y − x t − x − y x + y ﬁnd the initial velocity v0. at x = −1, y = 2.

v0 = v − at v0 = vat v0 = v + at v0 = at − v 8 10 2 4 − − − − 3 3 3 3 9000039302 (level 2): Find N, the number of the turns, as a function of the other 9000079202 (level 2): variables in the formula for the magnetic induction of a Find the set M of all the real x for which the following solenoid. NI expression is not a well deﬁned number. B = µ l x − 4 x3 − 16x Bl Blµ I Bl N = N = N = B − µ N = − I µI I l µ M = {−4; 0; 4} M = {−4; 4} M = {0; 4}

M = {0} 9000039303 (level 2): Find the time t as a function of the other variables in the formula for the distance of a motion in the form s = v t + s . 0 0 9000079203 (level 2): Find all real x for which the following expression equals zero. s − s s s + s v0 t = 0 t = t = 0 t = v t + s v s − s 2x + 1 0 0 0 0 1 − x − 1

9000039304 (level 2): Find the focus length f as a function of the other variables 1 x = −2 x = − x = 0 x = −1 from the following equation relating this distance with object 2 and image distances a and a0. 1 1 1 = + 9000079204 (level 2): f a a0

8 Find the domain of the following expression. 9000079209 (level 2): Evaluate the following expression at x = 4. x2 − x x2 − 1 : 2 − 1 x + 1 x + 2x + 1 x 2 x−2 − x−1

R \ {−1; 1} R \ {−1; 0; 1} R \ {−1} 8 31 8 − 6 R \ {−1; 0} 3 3 3

9000079205 (level 2): 9000079210 (level 2): Assuming x 6= 0 and x 6= 2, simplify the following expression. Consider the expression

x3 − x2 2 − x x 1 · V (x) = − . x − 2 x2 x − 1 1 − x Find the correct ordering of the values V (−2), V (0) and V (2). 1 − x x − 1 x + 1 x2 − 1 V (0) < V (−2) < V (2) V (−2) < V (0) < V (2)

9000079206 (level 2): V (0) < V (2) < V (−2) V (2) < V (0) < V (−2) Assuming x 6= 0, y 6= 0, x 6= y, simplify the following expression. 1 1 x2 − y2 9000083602 (level 2): 1 1 1 − + Evaluate the following expression at x = . y x 2 x2 − 2 x + y x + y 1 1 1 1 − − − 1 − 1 xy xy y x x y x

7 7 7 7 9000079207 (level 2): − − Assuming x 6∈ {0; 1; 3}, simplify the following expression. 4 4 2 2

x2 − 9 x2 − 3x−1 2 · 9000083603 (level 2): x − x x − 1 1 1 Evaluate the following expression at x = and y = − . 2 4 y x + 3 x − 3 x + 3 x + 3 x − x x2 x2 2x x x 1 + y

9000079208 (level 2): −1 3 4 1 Assuming x 6= 0 and y 6= 0, simplify the following expression.

x−2y2 −2 x2 : 9000083604 (level 2): x0y−8 x−4y7 Assuming x 6= −1, x 6= ±y, simplify the expression.

x2 + 2xy + y2 (x + 1)(y − x) 4 · 1 y13 y15 x 2x2 + 4x + 2 y2 − x2 x2y13 x2 x6 y27

9 x + y x + y 1 Find all the values of x ∈ for which the given expression x + y R 2x + 2 2 2 equals zero. x3 − x x − 1 9000083605 (level 2): Find the common denominator of the fractions. x = −1, x = 0 x = 0 3x x + 5 and x2 + 4x + 4 x2 − 4 x = 1 x = −1, x = 0, x = 1

(x + 2)2(x − 2), x 6= ±2 (x + 2)(x − 4), x 6= ±2 9000083704 (level 2): Find all the values of x ∈ R for which the given expression equals zero. (x + 2)2(x − 4), x 6= ±2 (x + 2)(x − 4), x 6= ±2 x2 − 4x + 4 x(x − 2) 9000083606 (level 2): Assuming x 6= 2, simplify the expression. The expression never equals zero. x2 + x − 6 x = 0 x3 − 8 x = 2 x + 3 x + 3 x + 3 x + 3 x2 + 2x + 4 x2 − 2x + 4 x2 + 4x + 4 x2 − 4 x = −2, x = 0

9000083701 (level 2): 9000083705 (level 2): Find all the values of x ∈ for which the given expression Find all the values of x ∈ R for which the given expression R equals zero. equals zero. x2 − 16 2x(x + 2)(x − 3) 2 2x − 8 x − 4

x = 0, x = 3 x = −2, x = 0, x = 3 x = −4 x = 4 x = ±4 x = 0

x = 0 x = ±2 9000083702 (level 2): Find all the values of x ∈ R for which the given expression equals zero. 9000083706 (level 2): x2 + 6x + 9 Find all the values of x ∈ R for which the given expression x2 − 9 equals zero. 4x2 − 36 4x2 + 24x + 36 The expression never equals zero.

x = 3 x = ±3 x = 4 x = 3 x = −3, x = 3 x = −3 The expression never equals zero. 9000083703 (level 2):

10 9000083707 (level 2): Assuming x ∈ R \ {−1}, ﬁnd the quotient of the polynomials. Find all the values of x ∈ for which the given expression R 2 equals zero. (3x + 2x + 7) : (x + 1) 4x3 + 20x2 + 25x x + 1 8 8 5 3x − 1 + 3x + 2 + 3x − 1 − x + 1 x + 1 x + 1 5 5 x = 0, x = − x = 0 x = − 2 2 5 3x + 2 − x + 1 x = −1

9000087502 (level 2): Assuming x ∈ R \ {±1}, ﬁnd the quotient of the polynomials. 9000083708 (level 2): Find all the values of x ∈ R for which the given expression (−2x4 − 3x2 + 3) : (x2 − 1) equals zero. x2 − (2x − 1)2 2 x − 4 2 2 −2x2 − 5 − −2x2 − 5 + x2 − 1 x2 − 1 1 1 x = , x = 1 x = − , x = 1 x = ±2 2 2 3 3 2x2 + 5 − 2x2 + 5 + x2 − 1 x2 − 1

x = 1 9000087503 (level 2): 3 Assuming x ∈ R \ − , ﬁnd the quotient of the polynomials. 9000083709 (level 2): 2 Find all the values of x ∈ for which the given expression R (x2 + x + 1) : (2x + 3) equals zero. (2x + 3)2 − (3x − 2)2 x − 5 1 1 7 1 1 7 7 x − + 4 x − + 4 x + 2 + 2 4 2x + 3 2 2 2x + 3 2x + 3 1 1 x = − x = 5 x = −5 x = 5 5 7 x − 2 + 2x + 3 9000083710 (level 2): Find all the values of x ∈ R for which the given expression 9000087504 (level 2): equals zero. 3 (4x + 3)2 − (5x − 2)2 Assuming x ∈ \ − , ﬁnd the quotient of the polynomials. R 5 5 + x (5x3 − 2x2 + x + 1) : (5x + 3) 1 5 x = 5, x = − x = −5 x = − , x = 1 9 9 4 7 4 7 x2 − x + − 5 x2 − x + + 5 5 5 5x + 3 5 5x + 3 x = 1, x = 9 4 9 4 9 x2 − x + − 5 x2 − x + + 5 5 5x + 3 5 5x + 3 9000087501 (level 2):

11 9000087505 (level 2): −61x2 + 63x − 4 1 −5x − 10 + Assuming x ∈ \ − , ﬁnd the quotient of the polynomials. x3 − 4x2 + 3x R 2

3 −16x2 + 23x − 36 (4x − 1) : (2x + 1) −5x − 10 + x3 − 4x2 + 3x

1 3 1 3 2x2 − x + − 2 2x2 + x + − 2 9000088801 (level 2): 2 2x + 1 2 2x + 1 Find the domain of the following expression.

1 3 1 3 2x + 1 2x2 − x + − 2 2x2 + x + − 2 4 2x + 1 4 2x + 1 6x2 + 3x

1 9000087506 (level 2): x 6= 0, x 6= − x 6= 0, x 6= −2 x 6= 0 Assuming x ∈ R \{1}, ﬁnd the quotient of the polynomials. 2 (2x + 2x2 − 3) : (x − 1) 1 x 6= 0, x 6= 2

1 2 1 2x + 4 + 2x + 4 + 2x + 2 + x − 1 x − 1 x − 1 9000088802 (level 2): Find the domain of the following expression. 2 2x + 2 + a a2 − 9 x − 1 · a2 + 9 a2 + 3a

9000087507 (level 2): a 6= 0, a 6= −3 a 6= 3, a 6= −3 a 6= 0, a 6= 3 Assuming x ∈ R, ﬁnd the quotient of two polynomials. (−x3 − x2 + x − 1) : (x2 + 1) a 6= −3

2x x x 9000088803 (level 2): −x − 1 + −x − 1 + x − 1 + 1 x2 + 1 x2 + 1 x2 + 1 Evaluate the following expression at x = . 2 2x x − 2 x − 1 + 1 − x2 + 1 2x + 1

9000087508 (level 2): 7 1 5 3 Assuming x ∈ R \{0,1,3}, ﬁnd the quotient of the polynomials. 4 4 4 4 (−5x4 + 4x2 + 3x − 4) : (x3 − 4x2 + 3x) 9000088804 (level 2): Simplify the following expression. −61x2 + 63x − 4 −5x − 20 + 2s − 8rs x3 − 4x2 + 3x 16r2 − 1 16x2 + 23x + 36 −5x − 20 + 3 2 x − 4x + 3x 2s 2s 2s 2s − 4r + 1 4r + 1 4r − 1 1 − 4r

12 m + n m(m + n) 0 2 9000088805 (level 2): m − n n(m − n) Simplify the following expression.

a4 − 1 9000088810 (level 2): 1 − a2 Simplify the following expression.

1 x x − · 1 − −a2 − 1 a2 + 1 a2 − 1 1 − a2 x x + 1

9000088806 (level 2): x − 1 x − 1 1 − x 1 − x Identify a correct expression which should be placed to the x x + 1 x + 1 x starred position in the following expression to obtain a valid statement. mn ∗ 9000101601 (level 2): 2 2 = 3 m + 2mn + n 2m(m + n) Expand (1 + x) x2 + x − 1 (1 − x).

2m2n(m + n) 2mn(m + n) 2m(m + n) −x4 − x3 + 2x2 + x − 1 x4 − x3 + 2x2 + x + 1

4 3 4 3 2 2m(m + n)2 −x + x − 1 x + x − 2x + x − 1

9000088807 (level 2): 9000101602 (level 2): 2 2 2 Identify a correct expression which should be placed to the Simplify the polynomial (x − 1)(x + 1) x + 1 − x − 1 starred position in the following expression to obtain a valid into one of the following forms. statement. 3 − 2x 3(4x2 − 12x + 9) 2 2 = 2 x − 1 0 2 x − 1 (x + 1) x − 2 ∗

x2 − 1 (3x − 6)(3 − 2x) (x − 2)(2x − 3) (x − 2)(9 − 4x)

(3x − 6)(2x − 3) 9000101603 (level 2): Simplify the polynomial (x + 1)(x − 1)2 − (x − 1)(x + 1)2 into one of the following forms. 9000088808 (level 2): Find the least common denominator of the following three −2 (x − 1) (x + 1) 2 (x − 1) (x + 1) 0 fractions. a −b 2b 2 , , a2 − ab a2 − b2 ab + b2

9000101604 (level 2): 2 2 22 ab(a2 − b2) ab(a − b) ab(a + b) ab(a + b)2 Expand the polynomial 2x + 4x − 4x − 2x .

32x3 0 32x3 − 8x 9000088809 (level 2): Simplify the following expression. 32x3 − 32x2 + 8x 1 1 m2 + 2mn + n2 − · m − n m + n 2n 9000101605 (level 2): 3 Expand 4x2y + 2xy2 .

13 64x6y3 + 96x5y4 + 48x4y5 + 8x3y6 Factor the following polynomial. 15xy − 10x − 3y + 2 16x2y3 + 24x3y3 + 8x3y6

6 3 3 3 4 5 3 6 64x y + 96x y + 96x y + 8x y (5x − 1) (3y − 2) 5x (3y − 2) 4x (3y − 2)

64x6y3 + 8x3y6 −5x (3y − 2)

9000101606 (level 2): 9000101702 (level 2): 3 2 Expand (x − y) − x (x + y) . Factor the following polynomial.

3x3 + 3x2y + 4xy + 4y2 −y3 − 5x2y + 2xy2 y3 − 5x2y + 2xy2

−y3 − 5x2y − 4xy2 −y3 − 5x2y + 4xy2 3x2 + 4y (x + y) (3x + y) x2 + y2 3x2 + 4 x + y2

2 9000101607 (level 2): 3x + y (x + y) 3 3 Expand x2 − y − y + x2 .

9000101703 (level 2): −6x4y − 2y3 −2y3 Factor the following polynomial.

−6x4y − 2y3 + 6x2y2 6x2y − 2y3 (5x − y)2 − (x − y)2

9000101608 (level 2): 4x (6x − 2y) x (5x − y) 6x (6x − 2y) −32x2 Expand (3x + y) 9x2 − 3xy + y2.

27x3 + y3 27x3 − y3 (3x + y)3 27x3 + 3y3 9000101704 (level 2): Factor the following polynomial.

16x2y4 − 25x4y2 9000101609 (level 2): Simplify using long division of polynomials.

2 2 2 2 2 2 3x3 + 17x2 + 23x + 5 : x2 + 4x + 1 4xy − 5x y 4xy + 5x y 4xy − 5x y 4xy + 5xy

2 2 2 2 2 4x y − 5xy 4x y + 5xy 4xy2 − 5x2y 3x + 5 3x − 5 3x + 1 3x − 1

9000101610 (level 2): 9000101705 (level 2): Simplify using long division of polynomials. Factor the following polynomial expression. 3 2 2x − x − 3x − 1 : (2x + 1) 16a2b2 − 4a2c2 − 16b2d2 + 4c2d2

x2 − x − 1 x2 − x + 1 x2 + x + 1 x2 − 2x − 1 4 (a − d)(a + d) (2b + c) (2b − c)

4 (a + b)2 (2b + c)2 9000101701 (level 2):

14 4 (a − b)(a + b) (2b + c) (2b − c) z 3x2 + 2y 9x4 − 6x2y + 4y2

4 (a − c)(a + c) (2b + d) (2b − d) z 3x2 − 2y 9x4 + 6x2y2 + 4y

9000101706 (level 2): 9000101710 (level 2): Factor the following polynomial. Factor the following polynomial.

8x4 − 48x3 + 72x2 x2y − x2z − 4xyz + 4xy2 + 4y3 − 4y2z

2 2 2 8x2 (x − 3)2 −8x2 (3 − x)2 8 x2 − 3 (y − z)(x + 2y) (y − z)(x − 2y)

2 (y − z) x2 + 4y + 4y2 2 8x x2 − 3 (y + z)(x − 2y)

9000146201 (level 2): 9000101707 (level 2): Expand the following expression. Factor the following polynomial. 3 2x3 − y2 x6 − 1

9 6 2 3 4 6 9 6 2 3 4 6 (x − 1) (x + 1) x2 + x + 1 x2 − x + 1 8x − 12x y + 6x y − y 8x − 4x y + 2x y − y

6 5 2 3 4 5 6 5 2 3 4 5 (x − 1) (x + 1) x2 + x + 1 x2 − x − 1 8x − 12x y + 6x y − y 8x − 4x y + 2x y − y

2 2 (x − 1) (x + 1) x + 2x + 1 x − 2x + 1 9000146202 (level 2): Expand the following expression. 2 2 (x − 1) (x + 1) x + x − 1 x − x + 1 √ 3 a2 + 3b

9000101708 (level 2): √ √ Factor the following polynomial. a6 + 3 3a4b + 9a2b2 + 3 3b3 8x3 − 27 √ √ a6 + 3a4b + 3a2b2 + 3 3b3 √ √ (2x − 3) 4x2 + 6x + 9 (2x − 3) 4x2 − 6x + 9 a5 + 3 3a4b + 9a2b2 + 3 3b3 √ √ 5 4 2 2 3 (2x + 9) 4x2 − 6x + 9 (2x − 3) 4x2 + 6x − 9 a + 3a b + 3a b + 3 3b

9000146203 (level 2): 9000101709 (level 2): Expand the following expression. Factor the following polynomial. √ 2 5 27x6z − 8y3z x − 2y

2 4 2 2 √ √ z 3x − 2y 9x + 6x y + 4y x10 − 2 2x5y + 2y2 x10 − 2x5y + 2y2

2 4 2 2 √ √ z 3x + 2y 9x + 6x y − 4y x10 − 2 2x5y − 2y2 x10 − 2x5y − 2y2

15 (x − 4) (3x + 2) (x − 4) (3x − 2) (x + 4) (3x + 2) 9000146204 (level 2): Expand the following expression. (x + 4) (3x − 2) a 2 + 4b3 2 9000146209 (level 2): Factor the following expression.

a2 a2 a2 27a3 − 8b9 + 4ab3 + 16b6 + 2ab3 + 16b6 + 4ab3 + 16b5 4 4 4

a2 3a − 2b3 9a2 + 6ab3 + 4b6 + 2ab3 + 16b5 4 3a + 2b3 9a2 − 6ab3 + 4b6

9000146205 (level 2): 3 2 3 6 Factor the following expression. 3a − 2b 9a + 12ab + 4b

6 2 9a − 4b 3a + 2b3 9a2 − 12ab3 + 4b6

3 3 3 3 3a − 2b 3a + 2b 3a − 2b 3a − 2b 9000146210 (level 2): Factor the following expression. 3a3 − 2b 3a2 + 2b 3a3 − 2b 3a2 − 2b 64x6 + 125

9000146206 (level 2): 2 4 2 Factor the following expression. 4x + 5 16x − 20x + 25

2 10 x y − 81 4x2 − 5 16x4 + 20x2 + 25

4x2 + 5 16x3 − 20x2 + 25 xy5 − 9 xy5 + 9 xy5 − 9 xy5 − 9

4x2 − 5 16x3 + 20x2 + 25 xy5 − 9 xy2 + 9 xy5 − 9 xy2 − 9

9000146701 (level 2): 9000146207 (level 2): Expand the following polynomial. Factor the following expression. 2 − (2x + 1) + x(5 − 2x) − 3(x − 2) 4a2 − (a − 1)2

−2x2 + 7 −2x2 + 9 −2x2 − 3 −2x2 − 5 (a + 1) (3a − 1) (a − 1) (3a − 1) (a + 1) (3a + 1)

(a − 1) (3a + 1) 9000146702 (level 2): Expand the following expression.

9000146208 (level 2): a − 4(2 − a) − a(5a + 1) + 2a(3 − 2a) Factor the following expression.

(2x − 1)2 − (x + 3)2

16 2 2 2 15 −9a + 10a − 8 −9a + 12a − 8 −9a + 2a − 8 3x + 2 3x − 2 3x − 7 + 2x − 3 −9a2 + 4a − 8 27 3x − 7 − 2x − 3 9000146703 (level 2): Expand the following expression. 9000146708 (level 2): (a − 2)(5a + 3) − (2a + 1)(3 − a) Divide the following two polynomials using long division.

2x3 + x2 − 17x + 5 : x2 + 3x − 1 7a2 − 12a − 9 3a2 − 12a − 9 7a2 − 2a − 9

3a2 − 2a − 9 2x − 5 2x + 5

2x − 2 12 − 40x 9000146704 (level 2): 2x + 7 + 2x + 7 + Expand the following polynomial. x2 + 3x − 1 x2 + 3x − 1

(3 − x)(x − 2) − (x + 1)(x − 3) 9000146709 (level 2): Divide the following two polynomials using long division. −2x2 + 7x − 3 −2x2 + 3x − 9 −2x2 + 3x − 3 4x2 − 10x − 1 :(x − 2) −2x2 + 7x − 9 5 3 5 4x − 2 − 4x − 2 + 4x + 2 − 9000146705 (level 2): x − 2 x − 2 x − 2 Expand the following expression. 3 4x + 2 + (3 − 2a)2 − (3a − 4)(3a + 4) x − 2

−5a2 − 12a + 25 −5a2 − 12a − 7 −5a2 + 25 9000146710 (level 2): Divide the following two polynomials using long division. 2 −5a − 7 x3 + 3x2 − x + 4 : x2 − x + 1

9000146706 (level 2): 2x 2x + 8 Expand the following polynomial. x + 4 + x + 4 + x2 − x + 1 x2 − x + 1 (2x + 3)2 − (2 − x)2 6 − 2x 2x + 2 x + 2 + x + 2 + x2 − x + 1 x2 − x + 1 3x2 + 16x + 5 3x2 + 8x + 13 3x2 + 13

2 9000083601 (level 3): 3x + 5 Find the conditions on x under which the expression

x−y x+y x+y − x−y 9000146707 (level 2): xy Divide the following two polynomials using long division. x2−y2 is well-deﬁned. 6x2 − 5x − 6 : (2x − 3)

17 x 6= 0, y 6= 0, x 6= ±y x 6= −y 4 Expressions with powers and roots

x 6= ±y x 6= 0, y 6= 0 9000010501 (level 1): For x ∈ R, x > 0, simplify the following expression. 9000083607 (level 3): √ Assuming x 6= 0, x 6= ±1, y 6= 0, simplify the expression. 3 x5 " # x 2 x − 12 2xy : : √ √ √ √ x + 1 y x2 − 1 x 3 x2 x2 3 x2 x3 3 x2 x2 5 x3

xy x2 − 1 x − 1 9000010502 (level 1): 2 4 For x ∈ , x > 0, simplify the following expression. 2 (x − 1) 4 4 R √ √ 3 x5 · 3 x4

9000083608 (level 3): √ √ √ Assuming xy 6= −1, simplify the expression. 3 3 x3 x12 3 x x x4 x−y 1+xy + y y(x−y) 1 − 1+xy 9000010503 (level 1): For x ∈ R, x > 0, simplify the following expression. √ √ 5 2 x(1 + y2) x · x x x − 1 x(1 + y2) 1 − y2 √ √ √ √ 10 x7 10 x 5 x2 10 x2 9000083609 (level 3): Assuming x 6= 0, x 6= ±y, y 6= 0, simplify the expression. 9000010504 (level 1): x2+y2 For x ∈ R, x > 0, simplify the following expression. x − 2y √ √ 1 1 xy 3 2 3 y2 − x2 · x+y x : x

√ √ 3 9 x − y x − y x x 1 x y(x − y) x(x − y) y x

9000010505 (level 1): 9000083610 (level 3): For x ∈ R, x > 0, simplify the following expression. √ Assuming x 6= ±y and y 6= 2x, simplify the expression. 5 √ x3 : 3 x 2x y y2 1 x + − : + x + y x − y x2 − y2 x + y x2 − y2 √ √ √ √ 15 x4 5 x 3 x2 5 x2

x x 2x − y 2x − y 1 9000010506 (level 1): For x ∈ R, x > 0, simplify the following expression. √ √ x · 2 x · 3 x

18 √ √ √ √ x 6 x5 6 x3 2 x x5 6 x5 9000013503 (level√ 2): Write the number 6 3−3 as a power with a rational exponent.

− 1 1 2 −2 9000010507 (level 1): 3 2 3 2 3 3 For x ∈ R, x > 0, simplify the following expression. √ x3 : 2 x 9000013504 (level 2): q√ Simplify 4 25. √ √ √ √ 2 2 3 2 2 6 √ √ √ √ x x x x 3 x 4 8 4 x 5 5 25 5

9000010508 (level 1): 9000013505 (level√ 2): 3 For x ∈ R, x > 0, simplify the following expression. Write the number 3 3 in an equivalent form involving just √ √ radical of a single number or radical of a single power. 3 x2 · 5 x4 √ √ √ √ 3 81 9 3 27 3 32 √ √ √ √ 15 x22 15 x6 15 x8 x3 15 x 9000013506 (level√ 2): Write the number 3 16000 in an equivalent form with smallest 9000010509 (level 1): possible number under the radical. For x ∈ , x > 0, simplify the following expression. R √ √ √ √ √ 3 3 3 3 3 20 2 10 4 100 2 4 10 x · x11

9000013507√ (level 2): √ √ √ √ 4 x4 3 x2 x11 3 x x12 3 x x 3 x Simplify 16 · 81. √ √ 6 36 6 4 6 9000010510 (level 1): For x ∈ R, x > 0, simplify the following expression. √ √ 9000013508 (level√ 2): 3 x : 6 x 7 Write the fraction √ in an equivalent form which does not 3 contain radical in the denominator.

√ √ √3 6 x 2 x x2 x √ √ √ √ 21 10 7 + 3 7 3 3 3 3 9000013501 (level 2): 3 Write the expression 2 4 in an equivalent form which does not 9000013509 (level 2): contain a rational exponent. √ Simplify the expression (1 + 2)2. √ √ √ √ 4 3 4 3 4 4 2 √ √ √ 2 2 2 3 3 + 2 2 3 3 − 2 2 3 + 2

9000013502 (level 2): 9000013510 (level 2): 6 − 5 1 Simplify the expression 0.5 7 · 0.5 14 and write it in an Write the fraction √ in an equivalent form which does equivalent form which does not contain a rational exponent. 1 + 2 not contain radical in the denominator. √ √ √ √ 7 14 14 0.5 0.5 0.511 0.5 √ √ 1 √ 2 − 1 2 √ 1 − 2 2

19 5 Absolute value −5x + 45 5x − 45 x − 45 x − 21

9000078505 (level 2): 9000081406 (level 2): Assuming x ∈ (0; ∞), simplify the following expression. For x ∈ R ﬁnd the correct relationship between |x| and | − x|.

3x − |2x| − | − x| |x| = | − x|

|x| > | − x| 0 2x 3x 4x |x| < | − x| 9000078506 (level 2): Assuming x ∈ (−∞; 0), simplify the following expression. None of them. The answer depends on the particular value of x. 3x − |2x| − | − x|

9000081407 (level 2): 6x 4x 2x 0 For x, y ∈ R ﬁnd the correct relationship between |x − y| and |y − x|.

9000078507 (level 2): |x − y| = |y − x| 1 Assuming x ∈ − ; 6 , simplify the following expression. 2 |x − y| > |y − x| 3 − |6 − x| + |2x + 1| |x − y| < |y − x|

3x − 2 x − 2 3x + 10 x + 8 None of them. The answer depends on the particular values of x, y.

9000078508 (level 2): Assuming x ∈ (1; ∞), simplify the following expression. 9000081408 (level 2): − For x ∈ R consider expressions |x|, | − x|, −|x| and −x. 3x − |2x + 1| + |x − 1| Which of these attains only negative values.

−|x| |x| | − x| −x 2x − 2 4x − 2 2x + 2 2x

9000081409 (level 2): 9000078509 (level 2): Among expressions 1 + |x|, |1 + x|, 1 − |x| and |1 − x| on the Evaluate the following expression. set x ∈ (−∞; −1) ﬁnd the expression which has smaller values than the other expressions from this list. |3 − 7| − |2(−4)| + |(−5)(−2)|

1 − |x| 1 + |x| |1 + x| 6 14 22 −2 |1 − x| None of them

9000078510 (level 2): Assuming x ∈ (6; 11), simplify the following expression.

3|x − 11| − 2|6 − x|

20 6 Percent problems 450 446 430 436

9000078908 (level 2): 9000078901 (level 2): A house price in the March was $259 000. The house has not If we increase an unknown number x by 5 %, we get 378. Find been sold for several months and the price dropped to x. $234 000 in August. Write the discount as percents of the original price? (Round to the tenth.) 360 359.1 396.9 350 9.7 % 10.7 % 10.5 % 9.5 % 9000078902 (level 2): If we decrease an unknown number x by 14 %, we get 602. Find x. 9000078909 (level 2): The salary of an employee is decreased by the tax at a ﬂat rate 14.5 %. An employee paid the tax $3 683. Find his/her 700 686.28 517.72 680 take home pay (i.e. the pre-tax income decreased by the tax).

9000078903 (level 2): $21 717 $25 400 $21 971 $23 352 The number 234 is by 20 % bigger than x. Find x.

195 187.2 280.8 205 9000078910 (level 2): There is 19 girls and 12 boys in the class. Find the diﬀerence between the percentage of the girls and boys in this class. 9000078904 (level 2): The number 87.5 is a 35 % from an unknown number x. Find 22.6 % 15.1 % 18.5 % 23.5 % x.

250 240 260 270 9000085301 (level 2): A bike was introduced to the market with the initial price $10 000 in April. As a consequence of bad sales values, the 9000078905 (level 2): price has been decreased by 20 % in June and by another 20 % A new laptop has been introduced to the marked with the in September. Find the price in September. original price $1 090. Soon after the introduction the price has been increased temporarily by 20 % and after several months $6 400 $6 000 $6 125 $6 500 decreased by 30 %. Find the final price for the laptop. Round your result to the nearest integer. 9000085302 (level 2): $916 $981 $952 $930 There are 32 students in the class (20 boys and 12 girls). One quarter of the boys and one quarter of the girls are excellent students. Suppose that one boy and one girl, both excellent 9000078906 (level 2): students, move to another school. Find the difference in the The price of a ship decreased in autumn by 16 % and in spring ratio of excellent students in the class before and after the increased by 4 %. This final price was $367 000. Find the change in the class. original price of the ship. (Round to the nearest thousand.) 5 % 6.25 % 7.5 % 8.25 % $420 000 $417 000 $409 000 $415 000

9000085303 (level 2): 9000078907 (level 2): The shop received 30 pieces of TV from the manufacturer The unknown number x has been increased by 20 % and then NoName. Five TVs in this delivery were defect. The shop also decreased by 5 %. The final value was 513. Find x. (Round to received TVs from the manufacturer GoodName. All these the closest integer.) TVs were perfect. The final inspection revealed that 10 % of TVs were defect in total. Use these information to find out

21 how many TVs have been supplied from the producer The edge of a cube is increased by 100 %. Find the increase of GoodName. the volume of the cube?

20 25 18 16 700 % 400 % 200 % 100 %

9000085304 (level 2): 9000085310 (level 2): The ﬁnal score of the hockey match between teams A and B The pad in the form of (not necessarily regular) octagon is was 2 : 2. The goaltender of A had success rate 90 % (he manufactured from a square. The side of the square is 4 cm. stopped 90 % of all shots on goal) and the goaltender of the An isosceles right triangle of the hypotenuse 1 cm is removed team B had fail rate 20 % (20 % of all shots on goal went into from each corner of the square which results in the desired his net). How many shots on goal has been in the match in octagon. How much of the material of the initial square goes total? to the waste?

30 25 35 40

9000085305 (level 2): The price for the ﬁrst issue of the book was $100, the price for the second issue $125. Find the rate of sale of the second issue if the target price for the second issue is the same as for the ﬁrst issue.

20 % 22,5 % 17,5 % 25 % 12.5 % 10 % 15 % 20 %

9000085306 (level 2): The ski has been sold for a reduced price in a sale after the season. The discount 18 % of the original price resumed in a price which was cheaper by $36. Find the original price of the ski.

$200 $250 $450 $500

9000085307 (level 2): A machine fills 2 000 bottles per hour. As a consequence of technical problems, the efficiency of the machine decreased by 10 %. How many bottles can be filled by the machine with the mentioned technical problems during a time period of 8 hours?

14 400 13 800 14 500 15 200

9000085308 (level 2): 1 1 1 1 The number + is bigger than − . Express the diﬀerence 2 3 2 3 between these numbers in percents of the smaller quantity.

400 % 500 % 300 % 600 %

9000085309 (level 2):