Tüm alıştırma soruları

Soru 181Soru →

An inventory manager is organizing items into storage boxes. When the items are packed into boxes of $24$ , $36$ , or $45$ items each, there are always $11$ items remaining unpacked. If the total number of items is between $700$ and $1,000$ , how many items are there in total?

Cevabı ve açıklamayı göster

Cevap: 731

Cevap

731

The number of items

N

satisfies

N \equiv 11 \pmod{24}

,

N \equiv 11 \pmod{36}

, and

N \equiv 11 \pmod{45}

. Consequently,

N - 11

is a multiple of

\text{LCM}(24, 36, 45)

. The prime factorizations are

24 = 2^3 \times 3

,

36 = 2^2 \times 3^2

, and

45 = 3^2 \times 5

. The LCM is

2^3 \times 3^2 \times 5 = 360

. Thus,

N = 360k + 11

for some positive integer

k

. Testing values of

k

within the range

700 < N < 1,000

yields

k = 2

, giving

N = 360(2) + 11 = 731

.

Adım Adım Çözüm

1

Find the Least Common Multiple (LCM) of the divisors 24, 36, and 45.

LCM(24, 36, 45) = 360

Any number that leaves a constant remainder when divided by a set of numbers must be a constant remainder greater than a common multiple of those numbers.

2

Formulate the general equation for the total number of items N.

N = 360k + 11

Adding the common remainder 11 to any multiple of the LCM yields a valid total number of items.

3

Evaluate integer values of k to satisfy the constraint 700 < N < 1,000.

k = 2 gives N = 731

k = 1 gives 371, which is too small, and k = 3 gives 1,091, which exceeds 1,000.

Anahtar Kavram

Least Common Multiple (LCM) with Constant Remainder

Soru 182Soru →

Let $N = 19^{43} + 11^{43} - 6^{43}$ . What is the remainder when $N$ is divided by $7$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

3

Reducing each base modulo 7 simplifies the expression significantly:

19 \equiv -2 \pmod{7}

,

11 \equiv 4 \pmod{7}

, and

6 \equiv -1 \pmod{7}

. Utilizing the cyclicity period of 3 for powers of 2 modulo 7 (

2^3 \equiv 1 \pmod{7}

),

19^{43} \equiv -(2^{43}) \equiv -2 \equiv 5 \pmod{7}

. Similarly,

11^{43} \equiv 4^{43} = 2^{86} = (2^3)^{28} \times 4 \equiv 4 \pmod{7}

. Lastly,

6^{43} \equiv (-1)^{43} = -1 \pmod{7}

. Combining these yields

N \equiv 5 + 4 - (-1) = 10 \equiv 3 \pmod{7}

.

Adım Adım Çözüm

1

Reduce each base modulo 7

19 \equiv -2 \pmod{7}

,

11 \equiv 4 \pmod{7}

, and

6 \equiv -1 \pmod{7}

Simplifying bases using modular arithmetic reduces large power evaluations to smaller equivalents.

2

Evaluate

19^{43} \pmod{7}

using cyclicity

19^{43} \equiv -2 \equiv 5 \pmod{7}

Since

2^3 \equiv 1 \pmod{7}

and

43 = 3(14) + 1

,

2^{43} \equiv 2 \pmod{7}

, so

(-2)^{43} = -(2^{43}) \equiv -2 \pmod{7}

.

3

Evaluate

11^{43} \pmod{7}

using powers of 2

11^{43} \equiv 4 \pmod{7}

11^{43} \equiv 4^{43} = 2^{86} = (2^3)^{28} \times 2^2 \equiv 1^{28} \times 4 = 4 \pmod{7}

.

4

Evaluate

6^{43} \pmod{7}

6^{43} \equiv -1 \pmod{7}

Any odd power of

-1

equals

-1

.

5

Combine all components to determine the final remainder

N \equiv 5 + 4 - (-1) = 10 \equiv 3 \pmod{7}

Summing the individual remainders modulo 7 gives a final positive remainder of 3.

Anahtar Kavram

Modular Arithmetic and Cyclicity of Powers

Soru 183Soru →

A positive integer $n$ is divisible by $20$ but is not divisible by $8$ . If $n$ has exactly $15$ positive divisors and $10n$ has exactly $24$ positive divisors, what is the value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 2500

Cevap

2500

To determine

n

, analyze its prime factorization. Divisibility by

20 = 2^2 \cdot 5^1

requires that

n

contains

2

raised to at least the power of

2

and

5

raised to at least the power of

1

. The condition that

n

is not divisible by

8 = 2^3

restricts the exponent of

2

to exactly

2

. Thus,

n = 2^2 \cdot 5^b \cdot K'

, where

b \ge 1

and

K'

is a product of powers of distinct prime factors other than

2

and

5

. The number of positive divisors of

n

is

d(n) = (2+1)(b+1)K = 3(b+1)K = 15

, which simplifies to

(b+1)K = 5

. Because

b \ge 1

, we have

b+1 \ge 2

. Since

5

is a prime number, its only divisor greater than or equal to

2

is

5

. Thus,

b+1 = 5

, giving

b = 4

, and

K = 1

, which means

n

has no prime factors other than

2

and

5

. Therefore,

n = 2^2 \cdot 5^4 = 4 \cdot 625 = 2500

. We verify that

10n = 2^3 \cdot 5^5

has

(3+1)(5+1) = 24

positive divisors, confirming the solution.

Adım Adım Çözüm

1

Determine the power of 2 in the prime factorization of

n

The exponent of

2

in

n

is exactly

2

Because

n

is a multiple of

20 = 2^2 \cdot 5

but not a multiple of

8 = 2^3

,

2^2

divides

n

but

2^3

does not.

2

Set up the divisor counting formula for

n

d(n) = (2+1)(b+1)K = 15 \implies (b+1)K = 5

, where

b \ge 1

is the exponent of

5

and

K

represents the product of terms from any additional prime factors

The total number of positive divisors of an integer

N = p_1^{a_1} p_2^{a_2} \dots

is given by

(a_1+1)(a_2+1)\dots

3

Solve for the exponents and prime factors of

n

b = 4

and

K = 1

, giving

n = 2^2 \cdot 5^4

Since

n

is divisible by

20

,

5

is a prime factor of

n

, so

b \ge 1

, which implies

b+1 \ge 2

. Since

5

is prime, its only factor greater than

1

is

5

, forcing

b+1 = 5

and

K = 1

.

4

Verify

d(10n)

and evaluate

n

10n = 2^3 \cdot 5^5

has

(3+1)(5+1) = 24

divisors, and

n = 4 \cdot 625 = 2500

Multiplying

n

by

10 = 2 \cdot 5

increases the exponent of

2

from

2

to

3

and the exponent of

5

from

4

to

5

.

Anahtar Kavram

Divisor Count Formula and Prime Factorization Constraints

Soru 184Soru →

If the quadratic equation $x^2 - kx + 36 = 0$ has two distinct positive integer roots, $r_1$ and $r_2$ , and $r_1 + 2$ and $r_2 + 2$ are the roots of the quadratic equation $x^2 - 19x + m = 0$ , what is the value of $m$ ?

Cevabı ve açıklamayı göster

Cevap: 70

Cevap

The value of

m

is 70.

By applying Vieta's formulas to both quadratic equations, we find that the sum of the original roots is

r_1 + r_2 = 15

and their product is

r_1 r_2 = 36

. Factoring

x^2 - 15x + 36 = (x-3)(x-12) = 0

confirms the original roots are

3

and

12

. The new roots are

3+2=5

and

12+2=14

. Their product

m = 5 \times 14 = 70

.

Adım Adım Çözüm

1

Apply Vieta's relations to the first quadratic equation

r_1 + r_2 = k

and

r_1 \cdot r_2 = 36

For any quadratic equation

x^2 + bx + c = 0

, the sum of roots is

-b

and the product of roots is

c

.

2

Apply Vieta's relations to the second quadratic equation

(r_1 + 2) + (r_2 + 2) = 19 \implies r_1 + r_2 = 15

The coefficient of

x

in

x^2 - 19x + m = 0

dictates that the sum of its roots equals

19

.

3

Determine the roots

r_1

and

r_2

and check integer constraints

r_1 = 3

and

r_2 = 12

The integer factors of

36

that sum to

15

are

3

and

12

, satisfying all conditions.

4

Compute the constant term

m

for the transformed equation

m = (3 + 2)(12 + 2) = 5 \times 14 = 70

The constant term

m

equals the product of the transformed roots

(r_1 + 2)

and

(r_2 + 2)

.

Anahtar Kavram

Vieta's Formulas and Quadratic Root Transformations

Tahmini Süre:2m 0s

Soru 185Soru →

Scanner X can scan a batch of archival documents in $6$ hours operating alone at a constant rate. Scanner Y can scan the exact same batch of documents in $3$ hours operating alone at a constant rate. Working together simultaneously at their respective constant rates, how many hours will it take Scanner X and Scanner Y to scan the entire batch of documents?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

It will take Scanner X and Scanner Y a total of

2

hours to scan the batch of documents together.

To calculate combined completion time, add the rates of each machine rather than their completion times. Scanner X completes

\frac{1}{6}

of the task per hour, and Scanner Y completes

\frac{1}{3}

(or

\frac{2}{6}

) of the task per hour. Together, they complete

\frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}

of the task per hour. The inverse of this rate gives the total time required:

2

hours.

Adım Adım Çözüm

1

Calculate individual work rates per hour.

Scanner X rate =

\frac{1}{6}

batch/hour, Scanner Y rate =

\frac{1}{3}

batch/hour.

Work rate is defined as

\text{Rate} = \frac{\text{Work}}{\text{Time}}

.

2

Add the individual rates together to find the total combined rate.

Combined rate =

\frac{1}{6} + \frac{1}{3} = \frac{3}{6} = \frac{1}{2}

batch/hour.

When entities work together simultaneously, their work rates are additive.

3

Compute total time taken for 1 whole batch.

Total Time =

2

hours.

Time is the reciprocal of the combined work rate:

\text{Time} = \frac{1}{\text{Combined Rate}}

.

Anahtar Kavram

Combined Work Rate

Tahmini Süre:45s

Soru 186Soru →

Three automated data-processing pipelines, $A$ , $B$ , and $C$ , process incoming dataset files at constant individual rates. Pipeline $A$ operating alone can complete a standard processing job in $10$ hours, while Pipeline $B$ operating alone can complete the same job in $15$ hours.

The processing begins with Pipelines $A$ and $B$ working together at their standard rates for $2$ hours. At that point, Pipeline $A$ encounters a network slowdown that reduces its processing rate by $50\%$ for the remainder of the job. Pipelines $A$ and $B$ continue working at these modified rates for another $2$ hours, after which Pipeline $C$ is brought online to assist them. If Pipeline $C$ operating alone at its constant rate could finish the entire job in $10$ hours, how many total hours from the start of the processing will it take for the job to be completed?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The total time required to complete the job is 6 hours.

To solve multi-stage work rate problems, determine the rate per unit of time for each participant, multiply by the duration of each operational phase, and track the remaining fraction of the job. In the first 2 hours, Pipelines

A

and

B

complete

\frac{1}{3}

of the job. In the next 2 hours with Pipeline

A

at half speed (rate

\frac{1}{20}

), they complete an additional

\frac{7}{30}

of the job, leaving

\frac{13}{30}

unfinished. When Pipeline

C

(rate

\frac{1}{10}

) joins, the combined rate becomes

\frac{13}{60}

per hour, taking exactly 2 more hours to complete the remaining

\frac{13}{30}

of the job. Adding all three intervals (

2 + 2 + 2

) yields a total time of 6 hours.

Adım Adım Çözüm

1

Determine the individual hourly work rates for each pipeline

Rate of Pipeline

A = \frac{1}{10}

job/hr, Rate of Pipeline

B = \frac{1}{15}

job/hr, Rate of Pipeline

C = \frac{1}{10}

job/hr.

Work rate is the reciprocal of the total time taken to complete one full job.

2

Calculate work completed during Stage 1 (first 2 hours)

Combined rate =

\frac{1}{10} + \frac{1}{15} = \frac{1}{6}

job/hr. Work done =

2 \times \frac{1}{6} = \frac{1}{3}

of the job.

Both Pipeline

A

and Pipeline

B

operate together at their standard rates for 2 hours.

3

Calculate work completed during Stage 2 (next 2 hours)

Modified rate of Pipeline

A = 0.50 \times \frac{1}{10} = \frac{1}{20}

job/hr. Combined rate =

\frac{1}{20} + \frac{1}{15} = \frac{7}{60}

job/hr. Work done =

2 \times \frac{7}{60} = \frac{7}{30}

of the job.

Pipeline

A

's efficiency drops by 50%, while Pipeline

B

remains unchanged.

4

Calculate the remaining work after 4 hours

Total work completed =

\frac{1}{3} + \frac{7}{30} = \frac{17}{30}

. Remaining work =

1 - \frac{17}{30} = \frac{13}{30}

.

Subtracting cumulative work done from the total job (1) leaves the remaining portion.

5

Calculate the duration of Stage 3 when Pipeline

C

joins

Combined rate =

\frac{1}{20} + \frac{1}{15} + \frac{1}{10} = \frac{13}{60}

job/hr. Time required =

\frac{13/30}{13/60} = 2

hours.

Dividing remaining work by the combined rate of all three active pipelines gives the remaining time.

6

Sum the durations of all stages to find total elapsed time

Total time =

2\text{ hours} + 2\text{ hours} + 2\text{ hours} = 6\text{ hours}

.

The question asks for total elapsed time from the start of processing.

Anahtar Kavram

Multi-stage combined work rate problems require tracking accumulated work completed across distinct intervals with changing worker rates.

Soru 187Soru →

A beverage recipe requires mixing orange juice and cranberry juice in a ratio of $4 : 1$ . If a caterer uses $12$ liters of orange juice, how many liters of cranberry juice must be added to maintain this ratio?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The caterer must add 3 liters of cranberry juice.

The ratio of orange juice to cranberry juice is

4:1

. Since

12

liters of orange juice represents

4

parts, each part equals

\frac{12}{4} = 3

liters. Therefore,

1

part of cranberry juice equals

3

liters.

Adım Adım Çözüm

1

Set up the ratio equation based on the recipe requirements.

Orange Juice : Cranberry Juice =

4 : 1

The problem states that for every 4 parts of orange juice, 1 part of cranberry juice is required.

2

Calculate the unknown amount of cranberry juice using the given quantity of orange juice.

Cranberry Juice =

\frac{12}{4} = 3

liters

Since 12 liters represents 4 equal parts, dividing 12 by 4 yields the size of 1 part.

Anahtar Kavram

Direct Ratio and Proportion Scaling

Soru 188Soru →

A corporate marketing team initially allocates its quarterly budget among Digital, Print, and Event advertising in the ratio $5 : 3 : 2$ , respectively. Midway through the quarter, $\$12,000$ is reallocated from Digital advertising to Event advertising, while the Print advertising budget remains unchanged. As a result of this transfer, the ratio of the Digital advertising budget to the Event advertising budget becomes $4 : 3$ . What was the total initial quarterly budget, in dollars, allocated across all three advertising categories?

Cevabı ve açıklamayı göster

Cevap: 120000

Cevap

The total initial quarterly budget allocated across all three categories was 120,000 dollars.

By representing the initial amounts as

5x

,

3x

, and

2x

, the total initial budget is

10x

. Accounting for the

\$12,000

shift gives

\frac{5x - 12,000}{2x + 12,000} = \frac{4}{3}

. Solving yields

x = 12,000

, which gives a total initial budget of

10 \times 12,000 = 120,000

dollars.

Adım Adım Çözüm

1

Define initial allocations using a multiplier variable

Digital =

5x

, Print =

3x

, Event =

2x

, Total Initial Budget =

10x

Representing ratio components algebraically allows set up of linear equations.

2

Adjust allocations according to the budget transfer

New Digital =

5x - 12,000

, New Event =

2x + 12,000

Reallocating

\$12,000

reduces Digital by 12,000 and increases Event by 12,000.

3

Set up and solve the ratio equation for

x

\frac{5x - 12,000}{2x + 12,000} = \frac{4}{3} \Rightarrow 15x - 36,000 = 8x + 48,000 \Rightarrow 7x = 84,000 \Rightarrow x = 12,000

Equating the altered ratio of Digital to Event to

4/3

allows solving for multiplier

x

.

4

Compute the total initial budget

Total =

10 \times 12,000 = 120,000

Substituting

x = 12,000

back into the total initial expression

10x

gives the final answer.

Anahtar Kavram

Altering non-isolated ratios through part-to-part modifications

Soru 189Soru →

For all real numbers $x$ and $y$ , the custom operation $\diamondsuit$ is defined by $x \diamondsuit y = 3x - 2y + 1$ . What is the value of $5 \diamondsuit 4$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The value of

5 \diamondsuit 4

is 8.

Substituting

x = 5

and

y = 4

into the operation

x \diamondsuit y = 3x - 2y + 1

yields

3(5) - 2(4) + 1 = 15 - 8 + 1 = 8

.

Adım Adım Çözüm

1

Substitute the operands into the custom operator rule

5 \diamondsuit 4 = 3(5) - 2(4) + 1

The first term

5

corresponds to

x

and the second term

4

corresponds to

y

in the formula.

2

Simplify the numeric expression

15 - 8 + 1 = 8

Multiply first, then perform subtraction and addition from left to right.

Anahtar Kavram

Function Definitions, Evaluation, and Custom Operators

Soru 190Soru →

An executive board must arrange 7 books side-by-side on a display shelf: 3 identical copies of a Finance textbook, 2 identical copies of an Economics textbook, and 2 distinct novels (Novel A and Novel B). How many distinct linear arrangements of these 7 books are possible such that Novel A and Novel B are not placed adjacent to each other?

Cevabı ve açıklamayı göster

Cevap: 300

Cevap

300

To find the number of distinct arrangements where Novel A and Novel B are not adjacent, use complementary counting. First, compute the total distinct arrangements of the 7 books with identical items: 7! / (3! * 2!) = 420. Next, find the number of arrangements where Novel A and Novel B ARE adjacent by bundling them as one unit [AB], which has 2 internal orderings (AB or BA). Arranging the 6 units ([AB], 3 Finance, 2 Economics) gives 6! / (3! * 2!) = 60 ways. Multiplying by 2 internal orderings gives 120 adjacent arrangements. Subtracting these from the total yields 420 - 120 = 300.

Adım Adım Çözüm

1

Calculate the unrestricted total number of distinct permutations of the 7 books.

420 distinct arrangements

There are 7 total items with 3 identical Finance books and 2 identical Economics books. Using the formula N! / (n1! * n2! ...), we get 7! / (3! * 2!) = 5,040 / 12 = 420.

2

Calculate the number of forbidden arrangements where Novel A and Novel B are adjacent.

120 forbidden arrangements

Treat Novel A and Novel B as a single composite item [AB]. There are 2! = 2 ways to order A and B inside this block (AB or BA). Then, arrange the 6 items ([AB] and the 5 textbooks with repetitions): 6! / (3! * 2!) = 60. Multiplying by 2 internal orderings gives 2 * 60 = 120.

3

Subtract the forbidden (adjacent) arrangements from the total unrestricted arrangements.

300 valid arrangements

By complementary counting, Valid = Total - Forbidden = 420 - 120 = 300.

Anahtar Kavram

Counting permutations with indistinguishable items and non-adjacency restrictions using complementary counting

Tahmini Süre:2m 0s

Soru 191Soru →

For all non-zero real numbers $x$ , the function $f$ is defined by $f(x) = \frac{x^2 - 4}{2x}$ , and the function $g$ is defined by $g(x) = 3x - 1$ . What is the positive value of $a$ for which $f(g(a)) = 0$ ?

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The positive value of

a

for which

f(g(a)) = 0

is 1.

To solve

f(g(a)) = 0

, first find the values of

y

where

f(y) = 0

. The numerator of

f(y) = \frac{y^2 - 4}{2y}

equals zero when

y^2 - 4 = 0

, yielding

y = 2

and

y = -2

. Setting

g(a) = 3a - 1

equal to these roots gives

3a - 1 = 2 \implies a = 1

and

3a - 1 = -2 \implies a = -1/3

. Since

a

must be positive, the correct value is 1.

Adım Adım Çözüm

1

Determine the values of the argument

y

that satisfy

f(y) = 0

.

y = 2

or

y = -2

.

A fraction equals zero when its numerator is zero and its denominator is non-zero. Setting

y^2 - 4 = 0

gives

y = \pm 2

.

2

Substitute

g(a) = 3a - 1

into

y

to solve for

a

.

Solving

3a - 1 = 2

yields

a = 1

; solving

3a - 1 = -2

yields

a = -1/3

.

Setting the expression for

g(a)

equal to each root of

f(y) = 0

identifies all potential values for

a

.

3

Select the value of

a

matching the positivity condition.

a = 1

.

The question explicitly specifies finding the positive value of

a

.

Anahtar Kavram

Function composition and evaluation of nested functional equations.

Soru 192Soru →

At a bio-tech research facility, the ratio of male to female scientists is initially $5 : 3$ . After $4$ female scientists leave and $12$ male scientists join the facility, the ratio of male to female scientists becomes $4 : 1$ . If a group of $N$ additional female scientists subsequently joins the facility, changing the ratio of male to female scientists to $8 : 5$ , what is the value of $N$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The value of

N

is 12.

By setting up ratio equations stage-by-stage using multiplier

k

, we determine that

k = 4

, yielding 32 male and 8 female scientists after the first change. Solving

\frac{32}{8+N} = \frac{8}{5}

yields

N = 12

.

Adım Adım Çözüm

1

Define initial variables using a common ratio multiplier

k

.

Male scientists =

5k

, Female scientists =

3k

.

Ratios preserve proportionality through a constant multiplier.

2

Formulate and solve the equation for the first population adjustment.

\frac{5k + 12}{3k - 4} = 4 \implies 5k + 12 = 12k - 16 \implies 7k = 28 \implies k = 4

.

Adding 12 males and removing 4 females sets the new ratio to

4 : 1

.

3

Determine the exact number of male and female scientists at the facility after the first adjustment.

Male scientists =

5(4) + 12 = 32

; Female scientists =

3(4) - 4 = 8

.

Substitute

k = 4

back into the adjusted quantity expressions.

4

Formulate and solve the equation for the second population adjustment involving

N

.

\frac{32}{8 + N} = \frac{8}{5} \implies 8(8 + N) = 160 \implies 64 + 8N = 160 \implies 8N = 96 \implies N = 12

.

The number of male scientists remains 32 while female scientists increase by

N

to achieve an

8 : 5

ratio.

Anahtar Kavram

Multi-step ratio manipulation and algebraic modeling

Tahmini Süre:2m 0s

Soru 193Soru →

For all non-zero real numbers $a$ and $b$ , the binary operator $\diamond$ is defined by $a \diamond b = a - \frac{b}{a}$ . If $k$ is a positive integer such that $(k \diamond 3) \diamond 3 = \frac{1}{2}$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The positive integer value of

k

is 3.

Applying the non-commutative custom operator definition

a \diamond b = a - \frac{b}{a}

to the nested expression

(k \diamond 3) \diamond 3 = \frac{1}{2}

requires working from the inside out. Letting

u = k \diamond 3 = k - \frac{3}{k}

, the outer equation becomes

u \diamond 3 = u - \frac{3}{u} = \frac{1}{2}

. Multiplying through by

2u

gives

2u^2 - u - 6 = 0

, which factors as

(2u + 3)(u - 2) = 0

, yielding

u = 2

or

u = -\frac{3}{2}

. Substituting

u = 2

back into

k - \frac{3}{k} = 2

produces

k^2 - 2k - 3 = 0

, which factors as

(k - 3)(k + 1) = 0

. Since

k

must be a positive integer,

k = 3

. Testing

u = -\frac{3}{2}

gives

2k^2 + 3k - 6 = 0

, which has no integer solutions.

Adım Adım Çözüm

1

Substitute a temporary variable for the inner custom operator expression.

Let

u = k \diamond 3 = k - \frac{3}{k}

.

Simplifies the nested expression

(k \diamond 3) \diamond 3 = \frac{1}{2}

to a single variable operation

u \diamond 3 = \frac{1}{2}

.

2

Apply the operator definition to

u \diamond 3

and solve the resulting quadratic equation for

u

.

u - \frac{3}{u} = \frac{1}{2} \implies 2u^2 - u - 6 = 0 \implies (2u + 3)(u - 2) = 0

, so

u = 2

or

u = -\frac{3}{2}

.

Clearing the denominator

u

transforms the custom operator equation into a standard quadratic equation.

3

Solve for positive integer

k

using the obtained values of

u

.

Setting

k - \frac{3}{k} = 2

yields

k^2 - 2k - 3 = 0 \implies (k - 3)(k + 1) = 0

. Since

k

must be a positive integer,

k = 3

. Setting

k - \frac{3}{k} = -\frac{3}{2}

yields

2k^2 + 3k - 6 = 0

, which has non-integer discriminant

\sqrt{57}

.

Only

u = 2

yields an integer solution for

k

, satisfying all problem constraints.

Anahtar Kavram

Custom Operator Evaluation and Nested Functional Equations

Soru 194Soru →

For all non-zero real numbers $x$ , the function $f$ satisfies the equation $2f(x) + f\left(\frac{1}{x}\right) = 6x$ . What is the value of $f(2)$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

f(2)

is 7.

Substituting

x = 2

into the functional equation gives

2f(2) + f\left(\frac{1}{2}\right) = 12

. Substituting

x = \frac{1}{2}

gives

2f\left(\frac{1}{2}\right) + f(2) = 3

. Multiplying the first equation by 2 and subtracting the second equation eliminates

f\left(\frac{1}{2}\right)

, producing

3f(2) = 21

, so

f(2) = 7

.

Adım Adım Çözüm

1

Substitute

x = 2

into the functional equation

2f(2) + f\left(\frac{1}{2}\right) = 12

This establishes a linear relation between

f(2)

and

f\left(\frac{1}{2}\right)

.

2

Substitute

x = \frac{1}{2}

into the functional equation

2f\left(\frac{1}{2}\right) + f(2) = 3

This establishes a second linear relation involving the reciprocal input.

3

Eliminate

f\left(\frac{1}{2}\right)

by solving the system of equations

f(2) = 7

Multiplying the first equation by 2 yields

4f(2) + 2f\left(\frac{1}{2}\right) = 24

. Subtracting the second equation gives

3f(2) = 21

, which simplifies to

f(2) = 7

.

Anahtar Kavram

Evaluating functional equations by reciprocal variable substitution to form and solve a linear system.

Tahmini Süre:2m 0s

Soru 195Soru →

How many distinct 8-digit positive integers can be formed by rearranging all of the digits $1, 1, 1, 2, 2, 3, 3,$ and $3$ such that the resulting integer is even and no two $1$ s are adjacent?

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

40

To ensure the 8-digit number is even, the units (8th) digit must be 2. This leaves 7 positions to fill with three 1s, one 2, and three 3s. By arranging the 4 non-1 digits ({2, 3, 3, 3}) first, there are 4! / (1! 3!) = 4 distinct arrangements. Placing 4 digits creates 5 distinct gaps where the 1s can be placed without being adjacent. Choosing 3 gaps out of 5 for the three identical 1s gives C(5, 3) = 10 ways. Thus, the total number of valid integers is 4 * 10 = 40.

Adım Adım Çözüm

1

Fix the last digit to satisfy the even integer restriction

The 8th digit is fixed as 2, leaving 7 positions to fill with the remaining digits {1, 1, 1, 2, 3, 3, 3}.

An integer is even if and only if its units digit is even. The digit 2 is the only even digit in the set.

2

Calculate the arrangements of the non-restricted digits

4! / (1! * 3!) = 4 distinct arrangements.

Arranging the four non-1 digits ({2, 3, 3, 3}) first sets up the framework for placing the restricted 1s.

3

Place the three identical 1s into the gaps using the combination formula

C(5, 3) = 10 distinct gap selections.

The 4 arranged digits create 5 gaps. Selecting 3 distinct gaps ensures no two 1s are adjacent.

4

Multiply the arrangements of non-1 digits by the number of gap choices

4 * 10 = 40.

By the Fundamental Counting Principle, each non-1 arrangement can be combined with any valid gap placement.

Anahtar Kavram

Permutations with Indistinguishable Objects and Non-Adjacency Restrictions (Gap Method)

Tahmini Süre:2m 0s

Soru 196Soru →

If 2 is a root of the quadratic equation $x^2 - kx + 24 = 0$ , where $k$ is a constant, and the quadratic equation $x^2 - (k + 2)x + m = 0$ has exactly one real solution, what is the value of $m$ ?

Cevabı ve açıklamayı göster

Cevap: 64

Cevap

The value of

m

is 64.

Substituting

x = 2

into

x^2 - kx + 24 = 0

gives

4 - 2k + 24 = 0

, which simplifies to

2k = 28

or

k = 14

. Substituting

k = 14

into

x^2 - (k + 2)x + m = 0

produces

x^2 - 16x + m = 0

. For a quadratic equation

ax^2 + bx + c = 0

to have exactly one real solution, its discriminant

b^2 - 4ac

must equal 0. Therefore,

(-16)^2 - 4(1)(m) = 0

, which means

256 - 4m = 0

, giving

m = 64

.

Adım Adım Çözüm

1

Substitute the known root

x = 2

into the equation

x^2 - kx + 24 = 0

.

k = 14

Since

x = 2

is a root of the quadratic equation, evaluating the expression at

x = 2

must equal zero.

2

Substitute

k = 14

into the second quadratic equation

x^2 - (k + 2)x + m = 0

.

x^2 - 16x + m = 0

This determines the linear coefficient of the second quadratic equation.

3

Set the discriminant of

x^2 - 16x + m = 0

to zero.

m = 64

A quadratic equation

ax^2 + bx + c = 0

has exactly one real solution if and only if its discriminant

b^2 - 4ac

equals zero.

Anahtar Kavram

Solving quadratic equations via root substitution and applying the discriminant condition for repeated roots.

Tahmini Süre:1m 30s

Soru 197Soru →

A bookshelf is to be arranged using 3 identical Mathematics books, 2 identical Physics books, and 1 Chemistry book. In how many distinct ways can all 6 books be arranged in a single row such that the Chemistry book is not adjacent to any Mathematics book?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

The single Chemistry book must not touch any of the 3 Mathematics books. This leaves three mutually exclusive possibilities for the Chemistry book: at the left end preceded by a Physics book (4 ways), at the right end followed by a Physics book (4 ways), or sandwiched between both Physics books as a single block (4 ways). Summing these gives 12 total valid arrangements.

Adım Adım Çözüm

1

Analyze the placement restriction on the Chemistry book

The Chemistry book (C) cannot be placed next to any Mathematics book (M). Thus, C can only be adjacent to Physics books (P) or placed at the boundary of the row touching a P.

Eliminating adjacency to M restricts C to touch only P or the ends of the shelf.

2

Count valid arrangements when C is at the far left (position 1)

The row begins with C-P. The remaining 4 spots must be filled with 3 identical M's and 1 P, yielding 4! / (3! 1!) = 4 distinct ways.

Position 1 has only one neighbor (position 2), which must be P.

3

Count valid arrangements when C is at the far right (position 6)

The row ends with P-C. The remaining 4 spots must be filled with 3 identical M's and 1 P, yielding 4! / (3! 1!) = 4 distinct ways.

Position 6 has only one neighbor (position 5), which must be P.

4

Count valid arrangements when C is in an interior position (positions 2 through 5)

C must be sandwiched between two P's, forming the block (P-C-P). Arranging this single block along with the 3 identical M's (total of 4 items) yields 4! / (3! 1!) = 4 distinct ways.

Any interior placement requires both adjacent neighbors of C to be P.

5

Sum the counts from all mutually exclusive cases

Total valid arrangements = 4 + 4 + 4 = 12.

The three cases cover all possible non-overlapping valid placements for C.

Anahtar Kavram

Counting permutations with identical elements and positional restrictions using case analysis and block formation.

Tahmini Süre:1m 30s

Soru 198Soru →

A commercial bakery has two industrial ovens, Oven A and Oven B. Working alone at its constant rate, Oven A can bake a full order of $1,800$ pastries in $6$ hours. Working alone at its constant rate, Oven B can bake the same order in $9$ hours. If Oven A begins baking the order alone and is joined by Oven B after $2$ hours, how many total hours will it take from the time Oven A starts until the full order of $1,800$ pastries is baked?

Cevabı ve açıklamayı göster

Cevap: 4.4

Cevap

The total time required to complete the entire order of pastries is 4.4 hours.

The total time of 4.4 hours is derived by determining Oven A's rate (1/6 job/hour) and Oven B's rate (1/9 job/hour). In the first 2 hours, Oven A completes 1/3 of the job, leaving 2/3. Operating together, their combined rate is 5/18 job/hour, which finishes the remaining 2/3 of the job in 2.4 hours. Summing 2 hours and 2.4 hours yields 4.4 total hours.

Adım Adım Çözüm

1

Determine the individual work rates per hour.

Oven A completes

\frac{1}{6}

of the job per hour; Oven B completes

\frac{1}{9}

of the job per hour.

Work rate is the reciprocal of the total time needed to complete one full job.

2

Calculate the fraction of work completed during the first 2 hours by Oven A.

Oven A completes

2 \times \frac{1}{6} = \frac{1}{3}

of the total job.

Work done equals rate multiplied by time spent working.

3

Find the remaining fraction of work to be done.

Remaining work is

1 - \frac{1}{3} = \frac{2}{3}

of the job.

Subtract the completed fraction from 1 (the whole job).

4

Calculate the combined rate of Oven A and Oven B working together.

Combined rate is

\frac{1}{6} + \frac{1}{9} = \frac{3}{18} + \frac{2}{18} = \frac{5}{18}

job per hour.

When entities work together, their individual rates add up.

5

Calculate the time spent by both ovens working together to complete the remaining work.

Time together is

\frac{2/3}{5/18} = \frac{12}{5} = 2.4

hours.

Time equals remaining work divided by the combined rate.

6

Calculate total time from the start.

Total time =

2 + 2.4 = 4.4

hours.

Add the initial single-agent work duration to the combined work duration.

Anahtar Kavram

Work Rate and Combined Work

Soru 199Soru →

For all real numbers $x$ except $0$ and $1$ , the function $f(x)$ satisfies the relation $f(x) + 2f\left(\frac{1}{1-x}\right) = 9x$ . What is the value of $f(2)$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

f(2)

is

6

.

Evaluating the functional equation

f(x) + 2f\left(\frac{1}{1-x}\right) = 9x

at the values

x = 2

,

x = -1

, and

x = \frac{1}{2}

creates a system of three linear equations in terms of

f(2)

,

f(-1)

, and

f\left(\frac{1}{2}\right)

. Solving this system yields

9f(2) = 54

, which simplifies directly to

f(2) = 6

.

Adım Adım Çözüm

1

Evaluate the functional equation at

x = 2

f(2) + 2f\left(\frac{1}{1-2}\right) = 9(2) \implies f(2) + 2f(-1) = 18

Applying the input

x = 2

creates an equation connecting

f(2)

and

f(-1)

.

2

Evaluate the functional equation at

x = -1

f(-1) + 2f\left(\frac{1}{1-(-1)}\right) = 9(-1) \implies f(-1) + 2f\left(\frac{1}{2}\right) = -9

Evaluating at the new input

x = -1

generates a second equation connecting

f(-1)

and

f\left(\frac{1}{2}\right)

.

3

Evaluate the functional equation at

x = \frac{1}{2}

f\left(\frac{1}{2}\right) + 2f\left(\frac{1}{1-1/2}\right) = 9\left(\frac{1}{2}\right) \implies f\left(\frac{1}{2}\right) + 2f(2) = 4.5

Evaluating at

x = \frac{1}{2}

completes the cyclic chain by linking back to

f(2)

.

4

Solve the system of three linear equations for

f(2)

From equation 3,

f\left(\frac{1}{2}\right) = 4.5 - 2f(2)

. Substituting into equation 2 yields

f(-1) = 4f(2) - 18

. Substituting into equation 1 gives

f(2) + 2(4f(2) - 18) = 18 \implies 9f(2) = 54 \implies f(2) = 6

.

Eliminating

f\left(\frac{1}{2}\right)

and

f(-1)

isolated

f(2)

to find its exact numerical value.

Anahtar Kavram

Cyclic Functional Equations and Substitution Systems

Soru 200Soru →

If $2x^2 + 5x - 3 = 0$ and $x > 0$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 0.5

Cevap

The value of

x

is 0.5.

Factoring the quadratic equation

2x^2 + 5x - 3 = 0

yields

(2x - 1)(x + 3) = 0

. Setting each linear factor equal to zero gives two possible solutions for

x

:

x = 0.5

and

x = -3

. Because the problem stipulates that

x > 0

, the negative solution is discarded, leaving

x = 0.5

.

Adım Adım Çözüm

1

Factor the quadratic expression

(2x - 1)(x + 3) = 0

Splitting the middle term

5x

into

6x - x

allows grouping to factor by grouping.

2

Find the roots of the equation

x = 0.5

or

x = -3

By the zero-product property, if the product of two factors is zero, at least one factor must be zero.

3

Apply the positivity constraint

x > 0

x = 0.5

The root

x = -3

violates the given condition that

x

must be strictly greater than zero.

Anahtar Kavram

Solving quadratic equations by factoring and applying domain constraints

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