Example 1: “As an investment banker, I….”

Example 2: “Managing a team to code a new software product for ABC Corp., I….”

In these brief examples, each candidate mistakenly introduces the reader to the very overrepresentation that he/she is trying to avoid. Many applicants feel it necessary to start their essays by offering their titles or company names, but this approach can immediately give the reader pause: “Here we go again.”

Overrepresented candidates need to consider their introductory lines quite carefully. Rather than stating the obvious, a candidate might immerse the reader in a situation or present a special aspect of his/her position.

Example 1 (launching into a story): “At 5:30 pm, I could rest easy. The deadline for all other offers had passed. At that point, I knew….”

Example 2: (stand out): “While managing a multinational team, half in Silicon Valley and half in Pakistan, I….”

In the first example here, the banker candidate avoids drab self-introduction and instead immerses the reader in an unraveling mystery. In the second example, the software engineer candidate introduces him-/herself not as a “coder” but as a multinational manager. While each applicant’s situation is different, every candidate can work with his/her story to avoid the pitfalls of overrepresentation.

First, draw the coordinate plane and add in point A, which has coordinates (d,d). Since d is greater than 0, this point lies in the first quadrant, as shown:

Notice that the origin, the point (0, d), the pint (d, 0), and point A (d,d) form the corners of a square:

Now, we draw the line segment between (0,0) and (d,d). This is the diameter of a circle, which we also draw:

Notice that the answer doe not depend on d. All that matters is that the circle contains a square. The fraction of the circle’s area in the first quadrant will be the same, no matter what. Thus, we can drop d as a variable and create a new variable r for the radius of the circle. It will be easier to compute areas in terms of r (which will also cancel out in the end).

The area of the circle is just ?r².

The fraction we are looking for is this:

So we need the shaded area of the circle, which consists of the square ( ) and 2 of the four “leaves of the table” (  ).

The square’s area can be found this way:

Now the area of the 4 leaves (  ) is the area of the circle (  ) minus the area of the square ( ). So the area of the 4 leaves is ?r² – 2r² = (? – 2)r².

This means that the area of 2 leaves is .

The whole shaded area is thus .

Finally, the desired fraction is .

The correct answer is D.

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The simplest way to approach this problem is to work backwards from the answer choices. Let’s construct a possible value of n for each choice, and then test those values against the given constraints.

Since we are asked for the remainder after division by 30, the easiest possible value of n for each choice is 30 more than the choice.

(A) 3 + 30 gives us n = 33

(B) 12 + 30 gives us n = 42

(C) 18 + 30 gives us n = 48

(D) 22 + 30 gives us n = 52

(E) 28 + 30 gives us n = 58

Now test those values of n against the constraints.

(A) n = 33 divided by 6 gives remainder 3 – FAIL

(B) n = 42 divided by 6 gives remainder 0 – FAIL

(C) n = 48 divided by 6 gives remainder 0 – FAIL

(D) n = 52 divided by 6 gives remainder 4 – PASS

(E) n = 58 divided by 6 gives remainder 4 – PASS

We can now just test the surviving choices for how they behave upon division by 5. To leave remainder 3 after division by 5, a number must end in either 3 or 5:

(D) n = 52 divided by 5 gives remainder 2 – FAIL

(E) n = 58 divided by 5 gives remainder 3 – PASS

The correct answer is therefore (E).

Another way to approach this problem is to translate the given language of remainders into the language of multiples. If n leaves a remainder of 4 after division by 6, then n is 4 more than a multiple of 6. Leaving aside the size requirement for a moment, we can see that n could be 4, 10, 16, 22, 28, 34, etc.

Likewise, if n leaves a remainder of 3 after division by 5, then n is 3 more than a multiple of 5. Again leaving aside the size requirement, we can see that n could be 3, 8, 13, 18, 23, 28, 33, etc. As we noted earlier, n must end in 3 or 8.

We might now spot 28 on both lists. Although n is not actually allowed to be 28 (because n must be larger than 30), we might try adding 30 to it to get 58. Since 30 is a multiple of 6, adding 30 to 28 won’t change the fact that after division by 6, we’ll get 4 as the remainder. The same idea holds true for 5: since 30 is a multiple of 5, adding 30 to 28 won’t change the fact that after division by 5, we’ll get 3 as the remainder. This way, we have constructed a possible n without using the answer choices.

Finally, the remainder after dividing 58 by 30 is 28.

Again, the correct answer is E.

Positive integer n leaves a remainder of 4 after division by 6 and a remainder of 3 after division by 5. If n is greater than 30, what is the remainder that n leaves after division by 30?

(A) 3

(B) 12

(C) 18

(D) 22

(E) 28

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With this application season nearing its end, the time has come for applicants to make some choices. While many are now in the fortunate position of choosing between schools, some are suddenly confronting a different and unexpected question: “Where did I go wrong?” For those seeking to analyze their applications, we have posed several questions and explored the answers to help you identify common errors, which you may have inadvertently let slip by.  

Did you write essays that clearly connect your candidacy with your target program?

Although connecting your goals and ambitions with your target school’s resources is always critical, in a competitive year—when a school can easily accept one candidate over the other for the most arbitrary of reasons—ensuring that your essays have profoundly linked you to your target programs is especially important. As you reanalyze your responses to questions like “What are your short-term and long-term post-MBA goals? How will Columbia Business School help you achieve these goals?” think to yourself, “How strong was my case for Columbia?”

Without realizing it, many candidates offer very trite and clichéd reasons for wanting to attend target schools. It is not enough to write at a high level: “Kellogg offers a world-renowned marketing program and a strong team environment.”  The prospective applicant must truly make his/her own personal case, offering details of classes, clubs, professors, research centers, etc. Then, the candidate must directly tie these resources to his/her specific career goals to create a compelling case. If your rationale for applying to a certain school was not truly compelling, then your target business school likely chose another candidate who exhibited a true passion for and connection with its program. 

Did you connect with your target program?

MBA programs do their best to manage their yields (number of acceptances/number of offers), as a high yield can help boost their position in certain rankings. So, the schools want to know that candidates are truly interested in and committed to their programs. While visiting your target campuses is not a prerequisite, it is strongly encouraged; when candidates cannot visit, it is advisable for them to connect in some way with alumni or current students to get to know a target program beyond its Web site. Candidates should ask themselves whether they proved at some point in their applications that they have firsthand knowledge of and experience with their target programs. The candidate who never visited the classroom and did not come to campus to interview may inadvertently send the wrong message and may simply face a relative disadvantage compared with another, similarly qualified candidate. While pinpointing this as “the reason” is difficult, it can be a factor within the bigger decision. 

Did you clearly, credibly and profoundly express your career goals?

As you reexamine your application, another potential trouble spot to consider is your goal statement. Without realizing it, many candidates offer superficial goals and do not convey the requisite passion for or knowledge of their prospective careers:

“In the short term, when I graduate from NYU-Stern, I want to become a Brand Manager. After three years, I will climb to the position of VP of Marketing, and then in the long term, I will become a Director of Marketing.”

Providing the most basic information is not enough to answer the question. As you reread your goal statement, ask yourself whether you showed an understanding of the demands of your future position and the value that you could bring to it.  Further, consider whether you connected your existing skills and the skills you will acquire via your MBA with your career goals. Finally, the most difficult task, try to discern whether you exhibited true passion for your goals—are they ambitious, but achievable? Is there a personal/professional imperative that you strive for these heights? Again, in such a competitive field, if your goal statement lacked passion, there were certainly many others out there which exhibited it. 

Did you offer a diversity of perspective on your candidacy?

Many candidates unwittingly offer a very narrow view of themselves. As you reread your application, ask yourself, “Did the reader learn something new about me in each essay?” While offering two stories from the same sphere (i.e., two stories from your professional life) is not “forbidden,” it is important to ensure that the reader experiences something different in each essay. So if, for example, in one essay, you position yourself as team leader for a major project and then, in another, you position yourself as an individual mentor to a single struggling teammate, you are still offering a new perspective on yourself, even though both stories come from the professional sphere. Of course, optimally, you will have a wealth of professional, community, personal, athletic, international and entrepreneurial experiences (and more) from which to draw and continuously offer standout stories; however, even if you do not have such “riches,” you can still creatively diversify your candidacy. If you only offered one dimension, or even two, it likely was not enough to sustain the interest of an admissions officer who has read hundreds of applications.   

Did you select the appropriate recommenders, and were your recommendations powerful?

While the MBA admissions process is largely based on judgment, one “fact” that candidates should be aware of is that admissions committees expect at least one recommendation from a current supervisor, unless the candidate has offered explicit and justifiable reasons (in an optional essay) for why this was not possible. Still, occasionally, some candidates do not explain their reasoning or mistakenly ask an indirect supervisor with a “better” title than that of their direct supervisor to write a recommendation. In assessing your recommendations, ask yourself, “Were my recommenders true supervisors, and did they know me intimately?” If you cannot answer with an emphatic “yes,” you may have missed a significant opportunity to market yourself to the admissions committee.

At this point, with the process concluded, it would be acceptable for you to speak with your recommenders and diplomatically (note: this will require individual judgment) ask to see your letters. If you have the opportunity to read your letters, ask yourself some questions: “Did my recommender capture the essence of my performance?” “Did my recommender make a statement that I am an elite performer and distinguish me from others?” “Are my letters honest and balanced?” and “Did my recommenders provide clear examples to back up any statements of praise?” If your answers to these questions are “no,” then you should consider new recommenders going forward or ensure that your recommenders are educated about what constitutes a compelling letter.

Did you apply to the “wrong” programs?

This is always a challenging question to answer, especially during a particularly competitive year. Just because you did not get into a target program, does that really mean that your decision to apply was a “mistake?” Hopefully, you got some encouraging feedback via interview requests—an indication that your application resonated with the admissions committees and that they considered you competitive. However, if you applied to several schools and did not get a single interview request, you may want to reassess whether you were targeting the right “portfolio” of schools.

In the MBA admissions world, candidates are (surprise!) obsessed with their stats—GPAs and GMAT scores. At mbaMission, we, like admissions officers, tell candidates that stats are just one piece of the puzzle. However, if both your GPA and GMAT scores were well below the published averages for your target schools and you did not get an interview, this may be an initial indication that you were reaching.

After all of this analysis, at the end of this piece, we leave open a possibility: maybe you didn’t do anything terribly wrong? In a competitive year, even candidates who put together very strong applications may find themselves on the outside looking in.  So, to some, our message is “Don’t be too critical of yourself.”  If you are determined to get your MBA, you may just need to readjust your risk profile and take a more conservative approach, by adding a safe school, expanding your list of competitive schools and reducing your “reaches.” In the end, you may achieve your desired results anyway, but, if you broaden your approach, you will also have greater peace of mind as you approach this process for the second time.

The question cannot be easily rephrased to incorporate the particular information given. However, of course we should take note that both variables are integers and that x is less than y. We are looking for the value of x + y.

Statement (1): SUFFICIENT. First, we should list out all the possible scenarios in which integers x and y fit the equation x^y = 4.

There are three possibilities, as we can find by trial and error: 2² = 4, (-2)² = 4, and 4¹ = 4. However, of these possibilities, there is only one for which x is less than y, namely (-2)² = 4. Thus, we can find the value of x + y, which is -2 + 2 = 0.

Statement (2): SUFFICIENT. Knowing that |x| = |y| does not tell us the values of the integers. However, since they have the same absolute value, but x is less than y, it must be the case that y is a positive integer and x is the negative of that integer. For instance, if y is 5, then x is -5. The sum of x and y must therefore be 0, no matter what.

The correct answer is (D).

If x and y are integers and x < y, what is the value of x + y?

(1) x^y = 4

(2) |x| = |y|

A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D: EACH statement ALONE is sufficient.

E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

We are asked which function does NOT obey the rule f(f(x)) = f(f(f(f(x)))). This rule looks intimidating, but all it means on the left side is that we put some number x into the function, get the output, and then put that output back into the function and see what we get out. We do the same thing (feeding the function its own output) two more times on the right side, then compare the two sides.

Now let’s look at the functions.

(A) f(x) = -|x|

This function takes the absolute value of x, then puts a negative sign on. For instance, if x = 7, then f(x) = -|7| = -7. Likewise, if x = -8, then f(x) = -|-8| = -8. In words, f(x) turns any number negative (it’s the “negative absolute value.” Applying this process twice gives you the same number as applying it 4 times. INCORRECT.

(B) f(x) = 2 – x

Let’s see what happens when we try to calculate f(f(x)). Work your way from the inside out:

f(f(x)) = f(2 – x) = 2 – (2 – x) = 2 – 2 + x = x. In other words, f(f(x)) just gives us x back. Therefore, applying two MORE f‘s to get f(f(f(f(x)))) will give us x again as well. INCORRECT.

(C) f(x) = 3x

If f(x) = 3x, then f(f(x)) = f(3x) = 3(3x) = 9x. Applying two MORE f‘s to get f(f(f(f(x)))) will give us 3(3(9x) = 81x. 9x does NOT equal 81x for any nonzero x, in fact. CORRECT.

(D) f(x) = 4/x

We should finish out the list, just to make sure.

f(f(x)) = f(4/x) = 4/(4/x) = x. As with the function in (B), this function brings us back to x if we apply it twice. Thus, if we apply it 4 times, we also get back to x. INCORRECT.

(E) f(x) = 5

f(f(x)) = f(5) = 5. This function may seem tricky, but it’s actually very simple: it gives you back a 5 no matter what you feed into it. If you give it a 5, in particular, you still get a 5 back, no matter how many times you go through that cycle. INCORRECT.

The correct answer is C.

For which of the following functions f(x) is the relation f(f(x)) = f(f(f(f(x)))) NOT true for at least some values of x not equal to zero?

(A) f(x) = -|x|
(B) f(x) = 2 – x
(C) f(x) = 3x
(D) f(x) = 4/x
(E) f(x) = 5