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Showing 301–316 of 316 problems
301.
Helena's vinyl record collection contains 33 LP records and 45 EP records. An LP record plays for three times as long as an EP record. How many more EP records must Helena buy so that the total playing time of her EP records equals the total playing time of her LP records?
(A) 45　 (B) 54　 (C) 63　 (D) 78　 (E) 87
302.
A sign in a parking lot states that the total price includes a 27% tax. To the nearest whole number, what percentage of the amount a customer pays is this tax?
(A)  21%\text{(A)}\;21\%(B)  22%\text{(B)}\;22\%(C)  27%\text{(C)}\;27\%(D)  37%\text{(D)}\;37\%(E)  73%\text{(E)}\;73\%
303.
Square ABCDABCD has side length 2424. Let EE be the midpoint of BC‾\overline{BC}, and let F‾\overline{F} be the intersection of AE‾\overline{AE} and BD‾\overline{BD}.
Square ABCD with E the midpoint of BC and F the intersection of AE and BD
What is the area of △BEF\triangle BEF?
(A)  36\text{(A)}\;36(B)  40\text{(B)}\;40(C)  45\text{(C)}\;45(D)  48\text{(D)}\;48(E)  52\text{(E)}\;52
304.
Half of the internal angles of a polygon measure 150∘150^\circ each, and the other half of its internal angles measure 190∘190^\circ each. How many sides does the polygon have?
(A)  36\text{(A)}\;36(B)  40\text{(B)}\;40(C)  60\text{(C)}\;60(D)  72\text{(D)}\;72(E)  180\text{(E)}\;180
305.
Let a1,a2,a3,…a_1,a_2,a_3,\ldots be a sequence with a1=94a_1=94, a2=190a_2=190, and ana_n equal to the arithmetic mean of the previous n−1n-1 terms for n≥3n\ge 3. What is the value of a10a_{10}?
(A)  47\text{(A)}\;47(B)  95\text{(B)}\;95(C)  142\text{(C)}\;142(D)  284\text{(D)}\;284(E)  564\text{(E)}\;564
306.
A triangle with sides 1616, 3030, and 3434 is inscribed in a circle, dividing the interior of the circle into four regions. Let AA, BB, and CC be the areas of the non-triangular regions, with CC being the greatest. What is the value of A+BA+B?
(A)  289π2−240\text{(A)}\;\frac{289\pi}{2}-240(B)  289π2−210\text{(B)}\;\frac{289\pi}{2}-210(C)  289π−480\text{(C)}\;289\pi-480(D)  289π−360\text{(D)}\;289\pi-360(E)  289π−240\text{(E)}\;289\pi-240
307.
How many ordered triples of positive integers (x,y,z)(x,y,z) satisfy both the following equations?
x+y+z=31x+y+z=31x+3y+3z=41x+3y+3z=41
(A)  4\text{(A)}\;4(B)  5\text{(B)}\;5(C)  6\text{(C)}\;6(D)  7\text{(D)}\;7(E)  8\text{(E)}\;8
308.
A driver sees an electronic highway sign stating that the next exit is 7 miles away and it will take 5 minutes to reach it. Assume both numbers have been rounded to the nearest integer. Among the following options, what is the lowest speed limit (in miles per hour) that would possibly allow the driver to legally reach the exit within the given time frame?
(A)  65\text{(A)}\;65(B)  70\text{(B)}\;70(C)  75\text{(C)}\;75(D)  80\text{(D)}\;80(E)  85\text{(E)}\;85
309.
Two similar but not congruent scalene triangles each have a side with length 99 and a side with length 1212. What is the length of the longest side of the larger triangle?
(A)  12\text{(A)}\;12(B)  14\text{(B)}\;14(C)  15\text{(C)}\;15(D)  16\text{(D)}\;16(E)  17\text{(E)}\;17
310.
For what odd positive integer nn is the mean of 1,4,9,16,…,n21,4,9,16,\ldots,n^2 exactly 32%32\% greater than its median?
(A)  19\text{(A)}\;19(B)  39\text{(B)}\;39(C)  49\text{(C)}\;49(D)  89\text{(D)}\;89(E)  169\text{(E)}\;169
311.
Gary writes down six distinct positive integers: 2,5,24,60,2402,5,24,60,240, and a three-digit number nn. One of those six numbers is the least common multiple of all six of them. What is the sum of all possible values of nn?
(A)  960\text{(A)}\;960(B)  1680\text{(B)}\;1680(C)  2160\text{(C)}\;2160(D)  2280\text{(D)}\;2280(E)  2400\text{(E)}\;2400
312.
A drawer has 100 red socks and 100 blue socks. Pippi randomly removes one sock at a time, without replacement, until all the socks of one color have been removed. What is the probability that there are exactly 2 socks left in the drawer once that happens?
(A)  27196\text{(A)}\;\frac{27}{196}(B)  49198\text{(B)}\;\frac{49}{198}(C)  247990\text{(C)}\;\frac{247}{990}(D)  50199\text{(D)}\;\frac{50}{199}(E)  2599\text{(E)}\;\frac{25}{99}
313.
Two bicycle clubs, with disjoint memberships, go on a tour together. On the morning of the tour, members greet each other with a handshake. Everyone shakes hands with everyone else exactly once. There were 119 handshakes between members of the same club and 112 handshakes between members of different clubs. How many more people came from the larger club than from the smaller club?
(A)  0\text{(A)}\;0(B)  4\text{(B)}\;4(C)  6\text{(C)}\;6(D)  7\text{(D)}\;7(E)  8\text{(E)}\;8
314.
Consider a triangle with positive area whose side lengths are functions of tt given by a=t2+1a=t^2+1, b=5t+7b=5t+7, and c=4t+4c=4t+4. How many different integer values of tt are possible?
(A)  0\text{(A)}\;0(B)  7\text{(B)}\;7(C)  9\text{(C)}\;9(D)  10\text{(D)}\;10(E)  infinitely many\text{(E)}\;\text{infinitely many}
315.
A pyramid of fractions starts with 12\frac{1}{2} in row 0. Each successive row has aa+b\frac{a}{a+b} and ba+b\frac{b}{a+b}; for instance, in row 1, 13\frac{1}{3} and 23\frac{2}{3} appear. In which row will 2027\frac{20}{27} appear?
Pyramid of fractions beginning with 1/2 in row 0
(A)  5\text{(A)}\;5(B)  8\text{(B)}\;8(C)  12\text{(C)}\;12(D)  26\text{(D)}\;26(E)  The node 2027 does not appear in the diagram.\text{(E)}\;\text{The node }\frac{20}{27}\text{ does not appear in the diagram.}
316.
An urn contains 6 red balls, 4 black balls, and 4 white balls. Suppose 7 balls are drawn at random without replacement and a score is computed as follows:
1 point is given for each red ball up to a maximum of 3 points.
1 point is given for each black ball up to a maximum of 2 points.
1 point is given for each white ball up to a maximum of 2 points.
For example, drawing 1 red ball, 4 black balls, and 2 white balls would earn a score of 1 + 2 + 2 = 5. What is the probability of getting the lowest possible score?
(A)  1858\text{(A)}\;\frac{1}{858}(B)  53432\text{(B)}\;\frac{5}{3432}(C)  1312\text{(C)}\;\frac{1}{312}(D)  2429\text{(D)}\;\frac{2}{429}(E)  5264\text{(E)}\;\frac{5}{264}