ARML Practice Problems Arvind Thiagarajan, May 7, 2006

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1 1 Geometry Problems ARML Practice Problems Arvind Thiagarajan, May 7, Find the coordinates of the point on the circle with equation (x 6) + (y 5) = 5 that is nearest the point (, 11). (TJ ARML 06 Practice 1). In rectangle ABCD, AB = 6 and BC = 8. Equilateral Triangles ADE and DCF are drawn on the exterior of the rectangle. If the area of triangle BEF is a 3 + b, find the ordered pair of rational numbers (a, b). (TJ ARML 06 Practice 1) 3. The ratio of the width of a rectangle to its length equals the ratio of its length to half its perimeter. If the area of the rectangle is 50( 5 1) and the width of the rectangle is k( 5 1), find the value of k. (TJ ARML 06 Practice ) 4. In ABC, AB = 5, BC = 6, and AC = 7. Points P on AB, Q on BC, and R on AC are located so that AP =, BQ =, CR = 3. If the area of ABC is x, then the area of PQR is kx. Find the value of k. (TJ ARML 06 Practice ) Point C lies on a circle one of whose diameters is AB. The bisector of CAB intersects BC at D and intersects the circle at E. If BD = 5 and CD = 7, find BE. (TJ ARML 06 Practice ) 6. In ABC, D is on AB so that AD:DB = 1 :, and G is on CD so that CG:GD = 3 :. If BG intersects AC at F, find BG:GF. (TJ ARML 06 Practice 4) 7. A chord of a triangle is a line segment whose endpoints lie on the sides of the triangle (but not at the vertices). For a triangle whose sides have lengths of 4, 5, and 6, find the length of the shortest chord that divides the triangle into two regions of equal area. (TJ ARML 06 Practice 4) 8. In right triangle ABC, AC = 4 and BC = 8. A square is drawn exterior to the triangle with AB as one side. Find the distance from C to the intersection of the diagonals of the square. (TJ ARML 06 Practice 5) 9. In trapezoid ABCD, the ratio of base AB to base CD is : 3. Diagonals AC and BD intersect at point X, and the line through X parallel to AB intersects AD at point P. Find the ratio of the area of PAX to the area of ABD. (TJ ARML 06 Practice 5) 1

2 10. The longer base of an isosceles trapezoid is a chord of a circle, and the shorter base is tangent to the circle. If the length of one leg is 5 and the lengths of the bases are 4 and 18, find the area of the circle. (TJ ARML 06 Practice 5) 11. Acute triangle ABC is inscribed in a circle. Altitudes AM and CN are extended to meet the circle again at P and Q respectively. If PQ:AC = 7 :, find the numerical value of sin B. (TJ ARML 06 Practice 5) 1. The length of each side of ABC is 6. If X is the trisection point of CA nearer C and if median AM intersects BX at U, then MU = k 3. Find k. (TJ ARML 06 Practice 6) 13. The length of a radius of a circle is Three congruent circles are drawn in the interior of the original circle, each internally tangent to the original circle and externally tangent to the others. Find the length of a radius of one of the three congruent circles. (TJ ARML 06 Practice 6) 14. In parallelogram ABCD, A is acute and AB = 5. Point E is on AD with AE = 4 and BE = 3. A line through B, perpendicular to CD, intersects CD at F. If BF = 5, find EF. (TJ ARML 06 Practice 6) 15. Find the numerical value of b for which the length of the path from A(0, ) to B(b, 0) to C(c, 10) to D(5, 9) will be a minimum. (TJ ARML 06 Practice 6) 16. In equilateral triangle ABC, points D, E, and F are on AB, BC, and CA, respectively, with AD = BE = CF = 1 and DB = EC = FA = 3. The area of DEF is a + b 3. Find the ordered pair of rational numbers (a, b). (TJ ARML 06 Practice 6) 17. Find the area of an equiangular octagon, the lengths of whose sides are alternately 1 and. 18. The length of the base of an isosceles triangle is 0. In the plane of the triangle, a point 4 units from this base is 10 units from each leg. Find the area of the triangle. Number Theory Problems x 1. Let [x] denote the greatest integer n such that n x. Let f(x) = [ 1 1 ][ 1 1 ]. If x 0 < x < 90, then the range of f consists of k elements. Find the value of k. (TJ ARML 06 Practice 1). Find all triples of real numbers (x, y, z) such that x + yz = 6 y + xz = 6 z + xy = 6 (TJ ARML 06 Practice 1)

3 3. In base eight, the four-digit numeral BBCC is the square of the two-digit numeral AA. Find the ordered triple of digits (A,B,C). (TJ ARML 06 Practice 1) 4. In arranging the ordered pairs of positive integers thusly: (1, 1), (1, ), (, 1), (1, 3), (, ), (3, 1), (1, 4), such that if two ordered pairs have a different element-sum, the one with the smaller element-sum comes first and if they have the same element-sum, the one with the smaller first element comes first. Find the 1978th ordered pair. (TJ ARML 06 Practice ) 5. The positive integer n, when divided by 3, 4, 5, 6, and 7, leaves remainders of, 3, 4, 5, and 6 respectively. Find the least possible value of n. (TJ ARML 06 Practice 4) 6. In base fifty, the integer x is represented by the numeral CC and x 3 is represented by the numeral ABBA. If C > 0, express all possible values of B in base ten. (TJ ARML 06 Practice 4) 7. The integers between 1 and 1000 inclusive are written in a row. Sam started at at 1 and circled every 4 th number in red. Janet started at at 1 and circled every 15 th number in blue. What is the smallest possible positive difference between a red number and a blue number? (TJ ARML 06 Practice 4) 8. The sum of 5 positive integers x, y, z, w, and u is equal to their product. If x y z w u, find the product xyzwu. 9. The sum of 19 consecutive positive integers equals p 3, where p is a prime number. Compute the smallest of the 19 integers. (TJ ARML 06 Practice 10) 10. Determine all positive primes p such that p p 1995 is a perfect square. (TJ ARML 06 Practice 10) 11. Compute the number of ordered triples (A, B, C) with A, B, C (0, 1,, 3, 4, 5, 6, 7, 8, 9) such that k is an integer if.abc +.ACB +.BAC +.BCA +.CAB +.CAB +.CBA k =..A +.B +.C (TJ ARML 06 Practice 10) 3 Logarithms and Exponents Problems 1. Find all positive numbers x that satisfy ( + log x) 3 + ( 1 + log x) 3 = (1 + log x ) 3. (TJ ARML 06 Practice 4). Of all ordered triples of positive integers (x, y, z) that satisfy 3x + 4y = 5z, find the smallest value of z. (TJ ARML 06 Practice 6) 3

4 3. Find all ordered pairs of real numbers (x, y) such that log x y 3 = 1 and log x y 3 = 7. 4 Sequences and Series 1. Express the following sum as a quotient of two integers: 15 n= (n 1)(n + 1).. The sum of 1999 positive numbers in an increasing arithmetic progression is 1. Compute the width of the smallest interval containing all possible values of the common difference. Do not leave your answer in factored form. (TJ ARML 06 Practice 10) 5 Combinatorics 1. A regular dodecahedron is a polyhedron whose 1 faces are congruent regular polygons. An interior diagonal of this polyhedron is a diagonal which is not wholly contained in one of the faces. A regular dodedecahedron has 30 edges and 0 vertices. How many interior diagonals does it have. 6 Trigonometry Problems 1. If sin 6 θ + cos 6 θ =, find all possible values of sin θ. (TJ ARML 06 Practice 1) 3. If 0 < x < π, find all values of x that satisfy sin 5x + sin 3x = 0. (TJ ARML 06 Practice 1) 3. Find the numerical value of sin 40 sin 80 + sin 80 sin sin 160 sin 30. (TJ ARML 06 Practice ) 4. If cos x+cos y +cos z = sin x+sin y +sin z = 0, find the numerical value of cos (x y)+ cos (y z) + cos (z x). (TJ ARML 06 Practice 4) 5. If x kπ, where k is an integer, then cot x tan x + cot x + tan x + sin x has a minimum value of m and maximum value of M, for all real number x. Find the ordered pair (m, M). (TJ ARML 06 Practice 6) 6. Find the numerical value of sin 18 cos 1 + cos 16 cos 10 sin cos 8 + cos 158 cos 98. 4

5 7 Polynomial Problems 1. a > 0, b > 0, a b, and a b + b a a b b a + a b b a a b + b a = ab Write an equation expressing a explicitly in terms of b. (TJ ARML 06 Practice 1). If a, b, and c are different numbers and if a 3 + 3a + 14 = 0, b 3 + 3b + 14 = 0 and c 3 + 3c + 14 = 0, find the value of 1 a + 1 b + 1. (TJ ARML 06 Practice ) c 3. Two drivers, A and B, were 5 km apart. They traveled towards each other at the same constant speed of x km per hour, with A having had a head start of 30 minutes. Upon meeting, each continued to the other s starting point at a constant speed of x 10 km per hour. If A completed the entire trip in 5 hours, find x. (TJ ARML 06 Practice ) 4. If x, y, z, a, b, and c are nonzero real numbers satisfying (4x + 9y + z )(a + b + c ) = (ax + 3by + cz), find the continued ratio x : y : z in terms of a, b, and c. (TJ ARML 06 Practice ) 5. One of the roots of x x x x + 1 = 9 10 is 1 + k, where k is a negative integer. Find k. 5 (TJ ARML 06 Practice 4) 6. If n > 1, find the two smallest integral values of n for which x + x + 1 is a factor of (x + 1) n x n 1, over the set of polynomials with integral coeffecients. (TJ ARML 06 Practice 5) 7. Every expression of the form a b +b c +c d +d a can be expressed as the sum of two squares in at least two different ways. Find any one of the three possible ordered pairs of positive integers (x, y), with x > y, that satisfies x +y = (TJ ARML 06 Practice 5) 8. Let A = xr 1 x, B = xs xr+s, and C = r 1 xs 1 x, where (1 r+s xr )(1 x s )(1 x r+s ) 0. Write an equation expressing C explicitly in terms of A and B. (TJ ARML 06 Practice 6) 9. Find all ordered pairs of integers (x, y) which satisfy x + 4x + y = The roots of ax + bx + c = 0 are irrational, but their calculator approximations are and If a, b, and c are integers whose greatest common 5

6 divisor is 1 and which satisfy a > 0, b 10 and c 10, compute the ordered triple (a, b, c). (TJ ARML 06 Practice 10) 8 Algorithmic Problems (Graph Theory, Greedy Algorithm, etc.) 1. One hundred pennies are arranged in seven stacks, of which no two stacks contain the same number of pennies. A student counts the number of pennies in each stack and takes 50 pennies in such a way as to disturb the fewest number of stacks. He ends up taking pennies from N stacks. For all such arrangements of pennies, what is the largest possible value of N that will be necessary. (TJ ARML 06 Practice 5) 6

For all questions, E. NOTA means none of the above answers is correct. Diagrams are NOT drawn to scale.

For all questions, E. NOTA means none of the above answers is correct. Diagrams are NOT drawn to scale. For all questions, means none of the above answers is correct. Diagrams are NOT drawn to scale.. In the diagram, given m = 57, m = (x+ ), m = (4x 5). Find the degree measure of the smallest angle. 5. The