MathDB

2016 AIME Problems

Part of AIME Problems

Subcontests

(15)
15
2

Circle Incident

Circles ω1\omega_1 and ω2\omega_2 intersect at points XX and YY. Line ℓ\ell is tangent to ω1\omega_1 and ω2\omega_2 at AA and BB, respectively, with line ABAB closer to point XX than to YY. Circle ω\omega passes through AA and BB intersecting ω1\omega_1 again at D≠AD \neq A and intersecting ω2\omega_2 again at C≠BC \neq B. The three points CC, YY, DD are collinear, XC=67XC = 67, XY=47XY = 47, and XD=37XD = 37. Find AB2AB^2.

2^215pro2^216me

For 1≤i≤2151\leq i\leq 215 let ai=12ia_i=\frac{1}{2^i} and a216=12215a_{216}=\frac{1}{2^{215}}. Let x1,x2,…,x216x_1,x_2,\ldots,x_{216} be positive real numbers such that \sum\limits_{i=1}^{216} x_i=1 \text{  and  } \sum\limits_{1\leq ix2=mnx_2=\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
14
2

Tiny Circles and Squares in R^2

Centered at each lattice point in the coordinate plane are a circle of radius 110\tfrac{1}{10} and a square with sides of length 15\tfrac{1}{5} whose sides are parallel to the coordinate axes. The line segment from (0,0)(0, 0) to (1001,429)(1001, 429) intersects mm of the squares and nn of the circles. Find m+nm + n.

All men are created equilateral

Equilateral △ABC\triangle ABC has side length 600600. Points PP and QQ lie outside of the plane of △ABC\triangle ABC and are on the opposite sides of the plane. Furthermore, PA=PB=PCPA=PB=PC, and QA=QB=QCQA=QB=QC, and the planes of △PAB\triangle PAB and △QAB\triangle QAB form a 120∘120^{\circ} dihedral angle (The angle between the two planes). There is a point OO whose distance from each of A,B,C,PA,B,C,P and QQ is dd. Find dd.
13
2

Kelvin the Frog

Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line y=24y = 24. A fence is located at the horizontal line y=0y = 0. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where y=0y=0, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where y<0y < 0. Freddy starts his search at the point (0,21)(0, 21) and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.

How about a nice game of chess?

Beatrix is going to place six rooks on a 6×66\times6 chessboard where both the rows and columns are labelled 11 to 66; the rooks are placed so that no two rooks are in the same row or the same column. The value of a square is the sum of its row number and column number. The score of an arrangement of rooks is the least value of any occupied square. The average score over all valid configurations is pq\frac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp+q.
12
2
11
2
10
2

Arithmetico-geometric... sequence?

A strictly increasing sequence of positive integers a1,a2,a3,…a_1, a_2, a_3, \ldots has the property that for every positive integer kk, the subsequence a2k−1,a2k,a2k+1a_{2k-1}, a_{2k}, a_{2k+1} is geometric and the subsequence a2k,a2k+1,a2k+2a_{2k}, a_{2k+1}, a_{2k+2} is arithmetic. Suppose that a13=2016a_{13} = 2016. Find a1a_1.

We meet again...

Triangle ABCABC is inscribed in circle ω\omega. Points PP and QQ are on side AB‾\overline{AB} with AP<AQAP<AQ. Rays CPCP and CQCQ meet ω\omega again at SS and TT (other than CC), respectively. If AP=4,PQ=3,QB=6,BT=5,AP=4,PQ=3,QB=6,BT=5, and AS=7AS=7, then ST=mnST=\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
9
2
7
2

Complex is Real

For integers aa and bb consider the complex number ab+2016ab+100−(∣a+b∣ab+100)i.\dfrac{\sqrt{ab+2016}}{ab+100} - \left(\frac{\sqrt{|a+b|}}{ab+100}\right)i. Find the number of ordered pairs of integers (a,b)(a, b) such that this complex number is a real number.

Be There or be Square

Squares ABCDABCD and EFGHEFGH have a common center and AB‾∥EF‾\overline{AB}\parallel \overline{EF}. The area of ABCDABCD is 20162016, and the area of EFGHEFGH is a smaller positive integer. Square IJKLIJKL is constructed so that each of its vertices lies on a side of ABCDABCD and each vertex of EFGHEFGH lies on a side of IJKLIJKL. Find the difference between the largest and smallest possible integer values of the area of IJKLIJKL.
6
2

The Fifth Fact

In △ABC\triangle ABC let II be the center of the inscribed circle, and let the bisector of ∠ACB\angle ACB intersect ABAB at LL. The line through CC and LL intersects the circumscribed circle of △ABC\triangle ABC at the two points CC and DD. If LI=2LI = 2 and LD=3LD = 3, then IC=pqIC = \tfrac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp + q.

p-addict

For polynomial P(x)=1−13x+16x2P(x)=1-\frac{1}{3}x+\frac{1}{6}x^2, define Q(x)=P(x)P(x3)P(x5)P(x7)P(x9)=∑i=050aixi. Q(x) = P(x)P(x^3)P(x^5)P(x^7)P(x^9) = \sum\limits_{i=0}^{50}a_ix^i. Then ∑i=050∣ai∣=mn\sum\limits_{i=0}^{50}|a_i|=\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
8
2
5
2

Anh read a book

Anh read a book. On the first day she read nn pages in tt minutes, where nn and tt are positive integers. On the second day Anh read n+1n + 1 pages in t+1t + 1 minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she completely read the 374374 page book. It took her a total of 319319 minutes to read the book. Find n+tn + t.

Bermuda Triangle

Triangle ABC0ABC_0 has a right angle at C0C_0. Its side lengths are pairwise relatively prime positive integers, and its perimeter is pp. Let C1C_1 be the foot of the altitude to AB‾\overline{AB}, and for n≥2n\geq 2, let CnC_n be the foot of the altitude to Cn−2B‾\overline{C_{n-2}B} in △Cn−2Cn−1B\triangle C_{n-2}C_{n-1}B. The sum ∑n=1∞Cn−1Cn=6p\sum\limits_{n=1}^{\infty}C_{n-1}C_n = 6p. Find pp.
4
2

Dihedral Angle in a Triangular Pyramid

A right prism with height hh has bases that are regular hexagons with sides of length 1212. A vertex AA of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain AA measures 60∘60^\circ. Find h2h^2.

Boxing Match

An a×b×ca\times b\times c rectangular box is built from a⋅b⋅ca\cdot b \cdot c unit cubes. Each unit cube is colored red, green, or yellow. Each of the aa layers of size 1×b×c1\times b \times c parallel to the (b×c)(b\times c)-faces of the box contains exactly 99 red cubes, exactly 12 green cubes, and some yellow cubes. Each of the bb layers of size a×1×ca\times 1 \times c parallel to the (a×c)(a\times c)-faces of the box contains exactly 20 green cubes, exactly 25 yellow cubes, and some red cubes. Find the smallest possible volume of the box.
3
2

Paths on an Icosahedron

A regular icosahedron is a 2020-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane, and a lower pentagon of five vertices all adjacent to the bottom vertex and all in another horizontal plane. Find the number of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no vertex is repeated.[asy] size(3cm); pair A=(0.05,0),B=(-.9,-0.6),C=(0,-0.45),D=(.9,-0.6),E=(.55,-0.85),F=(-0.55,-0.85),G=B-(0,1.1),H=F-(0,0.6),I=E-(0,0.6),J=D-(0,1.1),K=C-(0,1.4),L=C+K-A; draw(A--B--F--E--D--A--E--A--F--A^^B--G--F--K--G--L--J--K--E--J--D--J--L--K); draw(B--C--D--C--A--C--H--I--C--H--G--H--L--I--J--I--D^^H--B,dashed); dot(A^^B^^C^^D^^E^^F^^G^^H^^I^^J^^K^^L); [/asy]

L O G C E P T I O N

Let x,yx,y and zz be real numbers satisfying the system \begin{align*} \log_2(xyz-3+\log_5 x) &= 5 \\ \log_3(xyz-3+\log_5 y) &= 4 \\ \log_4(xyz-3+\log_5 z) &= 4. \end{align*} Find the value of ∣log⁡5x∣+∣log⁡5y∣+∣log⁡5z∣|\log_5 x|+|\log_5 y|+|\log_5 z|.
2
2
1
2