Found problems: 970
[u]Round 5[/u]
[b]p13.[/b] Simplify $\frac11+\frac13+\frac16+\frac{1}{10}+\frac{1}{15}+\frac{1}{21}$.
[b]p14.[/b] Given that $x + y = 7$ and $x^2 + y^2 = 29$, what is the sum of the reciprocals of $x$ and $y$?
[b]p15.[/b] Consider a rectangle $ABCD$ with side lengths $AB = 3$ and $BC = 4$. If circles are inscribeδ in triangles $ABC$ and $BCD$, how far are the centers of the circles from each other?
[u]Round 6[/u]
[b]p16.[/b] Evaluate $\frac{2!}{1!} +\frac{3!}{2!} +\frac{4!}{3!} + ... +\frac{99!}{98!}+\frac{100!}{99!}$ .
[b]p17.[/b] Let $ABCD$ be a square of side length $2$. A semicircle is drawn with diameter $\overline{AC}$ that passes through point $B$. Find the area of the region inside the semicircle but outside the square.
[b]p18.[/b] For how many positive integer values of $k$ is $\frac{37k - 30}{k}$ a positive integer?
[u]Round 7[/u]
[b]p19.[/b] Two parallel planar slices across a sphere of radius $25$ create cross sections of area $576\pi$ and $225\pi$. What is the maximum possible distance between the two slices?
[b]p20.[/b] How many positive integers cannot be expressed in the form $3\ell + 4m + 5t$, where $\ell$, $m$, and $t$ are nonnegative integers?
[b]p21.[/b] In April, a fool is someone who is fooled by a classmate. In a class of $30$ students, $14$ people were fooled by someone else and $29$ people fooled someone else. What is the largest positive integer $n$ for which we can guarantee that at least one person was fooled by at least $n$ other people?
[u]Round 8[/u]
[b]p22.[/b] Let $$S = 4 + \dfrac{12}{4 +\dfrac{ 12}{4 +\dfrac{ 12}{4+ ...}}}.$$ Evaluate $4 +\frac{ 12}{S}.$
[b]p23.[/b] Jonathan is buying bananagram sets for $\$11$ each and flip-flops for $\$17$ each. If he spends $\$227$ on purchases for bananagram sets and flip-flops, what is the total number of bananagram sets and flip-flops he bought?
[b]p24.[/b] Alan has a $3 \times 3$ array of squares. He starts removing the squares one at a time such that each time he removes one square, all remaining squares share a side with at least two other remaining squares. What is the maximum number of squares Alan can remove?
PS. You should use hide for answers. Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h2952214p26434209]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134133p28400917]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have?
In triangle $ABC$, $AB = AC = 3$ and $\angle A = 90^o$. Let $M$ be the midpoint of side $BC$. Points $D$ and $E$ lie on sides $AC$ and $AB$ respectively such that $AD > AE$ and $ADME$ is a cyclic quadrilateral. Given that triangle $EMD$ has area $2$, find the length of segment $CD$.
Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, while $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers where $c$ is not divisible by the square of any prime. Compute $a+b+c$.
[i]Proposed by Andrew Wen[/i]
In rectangle $ ABCD$, $ AB\equal{}100$. Let $ E$ be the midpoint of $ \overline{AD}$. Given that line $ AC$ and line $ BE$ are perpendicular, find the greatest integer less than $ AD$.
Let $ABC$ be a right triangle with $\angle{ACB}=90^{\circ}$. $D$ is a point on $AB$ such that $CD\perp AB$. If the area of triangle $ABC$ is $84$, what is the smallest possible value of $$AC^2+\left(3\cdot CD\right)^2+BC^2?$$
[i]2016 CCA Math Bonanza Lightning #2.3[/i]
Two circles, $\omega_1$ and $\omega_2,$ intersect at $X$ and $Y.$ The segment between their centers intersects $\omega_1$ and $\omega_2$ at $A$ and $B$ respectively, such that $AB=2.$ Given that the radii of $\omega_1$ and $\omega_2$ are $3$ and $4,$ respectively, find $XY.$
A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $\overline{AE}$, $\overline{BC}$, and $\overline{CD}$. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$.
Circles with centers $ (2,4)$ and $ (14,9)$ have radii 4 and 9, respectively. The equation of a common external tangent to the circles can be written in the form $ y \equal{} mx \plus{} b$ with $ m > 0$. What is $ b$?
[asy]
size(150); defaultpen(linewidth(0.7)+fontsize(8)); draw(circle((2,4),4));draw(circle((14,9),9)); draw((0,-2)--(0,20));draw((-6,0)--(25,0)); draw((2,4)--(2,4)+4*expi(pi*4.5/11)); draw((14,9)--(14,9)+9*expi(pi*6/7)); label("4",(2,4)+2*expi(pi*4.5/11),(-1,0)); label("9",(14,9)+4.5*expi(pi*6/7),(1,1)); label("(2,4)",(2,4),(0.5,-1.5));label("(14,9)",(14,9),(1,-1)); draw((-4,120*-4/119+912/119)--(11,120*11/119+912/119)); dot((2,4)^^(14,9));[/asy]
$ \textbf{(A) } \frac {908}{199}\qquad \textbf{(B) } \frac {909}{119}\qquad \textbf{(C) } \frac {130}{17}\qquad \textbf{(D) } \frac {911}{119}\qquad \textbf{(E) } \frac {912}{119}$
Let $O_1$ and $O_2$ be concentric circles with radii 4 and 6, respectively. A chord $AB$ is drawn in $O_1$ with length $2$. Extend $AB$ to intersect $O_2$ in points $C$ and $D$. Find $CD$.
Two circles are externally tangent. Lines $\overline{PAB}$ and $\overline{PA'B'}$ are common tangents with $A$ and $A'$ on the smaller circle and $B$ and $B'$ on the larger circle. If $PA = AB = 4$, then the area of the smaller circle is
[asy]
size(250);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair O=origin, Q=(0,-3sqrt(2)), P=(0,-6sqrt(2)), A=(-4/3,3.77-6sqrt(2)), B=(-8/3,7.54-6sqrt(2)), C=(4/3,3.77-6sqrt(2)), D=(8/3,7.54-6sqrt(2));
draw(Arc(O,2sqrt(2),0,360));
draw(Arc(Q,sqrt(2),0,360));
dot(A);
dot(B);
dot(C);
dot(D);
dot(P);
draw(B--A--P--C--D);
label("$A$",A,dir(A));
label("$B$",B,dir(B));
label("$A'$",C,dir(C));
label("$B'$",D,dir(D));
label("$P$",P,S);[/asy]
$ \textbf{(A)}\ 1.44\pi\qquad\textbf{(B)}\ 2\pi\qquad\textbf{(C)}\ 2.56\pi\qquad\textbf{(D)}\ \sqrt{8}\pi\qquad\textbf{(E)}\ 4\pi $
A circle passes through the three vertices of an isosceles triangle that has two sides of length $ 3$ and a base of length $ 2$. What is the area of this circle?
$ \textbf{(A)}\ 2\pi\qquad \textbf{(B)}\ \frac {5}{2}\pi\qquad \textbf{(C)}\ \frac {81}{32}\pi\qquad \textbf{(D)}\ 3\pi\qquad \textbf{(E)}\ \frac {7}{2}\pi$
To fold a paper airplane, Austin starts with a square paper $F OLD$ with side length $2$. First, he folds corners $L$ and $D$ to the square’s center. Then, he folds corner $F$ to corner $O$. What is the longest distance between two corners of the resulting figure?
Let $ABCD$ be a trapezoid with $AD=a, AB=2a, BC=3a$ and $\angle A=\angle B =90 ^o$. Let $E,Z$ be the midpoints of the sides $AB ,CD$ respectively and $I$ be the foot of the perpendicular from point $Z$ on $BC$. Prove that :
i) triangle $BDZ$ is isosceles
ii) midpoint $O$ of $EZ$ is the centroid of triangle $BDZ$
iii) lines $AZ$ and $DI$ intersect at a point lying on line $BO$
Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$.
[i]Proposed by C.R. Pranesachar, India [/i]
Point $E$ is on side $AB$ of square $ABCD$. If $EB$ has length one and $EC$ has length two, then the area of the square is
$\text{(A)}\ \sqrt{3} \qquad \text{(B)}\ \sqrt{5} \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 2\sqrt{3} \qquad \text{(E)}\ 5$
Square $ EFGH$ is inside the square $ ABCD$ so that each side of $ EFGH$ can be extended to pass through a vertex of $ ABCD$. Square $ ABCD$ has side length $ \sqrt {50}$ and $ BE \equal{} 1$. What is the area of the inner square $ EFGH$?
[asy]unitsize(4cm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
pair D=(0,0), C=(1,0), B=(1,1), A=(0,1);
pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0];
pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H);
draw(A--B--C--D--cycle);
draw(D--F);
draw(C--E);
draw(B--H);
draw(A--G);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,SE);
label("$D$",D,SW);
label("$E$",E,NNW);
label("$F$",F,ENE);
label("$G$",G,SSE);
label("$H$",H,WSW);[/asy]$ \textbf{(A)}\ 25\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 40\qquad \textbf{(E)}\ 42$
Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$. Given that the length of the hypotenuse $AB$ is 60, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$.
Let $ABCD$ be a trapezoid such that $AB||CD$ and let $P=AC\cap BD,AB=21,CD=7,AD=13,[ABCD]=168.$ Let the line parallel to $AB$ through $P$ intersect the circumcircle of $BCP$ in $X.$ Circumcircles of $BCP$ and $APD$ intersect at $P,Y.$ Let $XY\cap BC=Z.$ If $\angle ADC$ is obtuse, then $BZ=\frac{a}{b},$ where $a,b$ are coprime positive integers. Compute $a+b.$
The diameter $\overline{AB}$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and the line $ED$ is perpendicular to the line $AD$. Segment $\overline{AE}$ intersects the circle at point $C$ between $A$ and $E$. What is the area of $\triangle ABC$?
$\textbf{(A) \ } \frac{120}{37}\qquad \textbf{(B) \ } \frac{140}{39}\qquad \textbf{(C) \ } \frac{145}{39}\qquad \textbf{(D) \ } \frac{140}{37}\qquad \textbf{(E) \ } \frac{120}{31}$
A disk with radius $10$ and a disk with radius $8$ are drawn so that the distance between their centers is $3$. Two congruent small circles lie in the intersection of the two disks so that they are tangent to each other and to each of the larger circles as shown. The radii of the smaller circles are both $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[asy]
size(150);
defaultpen(linewidth(1));
draw(circle(origin,10)^^circle((3,0),8)^^circle((5,15/4),15/4)^^circle((5,-15/4),15/4));
[/asy]
In a right $\Delta ABC$ ($\angle C = 90^{\circ} $), $CD$ is the height. Let $r_1$ and $r_2$ be the radii of inscribed circles of $\Delta ACD$ and $\Delta DCB$. Find the radius of inscribed circle of $\Delta ABC$
Let $ABC$ be an acute triangle with circumcenter $O$ such that $AB=4$, $AC=5$, and $BC=6$. Let $D$ be the foot of the altitude from $A$ to $BC$, and $E$ be the intersection of $AO$ with $BC$. Suppose that $X$ is on $BC$ between $D$ and $E$ such that there is a point $Y$ on $AD$ satisfying $XY\parallel AO$ and $YO\perp AX$. Determine the length of $BX$.
[b]p1.[/b] A shape made by joining four identical regular hexagons side-to-side is called a hexo. Two hexos are considered the same if one can be rotated / reflected to match the other. Find the number of different hexos.
[b]p2.[/b] The sequence $1, 2, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6,... $ consists of numbers written in increasing order, where every even number $2n$ is written once, and every odd number $2n + 1$ is written $2n + 1$ times. What is the $2019^{th}$ term of this sequence?
[b]p3.[/b] On planet EMCCarth, months can only have lengths of $35$, $36$, or $42$ days, and there is at least one month of each length. Victor knows that an EMCCarth year has $n$ days, but realizes that he cannot figure out how many months there are in an EMCCarth year. What is the least possible value of $n$?
[b]p4.[/b] In triangle $ABC$, $AB = 5$ and $AC = 9$. If a circle centered at $A$ passing through $B$ intersects $BC$ again at $D$ and $CD = 7$, what is $BC$?
[b]p5.[/b] How many nonempty subsets $S$ of the set $\{1, 2, 3,..., 11, 12\}$ are there such that the greatest common factor of all elements in $S$ is greater than $1$?
[b]p6.[/b] Jasmine rolls a fair $6$-sided die, with faces labeled from $1$ to $6$, and a fair $20$-sided die, with faces labeled from $1$ to $20$. What is the probability that the product of these two rolls, added to the sum of these two rolls, is a multiple of $3$?
[b]p7.[/b] Let $\{a_n\}$ be a sequence such that $a_n$ is either $2a_{n-1}$ or $a_{n-1} - 1$. Given that $a_1 = 1$ and $a_{12} = 120$, how many possible sequences $a_1$, $a_2$, $...$, $a_{12}$ are there?
[b]p8.[/b] A tetrahedron has two opposite edges of length $2$ and the remaining edges have length $10$. What is the volume of this tetrahedron?
[b]p9.[/b] In the garden of EMCCden, there is a tree planted at every lattice point $-10 \le x, y \le 10$ except the origin. We say that a tree is visible to an observer if the line between the tree and the observer does not intersect any other tree (assume that all trees have negligible thickness). What fraction of all the trees in the garden of EMCCden are visible to an observer standing at the origin?
[b]p10.[/b] Point $P$ lies inside regular pentagon $\zeta$, which lies entirely within regular hexagon $\eta$. A point $Q$ on the boundary of pentagon $\zeta$ is called projective if there exists a point $R$ on the boundary of hexagon $\eta$ such that $P$, $Q$, $R$ are collinear and $2019 \cdot \overline{PQ} = \overline{QR}$. Given that no two sides of $\zeta$ and $\eta$ are parallel, what is the maximum possible number of projective points on $\zeta$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
[b]p1.[/b] Two players play the following game. On the lowest left square of an $8 \times 8$ chessboard there is a rook (castle). The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second layer is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy?
[b]p2.[/b] Find the smallest positive whole number that ends with $17$, is divisible by $17$, and the sum of its digits is $17$.
[b]p3.[/b] Three consecutive $2$-digit numbers are written next to each other. It turns out that the resulting $6$-digit number is divisible by $17$. Find all such numbers.
[b]p4.[/b] Let $ABCD$ be a convex quadrilateral (a quadrilateral $ABCD$ is called convex if the diagonals $AC$ and $BD$ intersect). Suppose that $\angle CBD = \angle CAB$ and $\angle ACD = \angle BDA$ . Prove that $\angle ABC = \angle ADC$.
[b]p5.[/b] A circle of radius $1$ is cut into four equal arcs, which are then arranged to make the shape shown on the picture. What is its area?
[img]https://cdn.artofproblemsolving.com/attachments/f/3/49c3fe8b218ab0a5378ecc635b797a912723f9.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].