Found problems: 603
We are given $n$ distinct rectangles in the plane. Prove that between the $4n$ interior angles formed by these rectangles at least $4\sqrt n$ are distinct.
Fix an integer $n \geq 2$. An $n\times n$ sieve is an $n\times n$ array with $n$ cells removed so that exactly one cell is removed from every row and every column. A stick is a $1\times k$ or $k\times 1$ array for any positive integer $k$. For any sieve $A$, let $m(A)$ be the minimal number of sticks required to partition $A$. Find all possible values of $m(A)$, as $A$ varies over all possible $n\times n$ sieves.
[i]Palmer Mebane[/i]
A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called [i]unstuck[/i] if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\sqrt{N}$, for a positive integer $N$. Find $N$.
[asy]
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draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((0,1)--(1,0));
draw((0,0)--(.5,sqrt(3)/2)--(1,0));
label("$A$",(0,0),SW);
label("$B$",(1,0),SE);
label("$C$",(1,1),NE);
label("$D$",(0,1),NW);
label("$E$",(.5,sqrt(3)/2),E);
label("$F$",intersectionpoint((0,0)--(.5,sqrt(3)/2),(0,1)--(1,0)),2W);
//Credit to chezbgone2 for the diagram[/asy]
Vertex $E$ of equilateral triangle $ABE$ is in the interior of square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and line segment $AE$. If length $AB$ is $\sqrt{1+\sqrt{3}}$ then the area of $\triangle ABF$ is
$\textbf{(A) }1\qquad\textbf{(B) }\frac{\sqrt{2}}{2}\qquad\textbf{(C) }\frac{\sqrt{3}}{2}$
$\qquad\textbf{(D) }4-2\sqrt{3}\qquad \textbf{(E) }\frac{1}{2}+\frac{\sqrt{3}}{4}$
Let $SABC$ be the pyramid where$ m(\angle ASB) = m(\angle BSC) = m(\angle CSA) = 90^o$. Show that, whatever the point $M \in AS$ is and whatever the point $N \in BC$ is, holds the relation
$$\frac{1}{MN^2} \le \frac{1}{SB^2} + \frac{1}{SC^2}.$$
On the $n\times n$ checker board, several cells were marked in such a way that lower left ($L$) and upper right($R$) cells are not marked and that for any knight-tour from $L$ to $R$, there is at least one marked cell. For which $n>3$, is it possible that there always exists three consective cells going through diagonal for which at least two of them are marked?
A semicircle of diameter $ 1$ sits at the top of a semicircle of diameter $ 2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.
[asy]unitsize(2.5cm);
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filldraw(Circle((0,.866),.5),grey,black);
label("1",(0,.866),S);
filldraw(Circle((0,0),1),white,black);
draw((-.5,.866)--(.5,.866),linetype("4 4"));
clip((-1,0)--(1,0)--(1,2)--(-1,2)--cycle);
draw((-1,0)--(1,0));
label("2",(0,0),S);[/asy]$ \textbf{(A)}\ \frac {1}{6}\pi \minus{} \frac {\sqrt {3}}{4} \qquad \textbf{(B)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{12}\pi \qquad \textbf{(C)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{24}\pi\qquad\textbf{(D)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{24}\pi$
$ \textbf{(E)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{12}\pi$
For a triangle $ ABC $ which $ \angle B \ne 90^{\circ} $ and $ AB \ne AC $, define $ P_{ABC} $ as follows ;
Let $ I $ be the incenter of triangle $ABC$, and let $ D, E, F $ be the intersection points with the incircle and segments $ BC, CA, AB $. Two lines $ AB $ and $ DI $ meet at $ S $ and let $ T $ be the intersection point of line $ DE $ and the line which is perpendicular with $ DF $ at $ F $. The line $ ST $ intersects line $ EF $ at $ R$. Now define $ P_{ABC} $ be one of the intersection points of the incircle and the circle with diameter $ IR $, which is located in other side with $ A $ about $ IR $.
Now think of an isosceles triangle $ XYZ $ such that $ XZ = YZ > XY $. Let $ W $ be the point on the side $ YZ $ such that $ WY < XY $ and Let $ K = P_{YXW} $ and $ L = P_{ZXW} $. Prove that $ 2 KL \le XY $.
Let $m$ and $n$ be fixed positive integers. Tsvety and Freyja play a game on an infinite grid of unit square cells. Tsvety has secretly written a real number inside of each cell so that the sum of the numbers within every rectangle of size either $m$ by $n$ or $n$ by $m$ is zero. Freyja wants to learn all of these numbers.
One by one, Freyja asks Tsvety about some cell in the grid, and Tsvety truthfully reveals what number is written in it. Freyja wins if, at any point, Freyja can simultaneously deduce the number written in every cell of the entire infinite grid (If this never occurs, Freyja has lost the game and Tsvety wins).
In terms of $m$ and $n$, find the smallest number of questions that Freyja must ask to win, or show that no finite number of questions suffice.
[i]Nikolai Beluhov[/i]
Triangle $ABC$ lies in the Cartesian Plane and has an area of 70. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20)$, respectively, and the coordinates of $A$ are $(p,q)$. The line containing the median to side $BC$ has slope $-5$. Find the largest possible value of $p+q$.
Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$
[i]Michael Ren and Ankan Bhattacharya, USA[/i]
Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.
Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$. The points $D,E$ and $F$ on the sides $BC,CA$ and $AB$ respectively are such that $BD+BF=CA$ and $CD+CE=AB$. The circumcircles of the triangles $BFD$ and $CDE$ intersect at $P \neq D$. Prove that $OP=OI$.
[b]p1.[/b] Find $20 \cdot 18 + 20 + 18 + 1$.
[b]p2.[/b] Suzie’s Ice Cream has $10$ flavors of ice cream, $5$ types of cones, and $5$ toppings to choose from. An ice cream cone consists of one flavor, one cone, and one topping. How many ways are there for Sebastian to order an ice cream cone from Suzie’s?
[b]p3.[/b] Let $a = 7$ and $b = 77$. Find $\frac{(2ab)^2}{(a+b)^2-(a-b)^2}$ .
[b]p4.[/b] Sebastian invests $100,000$ dollars. On the first day, the value of his investment falls by $20$ percent. On the second day, it increases by $25$ percent. On the third day, it falls by $25$ percent. On the fourth day, it increases by $60$ percent. How many dollars is his investment worth by the end of the fourth day?
[b]p5.[/b] Square $ABCD$ has side length $5$. Points $K,L,M,N$ are on segments $AB$,$BC$,$CD$,$DA$ respectively,such that $MC = CL = 2$ and $NA = AK = 1$. The area of trapezoid $KLMN$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$.
[b]p6.[/b] Suppose that $p$ and $q$ are prime numbers. If $p + q = 30$, find the sum of all possible values of $pq$.
[b]p7.[/b] Tori receives a $15 - 20 - 25$ right triangle. She cuts the triangle into two pieces along the altitude to the side of length $25$. What is the difference between the areas of the two pieces?
[b]p8.[/b] The factorial of a positive integer $n$, denoted $n!$, is the product of all the positive integers less than or equal to $n$. For example, $1! = 1$ and $5! = 120$. Let $m!$ and $n!$ be the smallest and largest factorial ending in exactly $3$ zeroes, respectively. Find $m + n$.
[b]p9.[/b] Sam is late to class, which is located at point $B$. He begins his walk at point $A$ and is only allowed to walk on the grid lines. He wants to get to his destination quickly; how many paths are there that minimize his walking distance?
[img]https://cdn.artofproblemsolving.com/attachments/a/5/764e64ac315c950367357a1a8658b08abd635b.png[/img]
[b]p10.[/b] Mr. Iyer owns a set of $6$ antique marbles, where $1$ is red, $2$ are yellow, and $3$ are blue. Unfortunately, he has randomly lost two of the marbles. His granddaughter starts drawing the remaining $4$ out of a bag without replacement. She draws a yellow marble, then the red marble. Suppose that the probability that the next marble she draws is blue is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positiveintegers. What is $m + n$?
[b]p11.[/b] If $a$ is a positive integer, what is the largest integer that will always be a factor of $(a^3+1)(a^3+2)(a^3+3)$?
[b]p12.[/b] What is the largest prime number that is a factor of $160,401$?
[b]p13.[/b] For how many integers $m$ does the equation $x^2 + mx + 2018 = 0$ have no real solutions in $x$?
[b]p14.[/b] What is the largest palindrome that can be expressed as the product of two two-digit numbers? A palindrome is a positive integer that has the same value when its digits are reversed. An example of a palindrome is $7887887$.
[b]p15.[/b] In circle $\omega$ inscribe quadrilateral $ADBC$ such that $AB \perp CD$. Let $E$ be the intersection of diagonals $AB$ and $CD$, and suppose that $EC = 3$, $ED = 4$, and $EB = 2$. If the radius of $\omega$ is $r$, then $r^2 =\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine $m + n$.
[b]p16.[/b] Suppose that $a, b, c$ are nonzero real numbers such that $2a^2 + 5b^2 + 45c^2 = 4ab + 6bc + 12ca$. Find the value of $\frac{9(a + b + c)^3}{5abc}$ .
[b]p17.[/b] Call a positive integer n spicy if there exist n distinct integers $k_1, k_2, ... , k_n$ such that the following two conditions hold:
$\bullet$ $|k_1| + |k_2| +... + |k_n| = n2$,
$\bullet$ $k_1 + k_2 + ...+ k_n = 0$.
Determine the number of spicy integers less than $10^6$.
[b]p18.[/b] Consider the system of equations $$|x^2 - y^2 - 4x + 4y| = 4$$
$$|x^2 + y^2 - 4x - 4y| = 4.$$ Find the sum of all $x$ and $y$ that satisfy the system.
[b]p19.[/b] Determine the number of $8$ letter sequences, consisting only of the letters $W,Q,N$, in which none of the sequences $WW$, $QQQ$, or $NNNN$ appear. For example, $WQQNNNQQ$ is a valid sequence, while $WWWQNQNQ$ is not.
[b]p20.[/b] Triangle $\vartriangle ABC$ has $AB = 7$, $CA = 8$, and $BC = 9$. Let the reflections of $A,B,C$ over the orthocenter H be $A'$,$B'$,$C'$. The area of the intersection of triangles $ABC$ and $A'B'C'$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ , where $b$ is squarefree and $a$ and $c$ are relatively prime. determine $a+b+c$. (The orthocenter of a triangle is the intersection of its three altitudes.)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $ABC$ be a acute angled triangle and let $AD,BE,CF$ be its altitudes. $X \not=A,B$ and $Y \not=A,C$ lie on sides $AB$ and $AC$, respectively, so that $ADXY$ is a cyclic quadrilateral. Let $H$ be the orthocenter of triangle $AXY$.
Prove that $H$ lies on line $EF$.
Let $n > 1$ be a given integer. An $n \times n \times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one colour. For each $n \times n \times 1$ box consisting of $n^2$ unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get $3n$ sets of colours, split into three groups according to the orientation.
It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of $n$, the maximal possible number of colours that are present.
If $ABCD$ is a $2\ X\ 2$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, then the area of quadrilateral $BEIH$ is
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label("A", A, NW);
label("B", B, SW);
label("C", C, SE);
label("D", D, NE);
label("E", E, W);
label("F", F, S);
label("H", (.8,0.6));
label("I", (0.4,1.4));
[/asy]
$ \textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{2}{5}\qquad\textbf{(C)}\ \frac{7}{15}\qquad\textbf{(D)}\ \frac{8}{15}\qquad\textbf{(E)}\ \frac{3}{5} $
Triangle $ ABC$ has $ AC\equal{}3$, $ BC\equal{}4$, and $ AB\equal{}5$. Point $ D$ is on $ \overline{AB}$, and $ \overline{CD}$ bisects the right angle. The inscribed circles of $ \triangle ADC$ and $ \triangle BCD$ have radii $ r_a$ and $ r_b$, respectively. What is $ r_a/r_b$?
$ \textbf{(A)}\ \frac{1}{28}\left(10\minus{}\sqrt{2}\right) \qquad
\textbf{(B)}\ \frac{3}{56}\left(10\minus{}\sqrt{2}\right) \qquad
\textbf{(C)}\ \frac{1}{14}\left(10\minus{}\sqrt{2}\right) \qquad
\textbf{(D)}\ \frac{5}{56}\left(10\minus{}\sqrt{2}\right) \\
\textbf{(E)}\ \frac{3}{28}\left(10\minus{}\sqrt{2}\right)$
Regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1$. Point $P$ lies inside the circle so that the region bounded by $\overline{PA_1}$, $\overline{PA_2}$, and the minor arc $\widehat{A_1A_2}$ of the circle has area $\tfrac17$, while the region bounded by $\overline{PA_3}$, $\overline{PA_4}$, and the minor arc $\widehat{A_3A_4}$ of the circle has area $\tfrac 19$. There is a positive integer $n$ such that the area of the region bounded by $\overline{PA_6}$, $\overline{PA_7}$, and the minor arc $\widehat{A_6A_7}$ is equal to $\tfrac18 - \tfrac{\sqrt 2}n$. Find $n$.
A frog trainer places one frog at each vertex of an equilateral triangle $ABC$ of unit sidelength. The trainer can make one frog jump over another along the line joining the two, so that the total length of the jump is an even multiple of the distance between the two frogs just before the jump. Let $M$ and $N$ be two points on the rays $AB$ and $AC$, respectively, emanating from $A$, such that $AM = AN = \ell$, where $\ell$ is a positive integer. After a fi nite number of jumps, the three frogs all lie in the triangle $AMN$ (inside or on the boundary), and no more jumps are performed.
Determine the number of final positions the three frogs may reach in the triangle $AMN$. (During the process, the frogs may leave the triangle $AMN$, only their nal positions are to be in that triangle.)
Let $T_1, T_2, T_3, T_4$ be pairwise distinct collinear points such that $T_2$ lies between $T_1$ and $T_3$, and $T_3$ lies between $T_2$ and $T_4$. Let $\omega_1$ be a circle through $T_1$ and $T_4$; let $\omega_2$ be the circle through $T_2$ and internally tangent to $\omega_1$ at $T_1$; let $\omega_3$ be the circle through $T_3$ and externally tangent to $\omega_2$ at $T_2$; and let $\omega_4$ be the circle through $T_4$ and externally tangent to $\omega_3$ at $T_3$. A line crosses $\omega_1$ at $P$ and $W$, $\omega_2$ at $Q$ and $R$, $\omega_3$ at $S$ and $T$, and $\omega_4$ at $U$ and $V$, the order of these points along the line being $P,Q,R,S,T,U,V,W$. Prove that $PQ + TU = RS + VW$
[i]Geza Kos, Hungary[/i]
An [i]animal[/i] with $n$ [i]cells[/i] is a connected figure consisting of $n$ equal-sized cells[1].
A [i]dinosaur[/i] is an animal with at least $2007$ cells. It is said to be [i]primitive[/i] it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
(1) Animals are also called [i]polyominoes[/i]. They can be defined inductively. Two cells are [i]adjacent[/i] if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
An acute triangle $ABC$ is given and $H$ and $O$ be its orthocenter and circumcenter respectively. Let $K$ be the midpoint of $AH$ and $\ell$ be a line through $O. $ Let $P$ and $Q$ be the projections of $B$ and $C$ on $\ell. $ Prove that$$KP+KQ\ge BC$$
In a triangle $ABC$, right angled at $A$, the altitude through $A$ and the internal bisector of $\angle A$ have lengths $3$ and $4$, respectively. Find the length of the median through $A$.
Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called [b]ideal[/b] if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n \plus{} p$ and $ n \plus{} q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$