Found problems: 85335
A deck consists of $n$ different cards. A move consists of taking out a group of cards in sequence from some place in the deck, and putting it back someplace else without changing the order within the group or turning any cards over. We are required to reverse the order of cards in the deck by such moves.
(a) Prove that for $n = 9$, this can be done in $5$ moves.
(b) Prove that for $n = 52$, this
i. can be done in $27$ moves,
ii. can’t be done in $17$ moves,
iii. can’t be done in $26$ moves.
(SM Voronin, Tchelyabinsk)
A trapezium has parallel sides of length equal to $a$ and $b$ ($a <b$), and the distance between the parallel sides is the altitude $h$. The extensions of the non-parallel lines intersect at a point that is a vertex of two triangles that have as sides the parallel sides of the trapezium. Express the areas of the triangles as functions of $a,b$ and $h$.
Problems 17, 18, and 19 refer to the following:
At Central Middle School the 108 students who take the AMC8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 1.5 cups flour, 2 eggs, 3 tablespoons butter, 3/4 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes.
They learn that a big concert is scheduled for the same night and attendance will be down $25\%$. How many recipes of cookies should they make for their smaller party?
$ \text{(A)}\ 6\qquad\text{(B)}\ 8\qquad\text{(C)}\ 9\qquad\text{(D)}\ 10\qquad\text{(E)}\ 11 $
On the sides $AD$ and $BC$ of a rhombus $ABCD$ we consider the points $M$ and $N$ respectively. The line $MC$ intersects the segment $BD$ in the point $T$, and the line $MN$ intersects the segment $BD$ in the point $U$. We denote by $Q$ the intersection between the line $CU$ and the side $AB$ and with $P$ the intersection point between the line $QT$ and the side $CD$.
Prove that the triangles $QCP$ and $MCN$ have the same area.
If $x, y, z$ are positive real numbers with sum $12$, prove that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+ 3 \ge \sqrt{x} +\sqrt{y }+\sqrt{z}$.
When equality is valid?
Given a board consists of $n \times n$ unit squares ($n \ge 3$). Each unit square is colored black and white, resembling a chessboard. In each step, TOMI can choose any $2 \times 2$ square and change the color of every unit square chosen with the other color (white becomes black and black becomes white). Find every $n$ such that after a finite number of moves, every unit square on the board has a same color.
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?
Let $ABC$ be a triangle with $\angle A=90^{\circ}$ and $AB=AC$. Let $D$ and $E$ be points on the segment $BC$ such that $BD:DE:EC = 1:2:\sqrt{3}$. Prove that $\angle DAE= 45^{\circ}$
A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?
Suppose Robin and Eddy walk along a circular path with radius $r$ in the same direction. Robin makes a revolution around the circular path every $3$ minutes and Eddy makes a revolution every minute. Jack stands still at a distance $R>r$ from the center of the circular path. At time $t=0$, Robin and Eddy are at the same point on the path, and Jack, Robin, and Eddy, and the center of the path are collinear. When is the next time the three people (but not necessarily the center of the path) are collinear?
In $\triangle ABC$, $CA=1960\sqrt{2}$, $CB=6720$, and $\angle C = 45^{\circ}$. Let $K$, $L$, $M$ lie on $BC$, $CA$, and $AB$ such that $AK \perp BC$, $BL \perp CA$, and $AM=BM$. Let $N$, $O$, $P$ lie on $KL$, $BA$, and $BL$ such that $AN=KN$, $BO=CO$, and $A$ lies on line $NP$. If $H$ is the orthocenter of $\triangle MOP$, compute $HK^2$.
[hide="Clarifications"]
[list]
[*] Without further qualification, ``$XY$'' denotes line $XY$.[/list][/hide]
[i]Evan Chen[/i]
In the right-angled trapezoid $AB CD$, $AB \parallel CD$, $m( \angle A) = 90°$, $AD = DC = a$ and $AB =2a$. On the perpendiculars raised in $C$ and $D$ on the plane containing the trapezoid one considers points $E$ and $F$ (on the same side of the plane) such that $CE = 2a$ and $DF = a$. Find the distance from the point $B$ to the plane $(AEF)$ and the measure of the angle between the lines $AF$ and $BE$.
Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow?
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
Calculate $\displaystyle{\sum_{n=1}^\infty\left(\lfloor\sqrt[n]{2010}\rfloor-1\right)}$ where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$.
Each of the players in a tennis tournament played one match against each of
the others. If every player won at least one match, show that there is a group
A; B; C of three players for which A beat B, B beat C and C beat A.
Let $A_1A_2\ldots A_8$ be a convex cyclic octagon, and for $i=1,2\ldots,8$ let $B_i=A_iA_{i+3}\cap A_{i+1}A_{i+4}$ (indices are meant modulo 8). Prove that points $B_1,\ldots, B_8$ lie on the same conic section.
The $ABCDE$ pentagon is inscribed in a circle and $AB = BC = CD$. Segments $AC$ and $BE$ intersect at $K$, and Segments $AD$ and $CE$ intersect at point$ L$. Prove that $AK = KL$.
The vertical axis indicates the number of employees, but the scale was accidentally omitted from this graph. What percent of the employees at the Gauss company have worked there for $5$ years or more?
[asy]
for(int a=1; a<11; ++a)
{
draw((a,0)--(a,-.5));
}
draw((0,10.5)--(0,0)--(10.5,0));
label("$1$",(1,-.5),S); label("$2$",(2,-.5),S); label("$3$",(3,-.5),S); label("$4$",(4,-.5),S);
label("$5$",(5,-.5),S); label("$6$",(6,-.5),S); label("$7$",(7,-.5),S); label("$8$",(8,-.5),S);
label("$9$",(9,-.5),S); label("$10$",(10,-.5),S); label("Number of years with company",(5.5,-2),S);
label("X",(1,0),N); label("X",(1,1),N); label("X",(1,2),N); label("X",(1,3),N); label("X",(1,4),N);
label("X",(2,0),N); label("X",(2,1),N); label("X",(2,2),N); label("X",(2,3),N); label("X",(2,4),N);
label("X",(3,0),N); label("X",(3,1),N); label("X",(3,2),N); label("X",(3,3),N);
label("X",(3,4),N); label("X",(3,5),N); label("X",(3,6),N); label("X",(3,7),N);
label("X",(4,0),N); label("X",(4,1),N); label("X",(4,2),N); label("X",(5,0),N); label("X",(5,1),N);
label("X",(6,0),N); label("X",(6,1),N); label("X",(7,0),N); label("X",(7,1),N);
label("X",(8,0),N); label("X",(9,0),N); label("X",(10,0),N);
label("Gauss Company",(5.5,10),N);
[/asy]
$\text{(A)}\ 9\% \qquad \text{(B)}\ 23\frac{1}{3}\% \qquad \text{(C)}\ 30\% \qquad \text{(D)}\ 42\frac{6}{7}\% \qquad \text{(E)}\ 50\% $
Consider a $2\times2$ square with one corner removed from $1\times1$ , leaving a shape in the form of $L$ .
[asy]
unitsize(0.5 cm);
draw((1,0)--(1,2)--(0,2)--(0,0)--(2,0)--(2,1)--(0,1));
[/asy]
We will call this figure [i]triomino[/i]. Determine all values of $m, n$ such that a rectangle of $m\times n$ can be exactly covered with triominos.
Let $ABCDEF$ be a hexagon having all interior angles equal to $120^o$ each. Let $P,Q,R, S, T, V$ be the midpoints of the sides of the hexagon $ABCDEF$. Prove the inequality $$p(PQRSTV ) \ge \frac{\sqrt3}{2} p(ABCDEF)$$, where $p(.)$ denotes the perimeter of the polygon.
Nguyễn Tiến Lâm
$\textbf{(Self-Isogonal Cubics)}$ Let $ABC$ be a triangle with $AB = 2$, $AC = 3$, $BC = 4$. The $\emph{isogonal conjugate}$ of a point $P$, denoted $P^\ast$, is the point obtained by intersecting the reflection of lines $PA$, $PB$, $PC$ across the angle bisectors of $\angle A$, $\angle B$, and $\angle C$, respectively.
Given a point $Q$, let $\mathfrak K(Q)$ denote the unique cubic plane curve which passes through all points $P$ such that line $PP^\ast$ contains $Q$. Consider:
[list]
[*] the M'Cay cubic $\mathfrak K(O)$,
where $O$ is the circumcenter of $\triangle ABC$,
[*] the Thomson cubic $\mathfrak K(G)$,
where $G$ is the centroid of $\triangle ABC$,
[*] the Napoleon-Feurerbach cubic $\mathfrak K(N)$,
where $N$ is the nine-point center of $\triangle ABC$,
[*] the Darboux cubic $\mathfrak K(L)$,
where $L$ is the de Longchamps point
(the reflection of the orthocenter across point $O$),
[*] the Neuberg cubic $\mathfrak K(X_{30})$,
where $X_{30}$ is the point at infinity along line $OG$,
[*] the nine-point circle of $\triangle ABC$,
[*] the incircle of $\triangle ABC$, and
[*] the circumcircle of $\triangle ABC$.
[/list]
Estimate $N$, the number of points lying on at least two of these eight curves. An estimate of $E$ earns $\left\lfloor 20 \cdot 2^{-|N-E|/6} \right\rfloor$ points.
Let $n\in\mathbb N$ and define $P(x,y)=x^n+xy+y^n.$
Show that we cannot obtain two non-constant polynomials $G(x,y)$ and $H(x,y)$ with real coefficients such that
$P(x,y)=G(x,y)\cdot H(x,y).$
Determine the number of $n$-tuples of integers $(x_1,x_2,\cdots ,x_n)$ such that $|x_i| \le 10$ for each $1\le i \le n$ and $|x_i-x_j| \le 10$ for $1 \le i,j \le n$.
Let $O$ be the circumcenter of triangle $ABC$: Points $D$ and $E$ are chosen at sides $AB$ and $AC$ respectively such that $\angle ADO = \angle AEO = 60^o$ and $BDEC$ is inscribed quadrangle. Prove or disprove that $ABC$ is isosceles triangle.
Find all positive integers, representable uniquely as \[\frac{x^{2}+y}{xy+1},\] where $x$ and $y$ are positive integers.