Found problems: 85335
Find the area of the figure bounded by the curves $y=1-x^2$, $|x|=1-|y|.$
At the beginning of the school year, $ 50\%$ of all students in Mr. Well's math class answered "Yes" to the question "Do you love math", and $ 50\%$ answered "No." At the end of the school year, $ 70\%$ answered "Yes" and $ 30\%$ answered "No." Altogether, $ x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $ x$?
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 80$
In the Archery with an especial gun, the probability of goal is $90 \%.$ If we continue our work until we goal.
[b]i)[/b] What is the probability which exactly $3$ balls consumed.
[b]ii)[/b] What is the probability which at least $3$ balls consumed.
What is the smallest integer multiple of 9997, other than 9997 itself, which contains only odd digits?
Consider a triangle $ ABC $ having incenter $ I $ and inradius $ r. $ Let $ D $ be the tangency of $ ABC $ 's incircle with $ BC, $ and $ E $ on the line $ BC $ such that $ AE $ is perpendicular to $ BC, $ and $ M\neq E $ on the segment $ AE $ such that $ AM=r. $
[b]a)[/b] Give an idenity for $ \frac{BD}{DC} $ involving only the lengths of the sides of the triangle.
[b]b)[/b] Prove that $ AB \cdot \overrightarrow{IC} +BC\cdot \overrightarrow{IA} +CA\cdot \overrightarrow{IB} =0. $
[b]c)[/b] Show that $ MI $ passes through the middle of the side $ BC. $
[i]Cătălin Zârnă[/i]
On Qingqing Grassland, there are 7 sheep numberd $1,2,3,4,5,6,7$ and 2017 wolves numberd $1,2,\cdots,2017$. We have such strange rules:
(1) Define $P(n)$: the number of prime numbers that are smaller than $n$. Only when $P(i)\equiv j\pmod7$, wolf $i$ may eat sheep $j$ (he can also choose not to eat the sheep).
(2) If wolf $i$ eat sheep $j$, he will immediately turn into sheep $j$.
(3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep.
Assume that all wolves are very smart, then how many wolves will remain in the end?
There are three colleges in a town. Each college has $n$ students. Any student of any college knows $n+1$ students of the other two colleges. Prove that it is possible to choose a student from each of the three colleges so that all three students would know each other.
Consider the set $S$ of integers $k$ which are products of four distinct primes. Such an integer $k=p_1p_2p_3p_4$ has $16$ positive divisors $1=d_1<d_2<\ldots <d_{15}<d_{16}=k$. Find all elements of $S$ less than $2002$ such that $d_9-d_8=22$.
A company sells detergent in three different sized boxes: small (S), medium (M) and large (L). The medium size costs $50\%$ more than the small size and contains $20\%$ less detergent than the large size. The large size contains twice as much detergent as the small size and costs $30\%$ more than the medium size. Rank the three sizes from best to worst buy.
$ \textbf{(A)}\ \text{SML}\qquad\textbf{(B)}\ \text{LMS}\qquad\textbf{(C)}\ \text{MSL}\qquad\textbf{(D)}\ \text{LSM}\qquad\textbf{(E)}\ \text{MLS} $
Does a parallelogram exist such that all pairwise meets of bisectors of its angles are situated outside it?
Let $ABCD$ be a convex quadrilateral. On the sides $AB$ and $CD$ there are interior points $K$ and $L$, respectively, such that $\angle BAL = \angle CDK$. Prove that the following statements are equivalent:
$i) \angle BLA= \angle CKD$
$ii) AD \parallel BC $
(a) Prove that if 3n stars are placed in $3n$ cells of a $2n \times 2n$ array, then it is possible to remove $n$ rows and $n$ columns in such away that all stars will be removed .
(b) Prove that it is possible to place $3n + 1$ stars in the cells of a $2n \times 2n$ array in such a way that after removing any $n$ rows and $n$ columns at least one star remains.
(K . P. Kohas, Leningrad)
In a triangle $ABC$, the circle through $C$ touching $AB$ at $A$ and the circle through $B$ touching $AC$ at $A$ have different radii and meet again at $D$. Let $E$ be the point on the ray $AB$ such that $AB = BE$. The circle through $A$, $D$, $E$ intersect the ray $CA$ again at $F$ . Prove that $AF = AC$.
A circle intersects a parabola at four distinct points. Let $M$ and $N$ be the midpoints of the arcs of the circle which are outside the parabola. Prove that the line $MN$ is perpendicular to the axis of the parabola.
I. Voronovich
Let $D$ be the foot of the altitude of triangle $ABC$ drawn from vertex $A$. Let $E$ and $F$ be points symmetric to $D$ w.r.t. lines $AB$ and $AC$, respectively. Let $R_1$ and $R_2$ be the circumradii of triangles $BDE$ and $CDF$, respectively, and let $r_1$ and $r_2$ be the inradii of the same triangles. Prove that $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$
Determine all positive integers $n$ for which there are positive real numbers $x,y$ and $z$ such that $\sqrt x +\sqrt y +\sqrt z=1$ and $\sqrt{x+n} +\sqrt{y+n} +\sqrt{z+n}$ is an integer.
Points $X$ and $Y$ are chosen on the sides $AB$ and $BC$ of the triangle $\triangle ABC$. The segments $AY$ and $CX$ intersect at the point $Z$. Given that $AY = YC$ and $AB = ZC$, prove that the points $B$, $X$, $Z$, and $Y$ lie on the same circle.
Suppose that each of the vertices of $ABC$ is a lattice point in the $xy$-plane and that there is exactly one lattice
point $P$ in the interior of the triangle. The line $AP$ is extended to meet $BC$ at $E$. Determine the largest possible value for the ratio of lengths of segments
$$\frac{|AP|}{|PE|}.$$
A positive integer of at least two digits written in the base $ 10 $ is called 'ascending' if the digits increase in value from left to right. For example, $ 123 $ is 'ascending', but $ 132 $ and $ 122 $ is not. How many 'ascending' numbers are there?
A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where $[x]$ denotes the smallest integer $\leq x)$
A [i]checkerboard[/i] is a rectangular grid of cells colored black and white such that the top-left corner is black and no two cells of the same color share an edge. Two checkerboards are [i]distinct[/i] if and only if they have a different number of rows or columns. For example, a $20 \times 25$ checkerboard and a $25 \times 20$ checkerboard are considered distinct.
Compute the number of distinct checkerboards that have exactly $41$ distinct black cells.
Given triangle $ ABC$. Points $ D,E,F$ outside triangle $ ABC$ are chosen such that triangles $ ABD$, $ BCE$, and $ CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.
A family of sets $ A_1, A_2, \ldots ,A_n$ has the following properties:
[b](i)[/b] Each $ A_i$ contains 30 elements.
[b](ii)[/b] $ A_i \cap A_j$ contains exactly one element for all $ i, j, 1 \leq i < j \leq n.$
Determine the largest possible $ n$ if the intersection of all these sets is empty.
Let $(a_n)_{n=1}^\infty$ be a real sequence such that \[a_n = (n-1)a_1 + (n-2)a_2 + \dots + 2a_{n-2} + a_{n-1}\] for every $n\geq 3$. If $a_{2011} = 2011$ and $a_{2012} = 2012$, what is $a_{2013}$?
$
\textbf{(A)}\ 6025
\qquad\textbf{(B)}\ 5555
\qquad\textbf{(C)}\ 4025
\qquad\textbf{(D)}\ 3456
\qquad\textbf{(E)}\ 2013
$
Let $ABC$ be scalene and acute triangle with $CA>CB$ and let $P$ be an internal point, satisfying $\angle APB=180^{\circ}-\angle ACB$; the lines $AP, BP$ meet $BC, CA$ at $A_1, B_1$. If $M$ is the midpoint of $A_1B_1$ and $(A_1B_1C)$ meets $(ABC)$ at $Q$, show that $\angle PQM=\angle BQA_1$.