Found problems: 85335
Six equal disks are placed on a plane so that their centers lie at the vertices of a regular hexagon with sides equal to the diameter of the disks. How many revolutions will a seventh disk of the same size make when rolling in the same plane externally over the disks before returning to its initial position?
Find all real numbers $x,y,z$ satisfying the following system of equations:
\begin{align*}
xy+z&=-30\\
yz+x &= 30\\
zx+y &=-18
\end{align*}
What is the largest number of squares on $9 \times 9$ square board that can be cut along their both diagonals so that the board does not fall apart into several pieces?
Bunbury the bunny is hopping on the positive integers. First, he is told a positive integer $n$. Then Bunbury chooses positive integers $a,d$ and hops on all of the spaces $a,a+d,a+2d,\dots,a+2013d$. However, Bunbury must make these choices so that the number of every space that he hops on is less than $n$ and relatively prime to $n$.
A positive integer $n$ is called [i]bunny-unfriendly[/i] if, when given that $n$, Bunbury is unable to find positive integers $a,d$ that allow him to perform the hops he wants. Find the maximum bunny-unfriendly integer, or prove that no such maximum exists.
Let $k$ be a circle with centre $M$ and let $AB$ be a diameter of $k$. Furthermore, let $C$ be a point on $k$ such that $AC = AM$. Let $D$ be the point on the line $AC$ such that $CD = AB$ and $C$ lies between $A$ and $D$. Let $E$ be the second intersection of the circumcircle of $BCD$ with line $AB$ and $F$ be the intersection of the lines $ED$ and $BC$. The line $AF$ cuts the segment $BD$ in $X$. Determine the ratio $BX/XD$.
If the four sides of a quadrilateral are $2, 3, 6$, and $x$, find the sum of all possible integral values for $x$.
Let $x,y,z$ be positive real numbers. Prove the following inequality: \[ \frac 1{x^2+yz} + \frac 1{y^2+zx } + \frac 1{z^2+xy} \leq \frac 12 \left( \frac 1{xy} + \frac 1{yz} + \frac 1{zx} \right). \]
Let $a>0$. Evaluate $\int_0^a x^2\left(1-\frac{x}{a}\right)^adx$.
[i]2011 Keio University entrance exam/Science and Technology[/i]
Let $ABC$ be a triangle with circumscribed and inscribed circles $(O)$ and $(I)$ respectively. $AA'$,$BB'$,$CC'$ are the bisectors of triangle $ABC$. Prove that $OI$ passes through the the isogonal conjugate of point $I$ with respect to triangle $A'B'C'$.
Let $ABCD$ be a parallelogram whose diagonals intersect in $M$. Suppose that the circumcircle of $ABM$ intersects the segment $AD$ in a point $E \ne A$ and that the circumcircle of $EMD$ intersects the segment $BE$ in a point $F \ne E$.
Show that $\angle ACB=\angle DCF$.
Tangents $ l_1$ and $ l_2$ common to circles $ c_1$ and $ c_2$ intersect at point $ P$, whereby tangent points remain to different sides from $ P$ on both tangent lines. Through some point $ T$, tangents $ p_1$ and $ p_2$ to circle $ c_1$ and tangents $ p_3$ and $ p_4$ to circle $ c_2$ are drawn. The intersection points of $ l_1$ with lines $ p_1, p_2, p_3, p_4$ are $ A_1, B_1, C_1, D_1$, respectively, whereby the order of points on $ l_1$ is: $ A_1, B_1, P, C_1, D_1$. Analogously, the intersection points of $ l_2$ with lines $ p_1, p_2, p_3, p_4$ are $ A_2, B_2, C_2, D_2$, respectively. Prove that if both quadrangles $ A_1A_2D_1D_2$ and $ B_1B_2C_1C_2$ are cyclic then radii of $ c_1$ and $ c_2$ are equal.
Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right.
Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors.
[i]Proposed by Gurgen Asatryan, Armenia[/i]
Every cell of a $m \times n$ chess board, $m\ge 2,n\ge 2$, is colored with one of four possible colors, e.g white, green, red, blue. We call such coloring good if the four cells of any $2\times 2$ square of the chessboard are colored with pairwise different colors. Determine the number of all good colorings of the chess board.
[i]Proposed by N. Beluhov[/i]
If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ in $\overline{AQ}$, and $\measuredangle QPC = 60^\circ$, then the length of $PQ$ divided by the length of $AQ$ is
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=(-1,0), B=(1,0), C=(0,1), D=(0,-1), Q=origin, P=(-0.5,0);
draw(P--C--D^^A--B^^Circle(Q,1));
label("$A$", A, W);
label("$B$", B, E);
label("$C$", C, N);
label("$D$", D, S);
label("$P$", P, S);
label("$Q$", Q, SE);
label("$60^\circ$", P+0.0.5*dir(30), dir(30));[/asy]
$ \textbf{(A)} \ \frac{\sqrt{3}}{2} \qquad \textbf{(B)} \ \frac{\sqrt{3}}{3} \qquad \textbf{(C)} \ \frac{\sqrt{2}}{2} \qquad \textbf{(D)} \ \frac12 \qquad \textbf{(E)} \ \frac23 $
Points $A, B, C, D$ lie on a line $l$, in that order. Find the locus of points $P$ in the plane for which $\angle{APB}=\angle{CPD}$.
[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide]
[u] Set 1[/u]
[b]R1.1 / P1.1[/b] Find $291 + 503 - 91 + 492 - 103 - 392$.
[b]R1.2[/b] Let the operation $a$ & $b$ be defined to be $\frac{a-b}{a+b}$. What is $3$ & $-2$?
[b]R1.3[/b]. Joe can trade $5$ apples for $3$ oranges, and trade $6$ oranges for $5$ bananas. If he has $20$ apples, what is the largest number of bananas he can trade for?
[b]R1.4[/b] A cone has a base with radius $3$ and a height of $5$. What is its volume? Express your answer in terms of $\pi$.
[b]R1.5[/b] Guang brought dumplings to school for lunch, but by the time his lunch period comes around, he only has two dumplings left! He tries to remember what happened to the dumplings. He first traded $\frac34$ of his dumplings for Arman’s samosas, then he gave $3$ dumplings to Anish, and lastly he gave David $\frac12$ of the dumplings he had left. How many dumplings did Guang bring to school?
[u]Set 2[/u]
[b]R2.6 / P1.3[/b] In the recording studio, Kanye has $10$ different beats, $9$ different manuscripts, and 8 different samples. If he must choose $1$ beat, $1$ manuscript, and $1$ sample for his new song, how many selections can he make?
[b]R2.7[/b] How many lines of symmetry does a regular dodecagon (a polygon with $12$ sides) have?
[b]R2.8[/b] Let there be numbers $a, b, c$ such that $ab = 3$ and $abc = 9$. What is the value of $c$?
[b]R2.9[/b] How many odd composite numbers are there between $1$ and $20$?
[b]R2.10[/b] Consider the line given by the equation $3x - 5y = 2$. David is looking at another line of the form ax - 15y = 5, where a is a real number. What is the value of a such that the two lines do not intersect at any point?
[u]Set 3[/u]
[b]R3.11[/b] Let $ABCD$ be a rectangle such that $AB = 4$ and $BC = 3$. What is the length of BD?
[b]R3.12[/b] Daniel is walking at a constant rate on a $100$-meter long moving walkway. The walkway moves at $3$ m/s. If it takes Daniel $20$ seconds to traverse the walkway, find his walking speed (excluding the speed of the walkway) in m/s.
[b]R3.13 / P1.3[/b] Pratik has a $6$ sided die with the numbers $1, 2, 3, 4, 6$, and $12$ on the faces. He rolls the die twice and records the two numbers that turn up on top. What is the probability that the product of the two numbers is less than or equal to $12$?
[b]R3.14 / P1.5[/b] Find the two-digit number such that the sum of its digits is twice the product of its digits.
[b]R3.15[/b] If $a^2 + 2a = 120$, what is the value of $2a^2 + 4a + 1$?
PS. You should use hide for answers. R16-30 /P6-10/ P26-30 have been posted [url=https://artofproblemsolving.com/community/c3h2786837p24497019]here[/url], and P11-25 [url=https://artofproblemsolving.com/community/c3h2786880p24497350]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
An angle is given in a plane. Using only a compass, one must find out
$(a)$ if this angle is acute. Find the minimal number of circles one must draw to be sure.
$(b)$ if this angle equals $31^{\circ}$.(One may draw as many circles as one needs).
Determine the number of distinct values which appear in the sequence \[\left\lfloor\frac{2025}{1}\right\rfloor,\left\lfloor\frac{2025}{2}\right\rfloor,\left\lfloor\frac{2025}{3}\right\rfloor,\dots,\left\lfloor\frac{2025}{2024}\right\rfloor,\left\lfloor\frac{2025}{2025}\right\rfloor.\]
Let $\triangle{ABC}$ be a scalene triangle with incentre $I$ and circumcircle $\omega$. Lines $AI, BI, CI$ intersect $\omega$ for the second time at points $D, E, F$, respectively. The parallel lines from $I$ to the sides $BC, AC, AB$ intersect $EF, DF, DE$ at points $K, L, M$, respectively. Prove that the points $K, L, M$ are collinear.
[i](Cyprus)[/i]
Let $F$ be a smooth (i.e. $C^{\infty}$) closed surface. Call a continuous map $f\colon F\rightarrow \mathbb{R}^2$ an [i]almost-immersion[/i] if there exists a smooth closed embedded curve $\gamma$ (possibly disconnected) in $F$ such that $f$ is smooth and of maximal rank (i.e., rank 2) on $F\backslash \gamma$ and each point $p\in\gamma$ admits local coordinate charts $(x,y)$ and $(u,v)$ about $p$ and $f(p)$, respectively, such taht the coordinates of $p$ and $f(p)$ are zero and the map $f$ is given by $(x,y)\rightarrow (u,v), u=|x|, v=y$.
Determine the genera of those smooth, closed, connected, orientable surfaces $F$ that admit an almost-immersion in the plane with the curve $\gamma$ having a given positive number $n$ of connected components.
In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$, let $M$ be the midpoint of $\overline{BC}$. Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$. Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$. Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$, and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$. Prove that $A$, $Y$, and $Z$ are collinear.
[i]Tiger Zhang[/i]
Three circles $O_1, \ O_2, \ O_3$ with radiii $r_1, \ r_2, \ r_3$ respectively are tangent extarnally in pairs. Let r be the radius of the inscrined circle of triangle $O_1O_2O_3$. Prove that $$ r=\sqrt{\dfrac{r_1r_2r_3}{r_1+r_2+r_3}}.$$
Let $ a,b,c,d$ be rational numbers with $ a>0$. If for every integer $ n\ge 0$, the number $ an^{3} \plus{}bn^{2} \plus{}cn\plus{}d$ is also integer, then the minimal value of $ a$ will be
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ \frac{1}{6} \qquad\textbf{(D)}\ \text{Cannot be found} \qquad\textbf{(E)}\ \text{None}$
Let $a_0, a_1, \ldots, a_n, a_{n+1}$ be a sequence of real numbers satisfying the following conditions:
\[a_0 = a_{n+1 }= 0,\]\[ |a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).\]
Prove that $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$
In the plane, four points are marked. It is known that these points are the centers of four circles, three of which are pairwise externally tangent, and all these three are internally tangent to the fourth one. It turns out, however, that it is impossible to determine which of the marked points is the center of the fourth (the largest) circle. Prove that these four points are the vertices of a rectangle.