Found problems: 85335
The Iphoon particle, of charge $q$, is accelerated from rest by a potential difference of $V$. This strange particle then enters a region with a uniform magnetic field, $B$, which is perpendicular to the particle's velocity. The Iphoon follows a circular path with radius $R$. If $ q = 1 \, \mu\text{C} $, $ V = 1 \, \text{kV} $, $ B = 1 \, \text{mT} $, and $ R = 2 \, \text{ft} $, let the weight of an Iphoon, in Newtons, be $ w $. If $ w \approx 10^p $, where $p$ is an integer, find $p$. That is, what is the order of magnitude of the weight?
[i](Proposed by Ahaan Rungta)[/i]
Find all positive integers $a$, $b$ and $c$ such that $ab$ is a square, and
\[a+b+c-3\sqrt[3]{abc}=1.\]
[i]Proposed by usjl[/i]
Justin throws a standard six-sided die three times in a row and notes the number of dots on the top face after each roll. How many different sequences of outcomes could he get?
Prove that for every triangle the following inequality holds:
\[\frac{ab+bc+ca}{4S} \geq \cot \frac{\pi}{6}.\]
where $a, b, c$ are lengths of the sides and $S$ is the area of the triangle.
Triangles $ABC$ and $DEF$, having a common incircle of radius $R$, intersect at points
$X_1, X_2, \ldots , X_6$ and form six triangles (see the figure below). Let $r_1, r_2,\ldots, r_6$ be the radii of the
inscribed circles of these triangles, and let $R_1, R_2, \ldots , R_6$ be the radii of the inscribed circles of the
triangles $AX_1F, FX_2B, BX_3D, DX_4C, CX_5E$ and $EX_6A$ respectively.
[img]https://i.ibb.co/BspgdHB/Image.jpg[/img]
Prove that \[ \sum_{i=1}^{6} \frac{1}{r_i} < \frac{6}{R}+\sum_{i=1}^{6} \frac{1}{R_i} \]
[i]U. Maksimenkau[/i]
Convex quadrilateral $ ABCD$ has $ AB\equal{}9$ and $ CD\equal{}12$. Diagonals $ AC$ and $ BD$ intersect at $ E$, $ AC\equal{}14$, and $ \triangle AED$ and $ \triangle BEC$ have equal areas. What is $ AE$?
$ \textbf{(A)}\ \frac{9}{2}\qquad \textbf{(B)}\ \frac{50}{11}\qquad \textbf{(C)}\ \frac{21}{4}\qquad \textbf{(D)}\ \frac{17}{3}\qquad \textbf{(E)}\ 6$
Let $J$ be the $A$-excenter of an acute triangle $ABC$. Let $X$, $Y$ be two points on the circumcircle of the triangle $ACJ$ such that $\overline{JX} = \overline{JY} < \overline{JC}$. Let $P$ be a point lies on $XY$ such that $PB$ is tangent to the circumcircle of the triangle $ABC$. Let $Q$ be a point lies on the circumcircle of the triangle $BXY$ such that $BQ$ is parallel to $AC$.
Prove that $\angle BAP = \angle QAC$.
[i]Proposed by Li4.[/i]
Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left( x^2+xf(y)\right)=xf(x+y)$$ for all reals $x,y$.
Compute the radius of the sphere inscribed in the tetrahedron with coordinates $(2,0,0)$, $(4,0,0)$, $(0,1,0)$, and $(0,0,3)$.
Let $ABCD$ be a square and $E$ a on point diagonal $(AC)$, different from its midpoint. $H$ and $K$ are the orthoceneters of triangles $ABE$ and $ADE$. Prove that $AH$ and $CK$ are parallel.
A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?
$\textbf{(A) }148\qquad\textbf{(B) }324\qquad\textbf{(C) }361\qquad\textbf{(D) }1296\qquad\textbf{(E) }1369$
Let $ABC$ be an acute triangle inscribed in a circle $(O)$ that is fixed, and two of the vertices $B$, $C$ are fixed while vertex $A$ varies on the circumference of the circle. Let $I$ be the center of the incircle, and $AD$ the angle bisector. Let $K$, $L$ be the circumcenters of $CAD$, $ABD$. A line through $O$ parallel to $DL$, $DK$ intersects the line that is through $I$ perpendicular to $IB$, $IC$ at $M$, $N$ respectively. Prove that $MN$ is tangent to a fixed circle when $A$ varies on the circle $(O)$.
Tran Quang Hung, Natural Science High School, National University, Hanoi
Show that a function $ f(x)\equal{}\int_{\minus{}1}^1 (1\minus{}|\ t\ |)\cos (xt)\ dt$ is continuous at $ x\equal{}0$.
Let $n \geq 2$. Let $a_1 , a_2 , a_3 , \ldots a_n$ be $n$ real numbers all less than $1$ and such that $|a_k - a_{k+1} | < 1$ for $1 \leq k \leq n-1$. Show that \[ \dfrac{a_1}{a_2} + \dfrac{a_2}{a_3} + \dfrac{a_3}{a_4} + \ldots + \dfrac{a_{n-1}}{a_n} + \dfrac{a_n}{a_1} < 2 n - 1 . \]
Nanna and Sofie move in the same direction along two parallel paths, which are $200$ meters apart. Nanna's speed is $3$ meters per second, Sofie's only $1$ meter per second. A tall, cylindrical building with a diameter of $100$ meters is located in the middle between the two paths. Since the building first once the line of sight breaks between the girls, their distance between them is $200$ metres. How long will it be before the two girls see each other again?
Determine all strictly increasing functions $f:\mathbb{N}_0\to\mathbb{N}_0$ which satisfy \[f(x)\cdot f(y)\mid (1+2x)\cdot f(y)+(1+2y)\cdot f(x)\]for all non-negative integers $x{}$ and $y{}$.
Points $C_0$ and $B_0$ are the respective midpoints of sides $AB$ and $AC$ of a non-isosceles acute triangle $ABC$, $O$ is its circumscenter and $H$ is the orthocenter. Lines $BH$ and $OC_0$ intersect at $P$, while lines $CH$ and $OB_0$ intersect at $Q$. $OPHQ$ is rhombus. Prove that points $A$, $P$ and $Q$ are collinear. (Author: L. Emelyanov)
Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$.
Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions:
[i](i)[/i] its side lengths are $ 1, 2, 3, \ldots, 1992$ in some order;
[i](ii)[/i] the polygon is circumscribable about a circle.
[i]Alternative formulation:[/i] Does there exist a 1992-gon with side lengths $ 1, 2, 3, \ldots, 1992$ circumscribed about a circle? Answer the same question for a 1990-gon.
[b]p1.[/b] There is a string of numbers $1234567891023...910134 ...91012...$ that concatenates the numbers
$1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, then $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $1$, then $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $1$, $2$, and so on. After $10$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, the string will be concatenated with $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$ again. What is the $2021$st digit?
[b]p2.[/b] Bob really likes eating rice. Bob starts eating at the rate of $1$ bowl of rice per minute. Every minute, the number of bowls of rice Bob eats per minute increases by $1$. Given there are $78$ bowls of rice, find number of minutes Bob needs to finish all the rice.
[b]p3.[/b] Suppose John has $4$ fair coins, one red, one blue, one yellow, one green. If John flips all $4$ coins at once, the probability he will land exactly $3$ heads and land heads on both the blue and red coins can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, Find $a + b$.
[b]p4.[/b] Three of the sides of an isosceles trapezoid have lengths $1$, $10$, $20$ Find the sum of all possible values of the fourth side.
[b]p5.[/b] An number two-three-delightful if and only if it can be expressed as the product of $2$ consecutive integers larger than $1$ and as the product of $3$ consecutive integers larger than $1$. What is the smallest two-three-delightful number?
[b]p6.[/b] There are $3$ students total in Justin's online chemistry class. On a $100$ point test, Justin's two classmates scored $4$ and $7$ points. The teacher notices that the class median score is equal to $gcd(x, 42)$, where the positive integer $x$ is Justin's score. Find the sum of all possible values of Justin's score.
[b]p7.[/b] Eddie's gym class of $10$ students decides to play ping pong. However, there are only $4$ tables and only $2$ people can play at a table. If $8$ students are randomly selected to play and randomly assigned a partner to play against at a table, the probability that Eddie plays against Allen is $\frac{a}{b}$ for relatively
prime positive integers $a$, $b$, Find $a + b$.
[b]p8.[/b] Let $S$ be the set of integers $k$ consisting of nonzero digits, such that $300 < k < 400$ and $k - 300$ is not divisible by $11$. For each $k$ in $S$, let $A(k)$ denote the set of integers in $S$ not equal to $k$ that can be formed by permuting the digits of $k$. Find the number of integers $k$ in $S$ such that $k$ is relatively prime to all elements of $A(k)$.
[b]p9.[/b] In $\vartriangle ABC$, $AB = 6$ and $BC = 5$. Point $D$ is on side $AC$ such that $BD$ bisects angle $\angle ABC$. Let $E$ be the foot of the altitude from $D$ to $AB$. Given $BE = 4$, find $AC^2$.
[b]p10.[/b] For each integer $1 \le n \le 10$, Abe writes the number $2^n + 1$ on a blackboard. Each minute, he takes two numbers $a$ and $b$, erases them, and writes $\frac{ab-1}{a+b-2}$ instead. After $9$ minutes, there is one number $C$ left on the board. The minimum possible value of $C$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$.
[b]p11.[/b] Estimation (Tiebreaker) Let $A$ and $B$ be the proportions of contestants that correctly answered Questions $9$ and $10$ of this round, respectively. Estimate $\left \lfloor \dfrac{1}{(AB)^2} \right \rfloor$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $ABC$ be a triangle, and $K$ and $L$ be points on $AB$ such that $\angle ACK = \angle KCL = \angle LCB$. Let $M$ be a point in $BC$ such that $\angle MKC = \angle BKM$. If $ML$ is the angle bisector of $\angle KMB$, find $\angle MLC$.
In a convex quadrilateral $ABCD$ the angles $BAD$ and $BCD$ are equal. Points $M$ and $N$ lie on the sides $(AB)$ and $(BC)$ such that the lines $MN$ and $AD$ are parallel and $MN=2AD$. The point $H$ is the orthocenter of the triangle $ABC$ and the point $K$ is the midpoint of $MN$. Prove that the lines $KH$ and $CD$ are perpendicular.
A semicircle has diameter $XY$. A square $PQRS$ with side length 12 is inscribed in the semicircle with $P$ and $S$ on the diameter. Square $STUV$ has $T$ on $RS$, $U$ on the semicircle, and $V$ on $XY$. What is the area of $STUV$?
Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that
\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]
[i]Hojoo Lee, Korea[/i]
Let $p_n$ denote the probability that, in $n$ tosses, a fair coin shows the head up $100$ consecutive times. Prove that the sequence $(p_n)$ converges and determine its limit.