Does there exist a nonisosceles triangle such that the altitude from one vertex, the bisectrix from the second one and the median from the third one are equal?

[b]p1.[/b] Eric is playing Brawl Stars. If he starts playing at $11:10$ AM, and plays for $2$ hours total, then how many minutes past noon does he stop playing? [b]p2.[/b] James is making a mosaic. He takes an equilateral triangle and connects the midpoints of its sides. He then takes the center triangle formed by the midsegments and connects the midpoints of its sides. In total, how many equilateral triangles are in James’ mosaic? [b]p3.[/b] What is the greatest amount of intersections that $3$ circles and $3$ lines can have, given that they all lie on the same plane? [b]p4.[/b] In the faraway land of Arkesia, there are two types of currencies: Silvers and Gold. Each Silver is worth $7$ dollars while each Gold is worth $17$ dollars. In Daniel’s wallet, the total dollar value of the Silvers is $1$ more than that of the Golds. What is the smallest total dollar value of all of the Silvers and Golds in his wallet? [b]p5.[/b] A bishop is placed on a random square of a $8$-by-$8$ chessboard. On average, the bishop is able to move to $s$ other squares on the chessboard. Find $4s$. Note: A bishop is a chess piece that can move diagonally in any direction, as far as it wants. [b]p6.[/b] Andrew has a certain amount of coins. If he distributes them equally across his $9$ friends, he will have $7$ coins left. If he apportions his coins for each of his $15$ classmates, he will have $13$ coins to spare. If he splits the coins into $4$ boxes for safekeeping, he will have $2$ coins left over. What is the minimum number of coins Andrew could have? [b]p7.[/b] A regular polygon $P$ has three times as many sides as another regular polygon $Q$. The interior angle of $P$ is $16^o$ greater than the interior angle of $Q$. Compute how many more diagonals $P$ has compared to $Q$. [b]p8.[/b] In an certain airport, there are three ways to switch between the ground floor and second floor that are 30 meters apart: either stand on an escalator, run on an escalator, or climb the stairs. A family on vacation takes 65 seconds to climb up the stairs. A solo traveller late for their flight takes $25$ seconds to run upwards on the escalator. The amount of time (in seconds) it takes for someone to switch floors by standing on the escalator can be expressed as $\frac{u}{v}$ , where $u$ and $v$ are relatively prime. Find $u + v$. (Assume everyone has the same running speed, and the speed of running on an escalator is the sum of the speeds of riding the escalator and running on the stairs.) [b]p9.[/b] Avanish, being the studious child he is, is taking practice tests to improve his score. Avanish has a $60\%$ chance of passing a practice test. However, whenever Avanish passes a test, he becomes more confident and instead has a $70\%$ chance of passing his next immediate test. If Avanish takes $3$ practice tests in a row, the expected number of practice tests Avanish will pass can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime. Find $a + b$. [b]p10.[/b] Triangle $\vartriangle ABC$ has sides $AB = 51$, $BC = 119$, and $AC = 136$. Point $C$ is reflected over line $\overline{AB}$ to create point $C'$. Next, point $B$ is reflected over line $\overline{AC'}$ to create point $B'$. If $[B'C'C]$ can be expressed in the form of $a\sqrt{b}$, where $b$ is not divisible by any perfect square besides $1$, find $a + b$. [b]p11[/b]. Define the following infinite sequence $s$: $$s = \left\{\frac{1}{1},\frac{1}{1 + 3},\frac{1}{1 + 3 + 6}, ... ,\frac{1}{1 + 3 + 6 + ...+ t_k},...\right\},$$ where $t_k$ denotes the $k$th triangular number. The sum of the first $2024$ terms of $s$, denoted $S$, can be expressed as $$S = 3 \left(\frac{1}{2}+\frac{1}{a}-\frac{1}{b}\right),$$ where $a$ and $b$ are positive integers. Find the minimal possible value of $a + b$. [b]p12.[/b] Omar writes the numbers from $1$ to $1296$ on a whiteboard and then converts each of them into base $6$. Find the sum of all of the digits written on the whiteboard (in base $10$), including both the base $10$ and base $6$ numbers. [b]p13.[/b] A mountain number is a number in a list that is greater than the number to its left and right. Let $N$ be the amount of lists created from the integers $1$ - $100$ such that each list only has one mountain number. $N$ can be expressed as $$N = 2^a(2^b - c^2),$$ where $a$, $b$ and $c$ are positive integers and $c$ is not divisible by $2$. Find $a + b+c$. (The numbers at the beginning or end of a list are not considered mountain numbers.)[hide]Original problem was voided because the original format of the answer didn't match the result's format. So I changed it in the wording, in order the problem to be correct[/hide] [b]p14.[/b] A circle $\omega$ with center $O$ has a radius of $25$. Chords $\overline{AB}$ and $\overline{CD}$ are drawn in $\omega$ , intersecting at $X$ such that $\angle BXC = 60^o$ and $AX > BX$. Given that the shortest distance of $O$ with $\overline{AB}$ and $\overline{CD}$ is $7$ and $15$ respectively, the length of $BX$ can be expressed as $x - \frac{y}{\sqrt{z}}$ , where $x$, $y$, and $z$ are positive integers such that $z$ is not divisible by any perfect square. Find $x + y + z.$ [hide]two answers were considered correct according to configuration[/hide] [b]p15.[/b] How many ways are there to split the first $10$ natural numbers into $n$ sets (with $n \ge 1$) such that all the numbers are used and each set has the same average? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Diagonal $AC$ is drawn in rectangle $ABCD$. Points $E$ and $F$ are placed on $BC$ such that $CE:EF:FB=2:1:1$. Let $G$ be the intersection of $DF$ with $AC$ and $H$ the intersection of $DE$ with $AC$. Given that $AD=4$ and $AB=8$, find the length of $GH$. Express your answer as a common fraction in simplest radical form. [center][img]https://cdn.artofproblemsolving.com/attachments/4/c/b69d79cd47bcb945e7a489533eb9761ccc7ccd.png[/img][/center] $\textbf{(A) } \dfrac{4\sqrt5}{21}\qquad\textbf{(B) } \dfrac{8\sqrt5}{21}\qquad\textbf{(C) } \dfrac{10\sqrt5}{21}\qquad\textbf{(D) } \dfrac{4\sqrt5}{5}\qquad\textbf{(E) } \sqrt5$

Find, with proof, the least positive integer $n$ having the following property: in the binary representation of $\frac 1n$, all the binary representations of $1, 2, \ldots, 1990$ (each consist of consecutive digits) are appeared after the decimal point.

Let set $A=\{1,2,\ldots,n\} ,$ and $X,Y$ be two subsets (not necessarily distinct) of $A.$ Define that $\textup{max} X$ and $\textup{min} Y$ represent the greatest element of $X$ and the least element of $Y,$ respectively. Determine the number of two-tuples $(X,Y)$ which satisfies $\textup{max} X>\textup{min} Y.$

Prove that for each vertex of a polyhedron it is possible to attach a natural number so that for each pair of vertices with a common edge, the attached numbers are not relatively prime (i.e. they have common divisors), and with each pair of vertices without a common edge the attached numbers are relatively prime. (Note: there are infinitely many prime numbers.)

How many ordered pairs of integers (x; y) satisfy the equation: 2$x^2$ + $y^2$ + xy = 2(x + y)?

Incircle of a triangle $ABC$ touches $AB$ and $AC$ at $D$ and $E$, respectively. Point $J$ is the excenter of $A$. Points $M$ and $N$ are midpoints of $JD$ and $JE$. Lines $BM$ and $CN$ cross at point $P$. Prove that $P$ lies on the circumcircle of $ABC$.

The triangle $ABC$ ($AB > BC$) is inscribed in the circle $\Omega$. On the sides $AB$ and $BC$, the points $M$ and $N$ are chosen, respectively, so that $AM = CN$, The lines $MN$ and $AC$ intersect at point $K$. Let $P$ be the center of the inscribed circle of triangle $AMK$, and $Q$ the center of the excircle of the triangle $CNK$ tangent to side $CN$. Prove that the midpoint of the arc $ABC$ of the circle $\Omega$ is equidistant from the $P$ and $Q$.

If distinct digits $D,E,L,M,Q$ (between $0$ and $9$ inclusive) satisfy \begin{tabular}{c@{\,}c@{\,}c@{\,}c} & & $E$ & $L$ \\ + & $M$ & $E$ & $M$ \\\hline & $Q$ & $E$ & $D$ \\ \end{tabular} what is the maximum possible value of the three digit integer $QED$? [i]2019 CCA Math Bonanza Individual Round #6[/i]

Let $ABC$ be an acute triangle, $M$ be the midpoint of $BC$ and $P$ be a point on line segment $AM$. Lines $BP$ and $CP$ meet the circumcircle of $ABC$ again at $X$ and $Y$ , respectively, and sides $AC$ at $D$ and $AB$ at $E$, respectively. Prove that the circumcircles of $AXD$ and $AYE$ have a common point $T \ne A$ on line $AM$.

Let $n$ be an even positive integer. Show that there is a permutation $\left(x_{1},x_{2},\ldots,x_{n}\right)$ of $\left(1,\,2,\,\ldots,n\right)$ such that for every $i\in\left\{1,\ 2,\ ...,\ n\right\}$, the number $x_{i+1}$ is one of the numbers $2x_{i}$, $2x_{i}-1$, $2x_{i}-n$, $2x_{i}-n-1$. Hereby, we use the cyclic subscript convention, so that $x_{n+1}$ means $x_{1}$.

Let $\alpha$ be the angle between two lines containing the diagonals of a regular $1996$-gon, and let $\beta\not= 0$ be another such angle. Prove that $\frac{\alpha}{\beta}$ is a rational number.

There are $20$ pupils in the Backwoods school. Any two of them have a common grandfather. Prove that there exist $14$ pupils all of whom have a common grandfather. (AV Shapovalov)

We call two simple polygons $P, Q$ $\textit{compatible}$ if there exists a positive integer $k$ such that each of $P, Q$ can be partitioned into $k$ congruent polygons similar to the other one. Prove that for every two even integers $m, n \geq 4$, there are two compatible polygons with $m$ and $n$ sides. (A simple polygon is a polygon that does not intersect itself.) [i]Proposed by Hesam Rajabzadeh[/i]

Suppose that each of $n$ people writes down the numbers $1, 2, 3$ in random order in one column of a $3\times n$ matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums $a, b, c$ of the resulting matrix be rearranged (if necessary) so that $a \le b \le c$. Show that for some $n \ge 1995$ ,it is at least four times as likely that both $b = a+1$ and $c = a+2$ as that $a = b = c$.

Seventeen football fans were planning to go to Korea to watch the World Cup football match. They selected 17 matches. The conditions of the admission tickets they booked were such that - One person should book at most one admission ticket for one match; - At most one match was same in the tickets booked by every two persons; - There was one person who booked six tickets. How many tickets did those football fans book at most?

The National Marriage Council wishes to invite $n$ couples to form 17 discussion groups under the following conditions: (1) All members of a group must be of the same sex; i.e. they are either all male or all female. (2) The difference in the size of any two groups is 0 or 1. (3) All groups have at least 1 member. (4) Each person must belong to one and only one group. Find all values of $n$, $n \leq 1996$, for which this is possible. Justify your answer.

Let $x_1$ be a given positive integer. A sequence $\{x_n\}_ {n\geq 1}$ of positive integers is such that $x_n$, for $n \geq 2$, is obtained from $x_ {n-1}$ by adding some nonzero digit of $x_ {n-1}$. Prove that a) the sequence contains an even term; b) the sequence contains infinitely many even terms.

Let $k\ge 0$ be an integer and suppose the integers $a_1,a_2,\dots,a_n$ give at least $2k$ different residues upon division by $(n+k)$. Show that there are some $a_i$ whose sum is divisible by $n+k$.

[u]Round 1[/u] [b]p1.[/b] If $2m = 200 cm$ and $m \ne 0$, find $c$. [b]p2.[/b] A right triangle has two sides of lengths $3$ and $4$. Find the smallest possible length of the third side. [b]p3.[/b] Given that $20(x + 17) = 17(x + 20)$, determine the value of $x$. [u]Round 2[/u] [b]p4.[/b] According to the Egyptian Metropolitan Culinary Community, food service is delayed on $\frac23$ of flights departing from Cairo airport. On average, if flights with delayed food service have twice as many passengers per flight as those without, what is the probability that a passenger departing from Cairo airport experiences delayed food service? [b]p5.[/b] In a positive geometric sequence $\{a_n\}$, $a_1 = 9$, $a_9 = 25$. Find the integer $k$ such that $a_k = 15$ [b]p6.[/b] In the Delicate, Elegant, and Exotic Music Organization, pianist Hans is selling two types of owers with different prices (per ower): magnolias and myosotis. His friend Alice originally plans to buy a bunch containing $50\%$ more magnolias than myosotis for $\$50$, but then she realizes that if she buys $50\%$ less magnolias and $50\%$ more myosotis than her original plan, she would still need to pay the same amount of money. If instead she buys $50\%$ more magnolias and $50\%$ less myosotis than her original plan, then how much, in dollars, would she need to pay? [u]Round 3[/u] [b]p7.[/b] In square $ABCD$, point $P$ lies on side $AB$ such that $AP = 3$,$BP = 7$. Points $Q,R, S$ lie on sides $BC,CD,DA$ respectively such that $PQ = PR = PS = AB$. Find the area of quadrilateral $PQRS$. [b]p8.[/b] Kristy is thinking of a number $n < 10^4$ and she says that $143$ is one of its divisors. What is the smallest number greater than $143$ that could divide $n$? [b]p9.[/b] A positive integer $n$ is called [i]special [/i] if the product of the $n$ smallest prime numbers is divisible by the sum of the $n$ smallest prime numbers. Find the sum of the three smallest special numbers. [u]Round 4[/u] [b]p10.[/b] In the diagram below, all adjacent points connected with a segment are unit distance apart. Find the number of squares whose vertices are among the points in the diagram and whose sides coincide with the drawn segments. [img]https://cdn.artofproblemsolving.com/attachments/b/a/923e4d2d44e436ccec90661648967908306fea.png[/img] [b]p11.[/b] Geyang tells Junze that he is thinking of a positive integer. Geyang gives Junze the following clues: $\bullet$ My number has three distinct odd digits. $\bullet$ It is divisible by each of its three digits, as well as their sum. What is the sum of all possible values of Geyang's number? [b]p12.[/b] Regular octagon $ABCDEFGH$ has center $O$ and side length $2$. A circle passes through $A,B$, and $O$. What is the area of the part of the circle that lies outside of the octagon? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2936505p26278645]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider skew lines $PP'$, $QQ'$ and points $X$, $Y$ lying on them, respectively. Initially, we have $X=P$, $Y=Q$. Both points $X$, $Y$ start moving simultaneously along the rays $PP'$, $QQ'$ with the speeds $c_1$, $c_2$, respectively. Show that midpoint $Z$ of segment $XY$ always lies on a fixed ray $RR'$, where $R$ is midpoint of $PQ$.

The side-lengths of two equilaterals $ABC$ and $KLM$ are $1$ and $1/4$, respectively. And triangle $KLM$ located inside triangle $ABC$. Denote by $\Sigma$ the sum of the distances from $A$ to lines $KL, LM$ and $MK$. Find the location of triangle $KLM$ when $\Sigma$ is maximal.

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Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$ respectively. What is its area? (A)$20\sqrt{3}$ (B)$20\sqrt{2}$ (C)$25\sqrt{3}$ (D)None of these