Found problems: 85335
If $ab+\sqrt{ab+1}+\sqrt{a^2+b}\sqrt{a+b^2}=0$, find the value of $b\sqrt{a^2+b}+a\sqrt{b^2+a}$
Let $O{}$ be the circumcenter of the triangle $ABC$. On the rays $AC$ and $BC$ consider the points $C_a$ and $C_b$ respectively, such that $AC_a$ and $BC_b$ are equal in length to $AB$. Let $O_c{}$ be the circumcenter of the triangle $CC_aC_b$. Define the points $O_a{}$ and $O_b{}$ similarly. Prove that $O{}$ is the orthocenter of the triangle $O_aO_bO_c$.
[i]Proposed by A. Zaslavsky[/i]
Consider an urn containing $51$ white and $50$ black balls. Every turn, we randomly pick a ball, record the color of the ball, and then we put the ball back into the urn. We stop picking when we have recorded $n$ black balls, where $n$ is an integer randomly chosen from $\{1, 2,... , 100\}$. What is the expected number of turns?
We call a positive integer [i]alternating[/i] if every two consecutive digits in its decimal representation are of different parity.
Find all positive integers $n$ such that $n$ has a multiple which is alternating.
Let $ABC$ be an isosceles triangle with $\measuredangle C=\measuredangle B=36$. The point $M$ is in interior of $ ABC$ such that $\measuredangle MBC=24^{\circ} , \measuredangle BCM=30^{\circ}$ $N = AM \cap BC.$. Find $\measuredangle MCB$ .
Given an acute triangle $ABC$ with $AB <AC$. The ex-circles of triangle $ABC$ opposite $B$ and $C$ are centered on $B_1$ and $C_1$, respectively. Let $D$ be the midpoint of $B_1C_1$. Suppose that $E$ is the point of intersection of $AB$ and $CD$, and $F$ is the point of intersection of $AC$ and $BD$. If $EF$ intersects $BC$ at point $G$, prove that $AG$ is the bisector of $\angle BAC$.
There are exactly 3 distinct 4-digit factors of 3212005. Find their sum.
[i]Team #3[/i]
Outside the square $ABCD$, the rhombus $BCMN$ is constructed with angle $BCM$ obtuse . Let $P$ be the intersection point of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and the triangle $DPM$ is right isosceles .
Given $m,n\in\mathbb N_+,$ define
$$S(m,n)=\left\{(a,b)\in\mathbb N_+^2\mid 1\leq a\leq m,1\leq b\leq n,\gcd (a,b)=1\right\}.$$
Prove that: for $\forall d,r\in\mathbb N_+,$ there exists $m,n\in\mathbb N_+,m,n\geq d$ and $\left|S(m,n)\right|\equiv r\pmod d.$
Choose a rational point $P_0(x_p,y_p)$ arbitrary on ellipse $C:x^2+2y^2=2098$. Define $P_1,P_2,\cdots$ recursively by the following rules:
$(1)$ Choose a lattice point $Q_i=(x_i,y_i)\notin C$ such that $|x_i|<50$ and $|y_i|<50$.
$(2)$ Line $P_iQ_i$ intersects $C$ at another point $P_{i+1}$.
Prove that for any point $P_0$ we can choose suitable points $Q_0,Q_1,\cdots$ such that $\exists k\in\mathbb{N}\cup\{0\}$, $\overline{OP_k}^2=2017$.
Let $a$, $b$, $c$ be real numbers such that $a^2 - 2 = 3b - c$, $b^2 + 4 = 3 + a$, and $c^2 + 4 = 3a - b$. Find $a^4 + b^4 + c^4$.
Let $M$ be a centre of gravity (the intersection point of the medians) of a triangle $ABC$. Under rotation by $120$ degrees about the point $M$, the point $B$ is taken to the point $P$; under rotation by $240$ degrees about $M$, the point $C$ is taken to the point $Q$. Prove that either $APQ$ is an equilateral triangle, or the points $A, P, Q$ coincide.
(Bykovsky, Khabarovsksk)
Let $ X_n\equal{}\{1,2,...,n\}$,where $ n \geq 3$.
We define the measure $ m(X)$ of $ X\subset X_n$ as the sum of its elements.(If $ |X|\equal{}0$,then $ m(X)\equal{}0$).
A set $ X \subset X_n$ is said to be even(resp. odd) if $ m(X)$ is even(resp. odd).
(a)Show that the number of even sets equals the number of odd sets.
(b)Show that the sum of the measures of the even sets equals the sum of the measures of the odd sets.
(c)Compute the sum of the measures of the odd sets.
Let $ABCD$ and $ABEF$ be two squares situated in two perpendicular planes and let $O$ be the intersection of the lines $AE$ and $BF$. If $AB=4$ compute:
a) the distance from $B$ to the line of intersection between the planes $(DOC)$ and $(DAF)$;
b) the distance between the lines $AC$ and $BF$.
Let \( M \) be the midpoint of side \( BC \) of triangle \( ABC \), and let \( D \) be an arbitrary point on the arc \( BC \) of the circumcircle that does not contain \( A \). Let \( N \) be the midpoint of \( AD \). A circle passing through points \( A \), \( N \), and tangent to \( AB \) intersects side \( AC \) at point \( E \). Prove that points \( C \), \( D \), \( E \), and \( M \) are concyclic.
[i]Proposed by Matthew Kurskyi[/i]
Let $ABC$ be a triangle with incenter $I$. The points $D$ and $E$ lie on the segments $CA$
and $BC$ respectively, such that $CD = CE$. Let $F$ be a point on the segment $CD$. Prove
that the quadrilateral $ABEF$ is circumscribable if and only if the quadrilateral $DIEF$
is cyclic.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
Let $m$ and $n$ be relatively prime positive integers. Determine all possible values of \[\gcd(2^m - 2^n, 2^{m^2+mn+n^2}- 1).\]
Solve for $g : \mathbb{Z}^+ \to \mathbb{Z}^+$ such that
$$g(m + n) = g(m) + mn(m + n) + g(n)$$
Show that $g(n)$ is of the form $\sum_{i=0}^{d} {c_i n^i}$ \\
and find necessary and sufficient conditions on $d$ and $c_0, c_1, \cdots , c_d$
Evaluate $\int_{1}^{3} ( (x - 1)(3 - x) )^{\dfrac{-1}{2}} dx$ and $\int_{1}^{\infty} (e^{x+1} + e^{3-x})^{-1} dx.$
For complex number constant $c$, and real number constants $p$ and $q$, there exist three distinct complex values of $x$ that satisfy $x^3 + cx + p(1 + qi) = 0$. Suppose $c$, $p$, and $q$ were chosen so that all three complex roots $x$ satisfy $\tfrac{5}{6} \leq \tfrac{\mathrm{Im}(x)}{\mathrm{Re}(x)} \leq \tfrac{6}{5}$, where $\mathrm{Im}(x)$ and $\mathrm{Re}(x)$ are the imaginary and real part of $x$, respectively. The largest possible value of $|q|$ can be expressed as a common fraction $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Two players $A$ and $B$ take turns taking stones from a pile of $N$ stones. They play in the order $A$, $B$, $A$, $B$, $A$, $....$, $A$ starts the game and the one who takes out the last stone loses.$ B$ can serve on each play $1$, $2$ or 3 stones, while$ A$ can draw $2, 3, 4$ stones or $1$ stone in each turn f it is the last one in the pile. Determine for what values of $N$ does $A$ have a winning strategy, and for what values the winning strategy is $B$'s.
The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only
\[a\le\cos A+\sqrt3\sin A.\]
[u]Round 1[/u]
[b]1.1.[/b] What is the area of a circle with diameter $2$?
[b]1.2.[/b] What is the slope of the line through $(2, 1)$ and $(3, 4)$?
[b]1.3.[/b] What is the units digit of $2^2 \cdot 4^4 \cdot 6^6$ ?
[u]Round 2[/u]
[b]2.1.[/b] Find the sum of the roots of $x^2 - 5x + 6$.
[b]2. 2.[/b] Find the sum of the solutions to $|2 - x| = 1$.
[b]2.3.[/b] On April $1$, $2018$, Mr. Dospinescu, Mr. Phaovibul and Mr. Pohoata all go swimming at the same pool. From then on, Mr. Dospinescu returns to the pool every 4th day, Mr. Phaovibul returns every $7$th day and Mr. Pohoata returns every $13$th day. What day will all three meet each other at the pool again? Give both the month and the day.
[u]Round 3[/u]
[b]3. 1.[/b] Kendall and Kylie are each selling t-shirts separately. Initially, they both sell t-shirts for $\$ 33$ each. A week later, Kendall marks up her t-shirt price by $30 \%$, but after seeing a drop in sales, she discounts her price by $30\%$ the following week. If Kim wants to buy $360$ t-shirts, how much money would she save by buying from Kendall instead of Kylie? Write your answer in dollars and cents.
[b]3.2.[/b] Richard has English, Math, Science, Spanish, History, and Lunch. Each class is to be scheduled into one distinct block during the day. There are six blocks in a day. How many ways could he schedule his classes such that his lunch block is either the $3$rd or $4$th block of the day?
[b]3.3.[/b] How many lattice points does $y = 1 + \frac{13}{17}x$ pass through for $x \in [-100, 100]$ ? (A lattice point is a point where both coordinates are integers.)
[u]Round 4[/u]
[b]4. 1.[/b] Unsurprisingly, Aaron is having trouble getting a girlfriend. Whenever he asks a girl out, there is an eighty percent chance she bursts out laughing in his face and walks away, and a twenty percent chance that she feels bad enough for him to go with him. However, Aaron is also a player, and continues asking girls out regardless of whether or not previous ones said yes. What is the minimum number of girls Aaron must ask out for there to be at least a fifty percent chance he gets at least one girl to say yes?
[b]4.2.[/b] Nithin and Aaron are two waiters who are working at the local restaurant. On any given day, they may be fired for poor service. Since Aaron is a veteran who has learned his profession well, the chance of him being fired is only $\frac{2}{25}$ every day. On the other hand, Nithin (who never paid attention during job training) is very lazy and finds himself constantly making mistakes, and therefore the chance of him being fired is $\frac{2}{5}$. Given that after 1 day at least one of the waiters was fired, find the probability Nithin was fired.
[b]4.3.[/b] In a right triangle, with both legs $4$, what is the sum of the areas of the smallest and largest squares that can be inscribed? An inscribed square is one whose four vertices are all on the sides of the triangle.
PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784569p24468582]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Determine all integers $n \geq 3$ for which there exists a con guration of $n$ points in the plane, no three collinear, that can be labelled $1$ through $n$ in two different ways, so that the following
condition be satis fied: For every triple $(i,j,k), 1 \leq i < j < k \leq n$, the triangle $ijk$ in one labelling has the same orientation as the triangle labelled $ijk$ in the other, except for $(i,j,k) = (1,2,3)$.