Found problems: 85335
2019 Taiwan APMO Preliminary Test, P4
We define a sequence ${a_n}$:
$$a_1=1,a_{n+1}=\sqrt{a_n+n^2},n=1,2,...$$
(1)Find $\lfloor a_{2019}\rfloor$
(2)Find $\lfloor a_{1}^2\rfloor+\lfloor a_{2}^2\rfloor+...+\lfloor a_{20}^2\rfloor$
2004 National Olympiad First Round, 31
For how many different values of integer $n$, one can find $n$ different lines in the plane such that each line intersects with exacly $2004$ of other lines?
$
\textbf{(A)}\ 12
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 9
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 1
$
2016 Fall CHMMC, 3
A gambler offers you a $2$ dollar ticket to play the following game: First, you pick a real number $0 \leq p \leq 1$, then you are given a weighted coin that comes up heads with probability $p$. If you receive $1$ dollar the [i]first[/i] time you flip a tail, and if you receive $2$ dollars [i]first[/i] time you flip a head, what is the optimal expected net winning of flipping the coin twice?
2019 Singapore MO Open, 1
In the acute-angled triangle $ABC$ with circumcircle $\omega$ and orthocenter $H$, points $D$ and $E$ are the feet of the perpendiculars from $A$ onto $BC$ and from $B$ onto $AC$ respecively. Let $P$ be a point on the minor arc $BC$ of $\omega$ . Points $M$ and $N$ are the feet of the perpendiculars from $P$ onto lines $BC$ and $AC$ respectively. Let $PH$ and $MN$ intersect at $R$. Prove that $\angle DMR=\angle MDR$.
2022 Princeton University Math Competition, A4 / B6
Find the number of ordered pairs $(x,y)$ of integers with $0 \le x < 2023$ and $0 \le y < 2023$ such that $y^3 \equiv x^2 \pmod{2023}.$
2015 Junior Balkan Team Selection Tests - Moldova, 3
Let $\Omega$ be the circle circumscribed to the triangle $ABC$. Tangents taken to the circle $\Omega$ at points $A$ and $B$ intersects at the point $P$ , and the perpendicular bisector of $ (BC)$ cuts line $AC$ at point $Q$. Prove that lines $BC$ and $PQ$ are parallel.
1989 IMO Longlists, 41
Let $ f(x) \equal{} a \sin^2x \plus{} b \sin x \plus{} c,$ where $ a, b,$ and $ c$ are real numbers. Find all values of $ a, b$ and $ c$ such that the following three conditions are satisfied simultaneously:
[b](i)[/b] $ f(x) \equal{} 381$ if $ \sin x \equal{} \frac{1}{2}.$
[b](ii)[/b] The absolute maximum of $ f(x)$ is $ 444.$
[b](iii)[/b] The absolute minimum of $ f(x)$ is $ 364.$
2022 Princeton University Math Competition, 8
Ryan Alweiss storms into the Fine Hall common room with a gigantic eraser and erases all integers $n$ in the interval $[2, 728]$ such that $3^t$ doesn’t divide $n!$, where $t = \left\lceil \frac{n-3}{2} \right\rceil$. Find the sum of the leftover integers in that interval modulo $1000$.
1977 AMC 12/AHSME, 17
Three fair dice are tossed at random (i.e., all faces have the same probability of coming up). What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one?
$\textbf{(A) }\frac{1}{6}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{1}{27}\qquad\textbf{(D) }\frac{1}{54}\qquad \textbf{(E) }\frac{7}{36}$
2014 Singapore Senior Math Olympiad, 25
Find the number of ordered pairs of integers (p,q) satisfying the equation $p^2-q^2+p+q=2014$.
1991 Czech And Slovak Olympiad IIIA, 2
A museum has the shape of a (not necessarily convex) 3$n$-gon.
Prove that $n$ custodians can be positioned so as to control all of the museum’s space.
2004 India Regional Mathematical Olympiad, 6
Let $p_1, p_2, \ldots$ be a sequence of primes such that $p_1 =2$ and for $n\geq 1, p_{n+1}$ is the largest prime factor of $p_1 p_2 \ldots p_n +1$ . Prove that $p_n \not= 5$ for any $n$.
1993 Iran MO (2nd round), 1
$G$ is a graph with $n$ vertices $A_1,A_2,\ldots,A_n,$ such that for each pair of non adjacent vertices $A_i$ and $A_j$ , there exist another vertex $A_k$ that is adjacent
to both $A_i$ and $A_j .$
[b](a) [/b]Find the minimum number of edges in such a graph.
[b](b) [/b]If $n = 6$ and $A_1,A_2,A_3,A_4,A_5,$ and $A_6$ form a cycle of length $6,$ find the number of edges that must be added to this cycle such that the above condition holds.
2006 AMC 12/AHSME, 13
Rhombus $ ABCD$ is similar to rhombus $ BFDE$. The area of rhombus $ ABCD$ is 24, and $ \angle BAD \equal{} 60^\circ$. What is the area of rhombus $ BFDE$?
[asy]
size(180);
defaultpen(linewidth(0.7)+fontsize(11));
pair A=origin, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/sqrt(3));
pair point=(3/2, sqrt(3)/2);
draw(B--C--D--A--B--F--D--E--B);
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));[/asy]
$ \textbf{(A) } 6 \qquad \textbf{(B) } 4\sqrt {3} \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 6\sqrt {3}$
2018 ABMC, Team
[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].
1997 Vietnam National Olympiad, 1
Let $ k \equal{} \sqrt[3]{3}$.
a, Find all polynomials $ p(x)$ with rationl coefficients whose degree are as least as possible such that $ p(k \plus{} k^2) \equal{} 3 \plus{} k$.
b, Does there exist a polynomial $ p(x)$ with integer coefficients satisfying $ p(k \plus{} k^2) \equal{} 3 \plus{} k$
1979 Yugoslav Team Selection Test, Problem 3
There are two circles of perimeter $1979$. Let $1979$ points be marked on the first circle, and several arcs with the total length of $1$ on the second. Show that it is possible to place the second circle onto the first in such a way that none of the marked points is covered by a marked arc.
2019 Ecuador Juniors, 4
Let $ABCD$ be a square. On the segments $AB$, $BC$, $CD$ and $DA$, choose points $E, F, G$ and $H$, respectively, such that $AE = BF = CG = DH$. Let $P$ be the intersection point of $AF$ and $DE$, $Q$ be the intersection point of $BG$ and $AF$, $R$ the intersection point of $CH$ and $BG$, and $S$ the point of intersection of $DE$ and $CH$. Prove that $PQRS$ is a square.
2010 Tournament Of Towns, 5
The quadrilateral $ABCD$ is inscribed in a circle with center $O$. The diagonals $AC$ and $BD$ do not pass through $O$. If the circumcentre of triangle $AOC$ lies on the line $BD$, prove that the circumcentre of triangle $BOD$ lies on the line $AC$.
2014 Singapore Senior Math Olympiad, 1
In the triangle $ABC$, the excircle opposite to the vertex $A$ with centre $I$ touches the side BC at D. (The circle also touches the sides of $AB$, $AC$ extended.) Let $M$ be the midpoint of $BC$ and $N$ the midpoint of $AD$. Prove that $I,M,N$ are collinear.
2024 ELMO Shortlist, C6
For positive integers $a$ and $b$, an $(a,b)$-shuffle of a deck of $a+b$ cards is any shuffle that preserves the relative order of the top $a$ cards and the relative order of the bottom $b$ cards. Let $n$, $k$, $a_1$, $a_2$, $\dots$, $a_k$, $b_1$, $b_2$, $\dots$, $b_k$ be fixed positive integers such that $a_i+b_i=n$ for all $1\leq i\leq k$. Big Bird has a deck of $n$ cards and will perform an $(a_i,b_i)$-shuffle for each $1\leq i\leq k$, in ascending order of $i$. Suppose that Big Bird can reverse the order of the deck. Prove that Big Bird can also achieve any of the $n!$ permutations of the cards.
[i]Linus Tang[/i]
2014 NIMO Problems, 2
Two points $A$ and $B$ are selected independently and uniformly at random along the perimeter of a unit square with vertices at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. The probability that the $y$-coordinate of $A$ is strictly greater than the $y$-coordinate of $B$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
[i]Proposed by Rajiv Movva[/i]
2024 All-Russian Olympiad, 1
Petya and Vasya only know positive integers not exceeding $10^9-4000$. Petya considers numbers as good which are representable in the form $abc+ab+ac+bc$, where $a,b$ and $c$ are natural numbers not less than $100$. Vasya considers numbers as good which are representable in the form $xyz-x-y-z$, where $x,y$ and $z$ are natural numbers strictly bigger than $100$. For which of them are there more good numbers?
[i]Proposed by I. Bogdanov[/i]
Kvant 2019, M2567
On sides $BC$, $CA$, $AB$ of a triangle $ABC$ points $K$, $L$, $M$ are chosen, respectively, and a point $P$ is inside $ABC$ is chosen so that $PL\parallel BC$, $PM\parallel CA$, $PK\parallel AB$. Determine if it is possible that each of three trapezoids $AMPL$, $BKPM$, $CLPK$ has an inscribed circle.
2010 Iran Team Selection Test, 5
Circles $W_1,W_2$ intersect at $P,K$. $XY$ is common tangent of two circles which is nearer to $P$ and $X$ is on $W_1$ and $Y$ is on $W_2$. $XP$ intersects $W_2$ for the second time in $C$ and $YP$ intersects $W_1$ in $B$. Let $A$ be intersection point of $BX$ and $CY$. Prove that if $Q$ is the second intersection point of circumcircles of $ABC$ and $AXY$
\[\angle QXA=\angle QKP\]