Found problems: 85335
2011 Today's Calculation Of Integral, 756
Let $a$ be real number. A circle $C$ touches the line $y=-x$ at the point $(a, -a)$ and passes through the point $(0,\ 1).$
Denote by $P$ the center of $C$. When $a$ moves, find the area of the figure enclosed by the locus of $P$ and the line $y=1$.
2017 APMO, 1
We call a $5$-tuple of integers [i]arrangeable[/i] if its elements can be labeled $a, b, c, d, e$ in some order so that $a-b+c-d+e=29$. Determine all $2017$-tuples of integers $n_1, n_2, . . . , n_{2017}$ such that if we place them in a circle in clockwise order, then any $5$-tuple of numbers in consecutive positions on the circle is arrangeable.
[i]Warut Suksompong, Thailand[/i]
India EGMO 2025 TST, 8
Let $ABCD$ be a trapezium with $AD||BC$; and let $X$ and $Y$ be the midpoints of $AC$ and $BD$ respectively. Prove that if $\angle DAY=\angle CAB$ then the internal bisectors of $\angle XAY$ and $\angle XBY$ meet on $XY$.
Proposed by Belur Jana Venkatachala
2021 Latvia Baltic Way TST, P14
Prove that there exist infinitely many triples of positive integers $(a,b,c)$ so that $a,b,c$ are pairwise coprime and
$$\bigg \lfloor \frac{a^2}{2021} \bigg \rfloor + \bigg \lfloor \frac{b^2}{2021} \bigg \rfloor = \bigg \lfloor \frac{c^2}{2021} \bigg \rfloor.$$
2022 Iran MO (2nd round), 3
Take a $n \times n$ chess page.Determine the $n$ such that we can put the numbers $1,2,3, \ldots ,n$ in the squares of the page such that we know the following two conditions are true:
a) for each row we know all the numbers $1,2,3, \ldots ,n$ have appeared on it and the numbers that are in the black squares of that row have the same sum as the sum of the numbers in the white squares of that row.
b) for each column we know all the numbers $1,2,3, \ldots ,n$ have appeared on it and the numbers that are in the black squares in that column have the same sum as the sum of the numbers in the white squares of that column.
2024 Taiwan TST Round 2, C
Let $k$ be a positive integer. The little one and the magician on the skywalk play a game. Initially, there are $N = 2^k$ distinct balls line up in a row, with each of the ball covered by a cup. On each turn, the little one chooses two cups, then the magician can either swap the balls in the two cups, or do a fake move so that the balls in the two cups stay the same. The little one cannot distinguish whether the magician fakes a move on not, nor can she observe the balls inside the cups. After $M = k \times 2^{k-1}$ turns, the magician opens all cups so the little one can check the ball in each of the cups. If the little one can identify whether the magician fakes a move or not for each of the $M$ turns, then the little one win. Prove that the little one has a winning strategy.
[i]
Proposed by usjl[/i]
2004 Junior Tuymaada Olympiad, 3
Point $ O $ is the center of the circumscribed circle of an acute triangle $ Abc $. A certain circle passes through the points $ B $ and $ C $ and intersects sides $ AB $ and $ AC $ of a triangle. On its arc lying inside the triangle, points $ D $ and $ E $ are chosen so that the segments $ BD $ and $ CE $ pass through the point $ O $. Perpendicular $ DD_1 $ to $ AB $ side and perpendicular $ EE_1 $ to $ AC $ side intersect at $ M $. Prove that the points $ A $, $ M $ and $ O $ lie on the same straight line.
2011 AIME Problems, 15
For some integer $m$, the polynomial $x^3-2011x+m$ has the three integer roots $a$, $b$, and $c$. Find $|a|+|b|+|c|$.
2021 Saudi Arabia Training Tests, 19
Let $ABC$ be a triangle with $AB < AC$ inscribed in $(O)$. Tangent line at $A$ of $(O)$ cuts $BC$ at $D$. Take $H$ as the projection of $A$ on $OD$ and $E,F$ as projections of $H$ on $AB,AC$.Suppose that $EF$ cuts $(O)$ at $R,S$. Prove that $(HRS)$ is tangent to $OD$
1973 AMC 12/AHSME, 27
Cars A and B travel the same distance. Care A travels half that [i]distance[/i] at $ u$ miles per hour and half at $ v$ miles per hour. Car B travels half the [i]time[/i] at $ u$ miles per hour and half at $ v$ miles per hour. The average speed of Car A is $ x$ miles per hour and that of Car B is $ y$ miles per hour. Then we always have
$ \textbf{(A)}\ x \leq y\qquad
\textbf{(B)}\ x \geq y \qquad
\textbf{(C)}\ x\equal{}y \qquad
\textbf{(D)}\ x<y\qquad
\textbf{(E)}\ x>y$
2023 Tuymaada Olympiad, 5
A graph contains $p$ vertices numbered from $1$ to $p$, and $q$ edges numbered from $p + 1$ to $p + q$. It turned out that for each edge the sum of the numbers of its ends and of the edge itself equals the same number $s$. It is also known that the numbers of edges starting in all vertices are equal. Prove that
\[s = \dfrac{1}{2} (4p+q+3).\]
2021/2022 Tournament of Towns, P7
A starship is located in a halfspace at the distance $a$ from its boundary. The crew knows this but does not know which direction to move to reach the boundary plane. The starship may travel through the space by any path, may measure the way it has already travelled and has a sensor that signals when the boundary is reached. Is it possible to reach the boundary for sure, having passed no more than:
$a)14a$
$b)13a$?
2020 AMC 12/AHSME, 20
Let $T$ be the triangle in the coordinate plane with vertices $\left(0,0\right)$, $\left(4,0\right)$, and $\left(0,3\right)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)
$\textbf{(A) } 12\qquad\textbf{(B) } 15\qquad\textbf{(C) }17 \qquad\textbf{(D) }20 \qquad\textbf{(E) }25$
2012 Denmark MO - Mohr Contest, 5
In the hexagon $ABCDEF$, all angles are equally large. The side lengths satisfy $AB = CD = EF = 3$ and $BC = DE = F A = 2$. The diagonals $AD$ and $CF$ intersect each other in the point $G$. The point $H$ lies on the side $CD$ so that $DH = 1$. Prove that triangle $EGH$ is equilateral.
2021 AMC 12/AHSME Fall, 19
Regular polygons with $5, 6, 7, $ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?
$\textbf{(A)}\ 52 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\
64 \qquad\textbf{(E)}\ 68$
2017 Balkan MO Shortlist, N4
Find all pairs of positive integers $(x,y)$ , such that $x^2$ is divisible by $2xy^2 -y^3 +1$.
2011 Belarus Team Selection Test, 1
$AB$ and $CD$ are two parallel chords of a parabola. Circle $S_1$ passing through points $A,B$ intersects circle $S_2$ passing through $C,D$ at points $E,F$. Prove that if $E$ belongs to the parabola, then $F$ also belongs to the parabola.
I.Voronovich
2015 ASDAN Math Tournament, 2
Heesu plays a game where he starts with $1$ piece of candy. Every turn, he flips a fair coin. On heads, he gains another piece of candy, unless he already has $5$ pieces of candy, in which case he loses $4$ pieces of candy and goes back to having $1$ piece of candy. On tails, the game ends. What is the expected number of pieces of candy that Heesu will have when the game ends?
2006 All-Russian Olympiad, 7
A $100\times 100$ chessboard is cut into dominoes ($1\times 2$ rectangles). Two persons play the following game: At each turn, a player glues together two adjacent cells (which were formerly separated by a cut-edge). A player loses if, after his turn, the $100\times 100$ chessboard becomes connected, i. e. between any two cells there exists a way which doesn't intersect any cut-edge. Which player has a winning strategy - the starting player or his opponent?
Kvant 2024, M2789
Let $n>100$ be a positive integer and originally the number $1$ is written on the blackboard. Petya and Vasya play the following game: every minute Petya represents the number of the board as a sum of two distinct positive fractions with coprime nominator and denominator and Vasya chooses which one to delete. Show that Petya can play in such a manner, that after $n$ moves, the denominator of the fraction left on the board is at most $2^n+50$, no matter how Vasya acts.
2023 Philippine MO, 6
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(2f(x)) = f(x - f(y)) + f(x) + y$$ for all $x, y \in \mathbb{R}$.
2011 Indonesia TST, 1
For all positive integer $n$, define $f_n(x)$ such that $f_n(x) = \sum_{k=1}^n{|x - k|}$.
Determine all solution from the inequality $f_n(x) < 41$ for all positive $2$-digit integers $n$ (in decimal notation).
2024 Al-Khwarizmi IJMO, 3
Find all $x, y, z \in \left (0, \frac{1}{2}\right )$ such that
$$
\begin{cases}
(3 x^{2}+y^{2}) \sqrt{1-4 z^{2}} \geq z; \\
(3 y^{2}+z^{2}) \sqrt{1-4 x^{2}} \geq x; \\
(3 z^{2}+x^{2}) \sqrt{1-4 y^{2}} \geq y.
\end{cases}
$$
[i]Proposed by Ngo Van Trang, Vietnam[/i]
1991 China National Olympiad, 1
We are given a convex quadrilateral $ABCD$ in the plane.
([i]i[/i]) If there exists a point $P$ in the plane such that the areas of $\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP$ are equal, what condition must be satisfied by the quadrilateral $ABCD$?
([i]ii[/i]) Find (with proof) the maximum possible number of such point $P$ which satisfies the condition in ([i]i[/i]).
1988 National High School Mathematics League, 8
In $\triangle ABC$, $\angle A=\alpha$, $CD,BE$ are height on sides $AB,AC$. Then$\frac{|DE|}{|BC|}=$________.