Found problems: 85335
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$\frac{|DE|}{|BC|}=$________.
2001 Moldova National Olympiad, Problem 3
Find all polynomials $P(x)$ with real coefficieints such that $P\left(x^2\right)=P(x)P(x-1)$ for all $x\in\mathbb R$.
2020 CMIMC Combinatorics & Computer Science, 8
Catherine has a plate containing $300$ circular crumbling mooncakes, arranged as follows:
[asy]
unitsize(10);
for (int i = 0; i < 16; ++i){
for (int j = 0; j < 3; ++j){
draw(circle((sqrt(3)*i,j),0.5));
draw(circle((sqrt(3)*(i+0.5),j-0.5),0.5));
}
}
dot((16*sqrt(3)+.5,.75));
dot((16*sqrt(3)+1,.75));
dot((16*sqrt(3)+1.5,.75));
[/asy]
(This continues for $100$ total columns). She wants to pick some of the mooncakes to eat, however whenever she takes a mooncake all adjacent mooncakes will be destroyed and cannot be eaten. Let $M$ be the maximal number of mooncakes she can eat, and let $n$ be the number of ways she can pick $M$ mooncakes to eat (Note: the order in which she picks mooncakes does not matter). Compute the ordered pair ($M$, $n$).
2007 Mathematics for Its Sake, 1
Find the number of extrema of the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as
$$ f(x)=\prod_{j=1}^n (x-j)^j, $$
where $ n $ is a natural number.
2007 National Olympiad First Round, 9
Let $|AB|=3$ and the length of the altitude from $C$ be $2$ in $\triangle ABC$. What is the maximum value of the product of the lengths of the other two altitudes?
$
\textbf{(A)}\ \frac{144}{25}
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 3\sqrt 2
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ \text{None of the above}
$
2001 Moldova National Olympiad, Problem 1
Find all real solutions of the equation
$$x^2+y^2+z^2+t^2=xy+yz+zt+t-\frac25.$$
2006 IMO Shortlist, 3
The sequence $c_{0}, c_{1}, . . . , c_{n}, . . .$ is defined by $c_{0}= 1, c_{1}= 0$, and $c_{n+2}= c_{n+1}+c_{n}$ for $n \geq 0$. Consider the set $S$ of ordered pairs $(x, y)$ for which there is a finite set $J$ of positive integers such that $x=\textstyle\sum_{j \in J}{c_{j}}$, $y=\textstyle\sum_{j \in J}{c_{j-1}}$. Prove that there exist real numbers $\alpha$, $\beta$, and $M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality \[m < \alpha x+\beta y < M\] if and only if $(x, y) \in S$.
[i]Remark:[/i] A sum over the elements of the empty set is assumed to be $0$.
2023 Belarusian National Olympiad, 8.3
In the triangle $ABC$ points $M$ and $N$ are the midpoints of sides $AC$ and $AB$ respectively. $I$ is the incenter of the triangle. It is known that the angle $MIC$ is a right angle.
Find the angle $NIB$.
2018 Junior Balkan Team Selection Tests - Romania, 1
Prove that a positive integer $A$ is a perfect square if and only if, for all positive integers $n$, at least one of the numbers $(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A,.., (A + n)^2- A$ is a multiple of $n$.
2011 China Western Mathematical Olympiad, 4
Find all pairs of integers $(a,b)$ such that $n|( a^n + b^{n+1})$ for all positive integer $n$