Found problems: 85335
1988 AMC 12/AHSME, 8
If $\frac{b}{a} = 2$ and $\frac{c}{b} = 3$, what is the ratio of $a + b$ to $b + c$?
$ \textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{3}{8}\qquad\textbf{(C)}\ \frac{3}{5}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{3}{4} $
2023 Quang Nam Province Math Contest (Grade 11), Problem 2
Given the sequence $(u_n)$ satisfying:$$\left\{ \begin{array}{l}
1 \le {u_1} \le 3\\
{u_{n + 1}} = 4 - \dfrac{{2({u_n} + 1)}}{{{2^{{u_n}}}}},\forall n \in \mathbb{Z^+}.
\end{array} \right.$$
Prove that: $1\le u_n\le 3,\forall n\in \mathbb{Z^+}$ and find the limit of $(u_n).$
2011 Austria Beginners' Competition, 3
Let $x, y$ be positive real numbers with $x + y + xy= 3$. Prove that$$x + y\ge 2.$$ When does equality holds?
(K. Czakler, GRG 21, Vienna)
2021 Princeton University Math Competition, 7
The roots of the polynomial $f(x) = x^8 +x^7 -x^5 -x^4 -x^3 +x+ 1 $ are all roots of unity. We say that a real number $r \in [0, 1)$ is nice if $e^{2i \pi r} = \cos 2\pi r + i \sin 2\pi r$ is a root of the polynomial $f$ and if $e^{2i \pi r}$ has positive imaginary part. Let $S$ be the sum of the values of nice real numbers $r$. If $S =\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.
1984 Putnam, A2
Express $\sum_{k=1}^\infty\frac{6^k}{(3^{k+1}-2^{k+1})(3^k-2^k)}$ as a rational number.
2014 Bosnia and Herzegovina Junior BMO TST, 4
It is given $5$ numbers $1$, $3$, $5$, $7$, $9$. We get the new $5$ numbers such that we take arbitrary $4$ numbers(out of current $5$ numbers) $a$, $b$, $c$ and $d$ and replace them with $\frac{a+b+c-d}{2}$, $\frac{a+b-c+d}{2}$, $\frac{a-b+c+d}{2}$ and $\frac{-a+b+c+d}{2}$. Can we, with repeated iterations, get numbers:
$a)$ $0$, $2$, $4$, $6$ and $8$
$b)$ $3$, $4$, $5$, $6$ and $7$
LMT Team Rounds 2021+, A5
In rectangle $ABCD$, points $E$ and $F$ are on sides $\overline{BC}$ and $\overline{AD}$, respectively. Two congruent semicircles are drawn with centers $E$ and $F$ such that they both lie entirely on or inside the rectangle, the semicircle with center $E$ passes through $C$, and the semicircle with center $F$ passes through $A$. Given that $AB=8$, $CE=5$, and the semicircles are tangent, find the length $BC$.
[i]Proposed by Ada Tsui[/i]
2014 Taiwan TST Round 3, 1
Consider a $6 \times 6$ grid. Define a [i]diagonal[/i] to be the six squares whose coordinates $(i,j)$ ($1 \le i,j \le 6)$ satisfy $i-j \equiv k \pmod 6$ for some $k=0,1,\dots,5$. Hence there are six diagonals.
Determine if it is possible to fill it with the numbers $1,2,\dots,36$ (each exactly once) such that each row, each column, and each of the six diagonals has the same sum.
Estonia Open Senior - geometry, 2001.1.1
Points $A, B, C, D, E$ and F are given on a circle in such a way that the three chords $AB, CD$ and $EF$ intersect in one point. Express angle $\angle EFA$ in terms of angles $\angle ABC$ and $\angle CDE$ (find all possibilities).
2012 Tournament of Towns, 5
In an $8\times 8$ chessboard, the rows are numbers from $1$ to $8$ and the columns are labelled from $a$ to $h$. In a two-player game on this chessboard, the fi rst player has a White Rook which starts on the square $b2$, and the second player has a Black Rook which starts on the square $c4$. The two players take turns moving their rooks. In each move, a rook lands on another square in the same row or the same column as its starting square. However, that square cannot be under attack by the other rook, and cannot have been landed on before by either rook. The player without a move loses the game. Which player has a winning strategy?
2024 Belarus Team Selection Test, 2.3
A right triangle $ABC$ ($\angle A=90$) is inscribed in a circle $\omega$. Tangent to $\omega$ at $A$ intersects $BC$ at $P$, $B$ lies between $P$ and $C$. Let $M$ be the midpoint of the minor arc $AB$. $MP$ intersects $\omega$ at $Q$. Point $X$ lies on a ray $PA$ such that $\angle XCB=90$. Prove that line $XQ$ passes through the orthocenter of the triangle $ABO$
[i]Mayya Golitsyna[/i]
2019 Mathematical Talent Reward Programme, SAQ: P 3
Suppose $a$, $b$, $c$ are three positive real numbers with $a + b + c = 3$. Prove that
$$\frac{a}{b^2 + c}+\frac{b}{c^2 + a}+\frac{c}{a^2 + b}\geq \frac{3}{2}$$
2023/2024 Tournament of Towns, 5
5. Nine farmers have a checkered $9 \times 9$ field. There is a fence along the boundary of the field. The entire field is completely covered with berries (there is a berry in every point of the field, except the points of the fence). The farmers divided the field along the grid lines in 9 plots of equal area (every plot is a polygon), however they did not demarcate their boundaries. Each farmer takes care of berries only inside his own plot (not on its boundaries). A farmer will notice a loss only if at least two berries disappeared inside his plot. There is a crow which knows all of the above, except the location of boundaries of plots. Can the crow carry off 8 berries from the field so that for sure no farmer will notice this?
Tatiana Kazitsina
2012-2013 SDML (Middle School), 2
If $\frac{a}{3}=b$ and $\frac{b}{4}=c$, what is the value of $\frac{ab}{c^2}$?
$\text{(A) }12\qquad\text{(B) }36\qquad\text{(C) }48\qquad\text{(D) }60\qquad\text{(E) }144$
1996 IMO Shortlist, 3
Let $O$ be the circumcenter and $H$ the orthocenter of an acute-angled triangle $ABC$ such that $BC>CA$. Let $F$ be the foot of the altitude $CH$ of triangle $ABC$. The perpendicular to the line $OF$ at the point $F$ intersects the line $AC$ at $P$. Prove that $\measuredangle FHP=\measuredangle BAC$.
2021 Junior Balkan Team Selection Tests - Moldova, 4
Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that
$73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$.
2016 BMT Spring, 6
Bob plays a game on the whiteboard. Initially, the numbers $\{1, 2, ...,n\}$ are shown. On each turn, Bob takes two numbers from the board $x$, $y$, erases them both, and writes down $2x + y$ onto the board. In terms of n, what is the maximum possible value that Bob can end up with?
2015 Korea - Final Round, 3
There are at least $3$ subway stations in a city.
In this city, there exists a route that passes through more than $L$ subway stations, without revisiting.
Subways run both ways, which means that if you can go from subway station A to B, you can also go from B to A.
Prove that at least one of the two holds.
$\text{(i)}$. There exists three subway stations $A$, $B$, $C$ such that there does not exist a route from $A$ to $B$ which doesn't pass through $C$.
$\text{(ii)}$. There is a cycle passing through at least $\lfloor \sqrt{2L} \rfloor$ stations, without revisiting a same station more than once.
2024 CCA Math Bonanza, T10
Find $$\sum_{a=1}^{9} \sum_{b=1}^{9} \sum_{c=1}^{9} \sum_{d=1}^{9} \min(2a + 0b + 2c + 4d, 4a + 1b + 4c + 3d) + \max(10, a + b + 2c + 2d).$$
[i]Team #10[/i]
2023 pOMA, 1
Let $n$ be a positive integer. Marc has $2n$ boxes, and in particular, he has one box filled with $k$ apples for each $k=1,2,3,\ldots,2n$. Every day, Marc opens a box and eats all the apples in it. However, if he eats strictly more than $2n+1$ apples in two consecutive days, he gets stomach ache. Prove that Marc has exactly $2^n$ distinct ways of choosing the boxes so that he eats all the apples but doesn't get stomach ache.
1972 Putnam, B6
Let $ n_1<n_2<n_3<\cdots <n_k$ be a set of positive integers. Prove that the polynomial $ 1\plus{}z^{n_1}\plus{}z^{n_2}\plus{}\cdots \plus{}z^{n_k}$ has no roots inside the circle $ |z|<\frac{\sqrt{5}\minus{}1}{2}$.
2005 JHMT, 1
A circle with diameter $23$ is cut by a chord $AC$. Two different circles can be inscribed between the large circle and $AC$. Find the sum of the two radii.
1983 National High School Mathematics League, 9
In $\triangle ABC,\sin A=\frac{3}{5},\cos B=\frac{5}{13}$, then $\cos C=$________.
1994 Korea National Olympiad, Problem 1
Consider the equation $ y^2\minus{}k\equal{}x^3$, where $ k$ is an integer.
Prove that the equation cannot have five integer solutions of the form
$ (x_1,y_1),(x_2,y_1\minus{}1),(x_3,y_1\minus{}2),(x_4,y_1\minus{}3),(x_5,y_1\minus{}4)$.
Also show that if it has the first four of these pairs as solutions, then $ 63|k\minus{}17$.
1990 Greece Junior Math Olympiad, 2
For which real values of $x,y$ the expression$\frac{2-\left(\dfrac{x+y}{3}-1\right)^2}{\left(\dfrac{x-3}{2}+\dfrac{2y-x}{3}\right)^2+4}$ becomes maximum? Which is that maximum value?