Found problems: 85335
2020 Tournament Of Towns, 2
Three legendary knights are fighting against a multiheaded dragon.
Whenever the first knight attacks, he cuts off half of the current number of heads plus one more. Whenever the second knight attacks, he cuts off one third of the current number of heads plus two more. Whenever the third knight attacks, he cuts off one fourth of the current number of heads plus three more. They repeatedly attack in an arbitrary order so that at each step an integer number of heads is being cut off. If all the knights cannot attack as the number of heads would become non-integer, the dragon eats them. Will the knights be able to cut off all the dragon’s heads if it has $41!$ heads?
Alexey Zaslavsky
2003 Romania Team Selection Test, 8
Two circles $\omega_1$ and $\omega_2$ with radii $r_1$ and $r_2$, $r_2>r_1$, are externally tangent. The line $t_1$ is tangent to the circles $\omega_1$ and $\omega_2$ at points $A$ and $D$ respectively. The parallel line $t_2$ to the line $t_1$ is tangent to the circle $\omega_1$ and intersects the circle $\omega_2$ at points $E$ and $F$. The line $t_3$ passing through $D$ intersects the line $t_2$ and the circle $\omega_2$ in $B$ and $C$ respectively, both different of $E$ and $F$ respectively. Prove that the circumcircle of the triangle $ABC$ is tangent to the line $t_1$.
[i]Dinu Serbanescu[/i]
2003 Tournament Of Towns, 4
A chocolate bar in the shape of an equilateral triangle with side of the length $n$, consists of triangular chips with sides of the length $1$, parallel to sides of the bar. Two players take turns eating up the chocolate. Each player breaks off a triangular piece (along one of the lines), eats it up and passes leftovers to the other player (as long as bar contains more than one chip, the player is not allowed to eat it completely).
A player who has no move or leaves exactly one chip to the opponent, loses. For each $n$, find who has a winning strategy.
2019 ELMO Shortlist, C4
Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card.
Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$.
[i]Proposed by Carl Schildkraut and Colin Tang[/i]
2013 Today's Calculation Of Integral, 862
Draw a tangent with positive slope to a parabola $y=x^2+1$. Find the $x$-coordinate such that the area of the figure bounded by the parabola, the tangent and the coordinate axisis is $\frac{11}{3}.$
2011 Purple Comet Problems, 10
The diagram shows a large circular dart board with four smaller shaded circles each internally tangent to the larger circle. Two of the internal circles have half the radius of the large circle, and are, therefore, tangent to each other. The other two smaller circles are tangent to these circles. If a dart is thrown so that it sticks to a point randomly chosen on the dart board, then the probability that the dart sticks to a point in the shaded area is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[asy]
size(150);
defaultpen(linewidth(0.8));
filldraw(circle((0,0.5),.5),gray);
filldraw(circle((0,-0.5),.5),gray);
filldraw(circle((2/3,0),1/3),gray);
filldraw(circle((-2/3,0),1/3),gray);
draw(unitcircle);
[/asy]
1985 AMC 12/AHSME, 22
In a circle with center $ O$, $ AD$ is a diameter, $ ABC$ is a chord, $ BO \equal{} 5$, and $ \angle ABO = \stackrel{\frown}{CD} = 60^{\circ}$. Then the length of $ BC$ is:
[asy]size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair O=origin, A=dir(35), C=dir(155), D=dir(215), B=intersectionpoint(dir(125)--O, A--C);
draw(C--A--D^^B--O^^Circle(O,1));
pair point=O;
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("$O$", O, dir(305));
label("$5$", B--O, dir(O--B)*dir(90));
label("$60^\circ$", dir(185), dir(185));
label("$60^\circ$", B+0.05*dir(-25), dir(-25));[/asy]
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 3 \plus{} \sqrt3 \qquad \textbf{(C)}\ 5 \minus{} \frac{\sqrt3}{2} \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{none of the above}$
2009 Ukraine National Mathematical Olympiad, 1
Compare the number of distinct prime divisors of $200^2 \cdot 201^2 \cdot ... \cdot 900^2$ and $(200^2 -1)(201^2 -1)\cdot ... \cdot (900^2 -1) .$
Estonia Open Senior - geometry, 1995.1.3
We call a tetrahedron a "trirectangular " if it has a vertex (we call this is called a "right-angled" vertex) in which the planes of the three sides of the tetrahedron intersect at right angles.
Prove the "three-dimensional Pythagorean theorem":
The square of the area of the opposite face of the "right-angled" vertex of the ""trirectangular " tetrahedron is equal to the sum of the squares of the areas of three other sides of the tetrahedron .
2005 Purple Comet Problems, 10
What is the $1000$ th digit to the right of the decimal point in the decimal representation of $\tfrac{37}{5500}$?
2022 Cyprus TST, 2
Let $n, m$ be positive integers such that
\[n(4n+1)=m(5m+1)\]
(a) Show that the difference $n-m$ is a perfect square of a positive integer.
(b) Find a pair of positive integers $(n, m)$ which satisfies the above relation.
Additional part (not asked in the TST): Find all such pairs $(n,m)$.
2018 Math Prize for Girls Problems, 13
A circle overlaps an equilateral triangle of side length $100\sqrt{3}$. The three chords in the circle formed by the three sides of the triangle have lengths 6, 36, and 60, respectively. What is the area of the circle?
2009 District Olympiad, 3
[b]a)[/b] For $ a,b\ge 0 $ and $ x,y>0, $ show that:
$$ \frac{a^3}{x^2} +\frac{b^3}{y^2}\ge \frac{(a+b)^3}{(x+y)^2} . $$
[b]b)[/b] For $ a,b,c\ge 0 $ and $ x,y,z>0 $ under the condition $ a+b+c=x+y+z, $ prove that:
$$ \frac{a^3}{x^2} +\frac{b^3}{y^2} +\frac{c^3}{z^2} \ge a+b+c. $$
2023 China Girls Math Olympiad, 8
Let $P_i(x_i,y_i)\ (i=1,2,\cdots,2023)$ be $2023$ distinct points on a plane equipped with rectangular coordinate system. For $i\neq j$, define $d(P_i,P_j) = |x_i - x_j| + |y_i - y_j|$. Define
$$\lambda = \frac{\max_{i\neq j}d(P_i,P_j)}{\min_{i\neq j}d(P_i,P_j)}$$.
Prove that $\lambda \geq 44$ and provide an example in which the equality holds.
2020 AMC 12/AHSME, 4
How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$
$\textbf{(A) } 80 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 125 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 500$
2007 South africa National Olympiad, 2
Consider the equation $ x^4 \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} 2007$, where $ a,b,c$ are real numbers. Determine the largest value of $ b$ for which this equation has exactly three distinct solutions, all of which are integers.
2021 Princeton University Math Competition, A6 / B8
Let $f$ be a polynomial. We say that a complex number $p$ is a double attractor if there exists a polynomial $h(x)$ so that $f(x)-f(p) = h(x)(x-p)^2$ for all x \in R. Now, consider the polynomial $$f(x) = 12x^5 - 15x^4 - 40x^3 + 540x^2 - 2160x + 1,$$ and suppose that it’s double attractors are $a_1, a_2, ... , a_n$. If the sum $\sum^{n}_{i=1}|a_i|$ can be written as $\sqrt{a} +\sqrt{b}$, where $a, b$ are positive integers, find $a + b$.
2008 Sharygin Geometry Olympiad, 5
(N.Avilov) Can the surface of a regular tetrahedron be glued over with equal regular hexagons?
2018 Thailand TST, 2
Call a rational number [i]short[/i] if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m-$[i]tastic[/i] if there exists a number $c\in \{1,2,3,\ldots ,2017\}$ such that $\dfrac{10^t-1}{c\cdot m}$ is short, and such that $\dfrac{10^k-1}{c\cdot m}$ is not short for any $1\le k<t$. Let $S(m)$ be the set of $m-$tastic numbers. Consider $S(m)$ for $m=1,2,\ldots{}.$ What is the maximum number of elements in $S(m)$?
2013 LMT, Team Round
[b]p1.[/b] Alan leaves home when the clock in his cardboard box says $7:35$ AM and his watch says $7:41$ AM. When he arrives at school, his watch says $7:47$ AM and the $7:45$ AM bell rings. Assuming the school clock, the watch, and the home clock all go at the same rate, how many minutes behind the school clock is the home clock?
[b]p2.[/b] Compute $$\left( \frac{2012^{2012-2013} + 2013}{2013} \right) \times 2012.$$
Express your answer as a mixed number.
[b]p3.[/b] What is the last digit of $$2^{3^{4^{5^{6^{7^{8^{9^{...^{2013}}}}}}}}} ?$$
[b]p4.[/b] Let $f(x)$ be a function such that $f(ab) = f(a)f(b)$ for all positive integers $a$ and $b$. If $f(2) = 3$ and $f(3) = 4$, find $f(12)$.
[b]p5.[/b] Circle $X$ with radius $3$ is internally tangent to circle $O$ with radius $9$. Two distinct points $P_1$ and $P_2$ are chosen on $O$ such that rays $\overrightarrow{OP_1}$ and $\overrightarrow{OP_2}$ are tangent to circle $X$. What is the length of line segment $P_1P_2$?
[b]p6.[/b] Zerglings were recently discovered to use the same $24$-hour cycle that we use. However, instead of making $12$-hour analog clocks like humans, Zerglings make $24$-hour analog clocks. On these special analog clocks, how many times during $ 1$ Zergling day will the hour and minute hands be exactly opposite each other?
[b]p7.[/b] Three Small Children would like to split up $9$ different flavored Sweet Candies evenly, so that each one of the Small Children gets $3$ Sweet Candies. However, three blind mice steal one of the Sweet Candies, so one of the Small Children can only get two pieces. How many fewer ways are there to split up the candies now than there were before, assuming every Sweet Candy is different?
[b]p8.[/b] Ronny has a piece of paper in the shape of a right triangle $ABC$, where $\angle ABC = 90^o$, $\angle BAC = 30^o$, and $AC = 3$. Holding the paper fixed at $A$, Ronny folds the paper twice such that after the first fold, $\overline{BC}$ coincides with $\overline{AC}$, and after the second fold, $C$ coincides with $A$. If Ronny initially marked $P$ at the midpoint of $\overline{BC}$, and then marked $P'$ as the end location of $P$ after the two folds, find the length of $\overline{PP'}$ once Ronny unfolds the paper.
[b]p9.[/b] How many positive integers have the same number of digits when expressed in base $3$ as when expressed in base $4$?
[b]p10.[/b] On a $2 \times 4$ grid, a bug starts at the top left square and arbitrarily moves north, south, east, or west to an adjacent square that it has not already visited, with an equal probability of moving in any permitted direction. It continues to move in this way until there are no more places for it to go. Find the expected number of squares that it will travel on. Express your answer as a mixed number.
PS. You had better use hide for answers.
2014 ASDAN Math Tournament, 4
Let $f(x)=\sum_{i=1}^{2014}|x-i|$. Compute the length of the longest interval $[a,b]$ such that $f(x)$ is constant on that interval.
1995 Chile National Olympiad, 1
Let $a,b,c,d$ be integers. Prove that $ 12$ divides $ (a-b) (a-c) (a-d) (b- c) (b-d) (c-d)$.
1999 All-Russian Olympiad, 4
A frog is placed on each cell of a $n \times n$ square inside an infinite chessboard (so initially there are a total of $n \times n$ frogs). Each move consists of a frog $A$ jumping over a frog $B$ adjacent to it with $A$ landing in the next cell and $B$ disappearing (adjacent means two cells sharing a side). Prove that at least $ \left[\frac{n^2}{3}\right]$ moves are needed to reach a configuration where no more moves are possible.
1987 Poland - Second Round, 4
Determine all pairs of real numbers $ a, b $ for which the polynomials $ x^4 + 2ax^2 + 4bx + a^2 $ and $ x^3 + ax - b $ have two different common real roots.
2005 Estonia Team Selection Test, 4
Find all pairs $(a, b)$ of real numbers such that the roots of polynomials $6x^2 -24x -4a$ and $x^3 + ax^2 + bx - 8$ are all non-negative real numbers.