Found problems: 85335
A $20 \times 20$ rectangular grid has been given. It is known that one of the grid's unit squares contains a hidden treasure. To find the treasure, we have been given an opportunity to order several scientific studies at the same time, results of which will be known only after some time. For each study we must choose one $1 \times 4$ rectangle, and the study will tell whether the rectangle contains the treasure. The $1 \times 4$ rectangle can be either horizontal or vertical, and it can extend over a side of the $20 \times 20$ grid, coming back in at the opposite side (you can think of the $20 \times 20$ grid as a torus - the opposite sides are connected).
What is the minimal amount of studies that have to ordered for us to precisely determine the unit square containing the treasure?
$2017$ chess players participated in the chess tournament, each of them played exactly one chess game with each other. Let's call a trio of chess players $A, B, C$ a [i]principled [/i]one, if $A$ defeated $B$, $B$ defeated $C$, and $C$ defeated $A$. What is the largest possible number of threes of principled chess players?
We are given triangles $ABC$ and $DEF$ such that $D\in BC, E\in CA, F\in AB$, $AD\perp EF, BE\perp FD, CF\perp DE$. Let the circumcenter of $DEF$ be $O$, and let the circumcircle of $DEF$ intersect $BC,CA,AB$ again at $R,S,T$ respectively. Prove that the perpendiculars to $BC,CA,AB$ through $D,E,F$ respectively intersect at a point $X$, and the lines $AR,BS,CT$ intersect at a point $Y$, such that $O,X,Y$ are collinear.
[i]Proposed by Sammy Luo[/i]
Suppose that $ p$ is a prime number and $ a$ and $ n$ positive integers such that: $ 2^p\plus{}3^p\equal{}a^n$. Prove that $ n\equal{}1$.
The Konigsberg School has assigned grades $1$ through $7$ to pods $A$ through $G$, one grade per pod. The school noticed that each pair of connected pods has been assigned grades differing by $2$ or more grade levels. (For example, grades $1$ and $2$ will not be in pods directly connected by a walkway.) What is the sum of the grade levels assigned to pods $C, E,$ and $F$?
$\textbf{(A)}\ 12\qquad \textbf{(B)}\ 13\qquad \textbf{(C)}\ 14\qquad \textbf{(D)}\ 15\qquad \textbf{(E)}\ 16$\\
Alice has six boxes labelled 1 through 6. She secretly chooses exactly two of the boxes and places a coin inside each. Bob is trying to guess which two boxes contain the coins. Each time Bob guesses, he does so by tapping exactly two of the boxes. Alice then responds by telling him the total number of coins inside the two boxes that he tapped. Bob successfully finds the two coins when Alice responds with the number 2.
What is the smallest positive integer $n$ such that Bob can always find the two coins in at most $n$ guesses?
[b]5.[/b] Determine the functions $G$ defined on the set of all non-zero real numbers the values of which are regular matrices of order $2$, and the functions $f$ mapping the two-dimensional real vector space $E_2$ into itself, such that for any vector $y \in E_2$ and for any regular matrix $X$ of order $2$, $f(X_y)= G(det X)Xf(y)$ ($det X $ denotes the determinant of $X$).[b](A. 5)[/b]
Consider a parameterized curve $C: x=e^{-t}\cos t,\ y=e^{-t}\sin t\left (0\leq t\leq \frac{\pi}{2}\right).$
(1) Find the length $L$ of $C$.
(2) Find the area $S$ of the region enclosed by the $x,\ y$ axis and $C$.
Please solve the problem without using the formula of area for polar coordinate for Japanese High School Students who don't study it in High School.
[i]1997 Kyoto University entrance exam/Science[/i]
We say that the rank of a group $ G$ is at most $ r$ if every subgroup of $ G$ can be generated by at most $ r$ elements. Prove
that here exists an integer $ s$ such that for every finite group $ G$ of rank $ 2$ the commutator series of $ G$ has length less than $ s$.
[i]J. Erdos[/i]
For positive integers $m$ and $n$, let $d(m, n)$ be the number of distinct primes that divide both $m$ and $n$. For instance, $d(60, 126) = d(2^2 \cdot 3 \cdot 5, 2 \cdot 3^2 \cdot 7) = 2.$ Does there exist a sequence $(a_n)$ of positive integers such that: [list]
[*] $a_1 \geq 2018^{2018};$
[*] $a_m \leq a_n$ whenever $m \leq n$;
[*] $d(m, n) = d(a_m, a_n)$ for all positive integers $m\neq n$?
[/list]
[i](Dominic Yeo, United Kingdom)[/i]
The positive integers $a$ and $b$ are such that $a+k$ is divisible by $b+k$ for all positive integers numbers $k<b$. Prove that $a-k$ is divisible by $b-k$ for all positive integers $k<b$.
(a) Is there a natural number $n$ such that the number $2^n$ has last digit $6$ and the sum of the other digits is $2$?
b) Are there natural numbers $a$ and $m\ge 3$ such that the number $a^m$ has last digit $6$ and the sum of the other digits is 3?
My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, what is the expected score of the losing team?
$\textbf{(A) }2/3\hspace{14em}\textbf{(B) }1\hspace{14.8em}\textbf{(C) }3/2$
$\textbf{(D) }8/5\hspace{14em}\textbf{(E) }5/8\hspace{14em}\textbf{(F) }2$
$\textbf{(G) }0\hspace{14.9em}\textbf{(H) }5/2\hspace{14em}\textbf{(I) }2/5$
$\textbf{(J) }3/4\hspace{14em}\,\textbf{(K) }4/3\hspace{13.9em}\textbf{(L) }2007$
Alice guessed that there were $5$ skittles in a jar, while Kayla guessed $19$. It turns out that they were equally close to the true number of skittles. How many skittles are in the jar?
$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14$
There are three empty jugs on a table. Winnie the Pooh, Rabbit, and Piglet put walnuts in the jugs one by one. They play successively, with the initial determined by a draw. Thereby Winnie the Pooh plays either in the first or second jug, Rabbit in the second or third, and Piglet in the first or third. The player after whose move there are exactly 1999 walnuts loses the games. Show that Winnie the Pooh and Piglet can cooperate so as to make Rabbit lose.
Let $ x$, $ y$ and $ z$ be integers such that $ S = x^{4}+y^{4}+z^{4}$ is divisible by 29. Show that $ S$ is divisible by $ 29^{4}$.
Prove that the equation $ (x\plus{}y)^n\equal{}x^m\plus{}y^m$ has a unique solution in integers with $ x>y>0$ and $ m,n>1$.
$\omega$ is circumcirlce of triangle $ABC$. We draw a line parallel to $BC$ that intersects $AB,AC$ at $E,F$ and intersects $\omega$ at $U,V$. Assume that $M$ is midpoint of $BC$. Let $\omega'$ be circumcircle of $UMV$. We know that $R(ABC)=R(UMV)$. $ME$ and $\omega'$ intersect at $T$, and $FT$ intersects $\omega'$ at $S$. Prove that $EF$ is tangent to circumcircle of $MCS$.
For every pair of reals $0 < a < b < 1$, we define sequences $\{x_n\}_{n \ge 0}$ and $\{y_n\}_{n \ge 0}$ by $x_0 = 0$, $y_0 = 1$, and for each integer $n \ge 1$: \begin{align*}
x_n & = (1 - a) x_{n - 1} + a y_{n - 1},
\\ y_n & = (1 - b) x_{n - 1} + b y_{n - 1}.
\end{align*} The [i]supermean[/i] of $a$ and $b$ is the limit of $\{x_n\}$ as $n$ approaches infinity. Over all pairs of real numbers $(p, q)$ satisfying $\left (p - \textstyle\frac{1}{2} \right)^2 + \left (q - \textstyle\frac{1}{2} \right)^2 \le \left(\textstyle\frac{1}{10}\right)^2$, the minimum possible value of the supermean of $p$ and $q$ can be expressed as $\textstyle\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.
[i]Proposed by Lewis Chen[/i]
The vertices of a regular $13$-gon are colored in three different colors. Show that there are three vertices which have the same color and are also the vertices of an isosceles triangle.
Let $AB$ be a chord, not a diameter, of a circle with center $O$. The smallest arc $AB$ is divided into three congruent arcs $AC$, $CD$, $DB$. The chord $AB$ is also divided into three equal segments $AC'$, $C'D'$, $D'B$. Let $P$ be the intersection point of between the lines $CC'$ and $DD'$. Prove that $\angle APB = \frac13 \angle AOB$.
In a tennis tournament, $n$ players want to make $2$ vs $2$ matches such that each player has each of the other players as opponents exactly once. Find all possible values of $n$.
A chord is drawn through the intersection point of the diagonals of an inscribed quadrilateral. It is known that the parts of this chord located outside the quadrilateral have lengths equal to $\frac13$ and $\frac14$ of this chord. In what ratio is this chord divided by the intersection point of the diagonals of the quadrilateral?
The roots of $x^3 + a x^2 + b x + c = 0$ are $\alpha, \beta$ and $\gamma.$ Find the cubic whose roots are $\alpha^3, \beta^3, \gamma^3.$
Suppose that $x$, $y$, $z$ are real numbers satisfying \[x+y+z=12,\qquad\text{and}\qquad x^2+y^2+z^2=54.\] Prove that:[list](a) Each of the numbers $xy$, $yz$, $zx$ is at least $9$, but at most $25$.
(b) One of the numbers $x$, $y$, $z$ is at most $3$, and another one is at least $5$.[/list]