Found problems: 85335
Nair and Yuli play the following game:
$1.$ There is a coin to be moved along a horizontal array with $203$ cells.
$2.$ At the beginning, the coin is at the first cell, counting from left to right.
$3.$ Nair plays first.
$4.$ Each of the players, in their turns, can move the coin $1$, $2$, or $3$ cells to the right.
$5.$ The winner is the one who reaches the last cell first.
What strategy does Nair need to use in order to always win the game?
Two non-zero real numbers, $ a$ and $ b,$ satisfy $ ab \equal{} a \minus{} b$. Which of the following is a possible value of $ \frac {a}{b} \plus{} \frac {b}{a} \minus{} ab$?
$ \textbf{(A)}\minus{}\!2 \qquad \textbf{(B)}\minus{}\!\frac {1}{2} \qquad \textbf{(C)}\ \frac {1}{3} \qquad \textbf{(D)}\ \frac {1}{2} \qquad \textbf{(E)}\ 2$
For which positive integer n can you color the numbers 1,2...2n with n colors, such that every color is used twice and the numbers 1,2,3...n occur as difference of two numbers of the same color exatly once.
Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? (L-trominos are made up of three unit squares sharing a corner, as shown below.)
[asy]
draw((0,0) -- (2,0) -- (2,1) -- (0,1));
draw((0,0) -- (0,2) -- (1,2) -- (1,0));
draw((4,1) -- (6,1) -- (6,2) -- (4,2));
draw((4,2) -- (4,0) -- (5,0) -- (5,2));
draw((10,0) -- (8,0) -- (8,1) -- (10,1));
draw((9,0) -- (9,2) -- (10,2) -- (10,0));
draw((14,1) -- (12,1) -- (12,2) -- (14,2));
draw((13,2) -- (13,0) -- (14,0) -- (14,2));
[/asy]
[i]Proposed by Andrew Lin.[/i]
Amandine and Brennon play a turn-based game, with Amadine starting.
On their turn, a player must select a positive integer which cannot be represented as a sum of multiples of any of the previously selected numbers.
For example, if $3, 5$ have been selected so far, only $1, 2, 4, 7$ are available to be picked;
if only $3$ has been selected so far, all numbers not divisible by three are eligible.
A player loses immediately if they select the integer $1$.
Call a number $n$ [i]feminist[/i] if $\gcd(n, 6) = 1$ and if Amandine wins if she starts with $n$. Compute the sum of the [i]feminist[/i] numbers less than $40$.
[i]Proposed by Ashwin Sah[/i]
Find a three-digit number with the property that the number represented by these digits and in the same order, but with a numbering base different than $ 10 $, is twice as large as the given number.
Let $a,b$ be coprime integers. Show that the equation $ax^2 + by^2 =z^3$ has an infinite set of solutions $(x,y,z)$ with $\{x,y,z\}\in\mathbb{Z}$ and each pair of $x,y$ mutually coprime.
Let $ABC$ be an acute triangle and let $X$ be a variable point on $AC$. The incircle of $\triangle ABX$ touches $AX, BX$ at $K, P$, respectively. The incircle of $\triangle BCX$ touches $CX, BX$ at $L, Q$, respectively. Find the locus of $KP \cap LQ$.
Let $A, B, C$ and $D$ be four different points lying on a common circle in this order. Assume that the line segment $AB$ is the (only) longest side of the inscribed quadrilateral $ABCD$. Prove that the inequality $AB + BD > AC + CD$ holds.
[i](Proposed by Karl Czakler)[/i]
In triangle $ABC$, the angle $\angle BCA$ is obtuse and $\angle BAC = 2\angle ABC\,.$ The line through $B$ and perpendicular to $BC$ intersects line $AC$ in $D$. Let $M$ be the midpoint of $AB$. Prove that $\angle AMC=\angle BMD$.
source : http://cage.ugent.be/~hvernaev/Olympiade/PMO982.pdf
Show that the triangle whose angles satisfy the equality
\[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\]
is right angled
Suppose $f: \mathbb{R}^{+} \mapsto \mathbb{R}^{+}$ is a function such that $\frac{f(x)}{x}$ is increasing on $\mathbb{R}^{+}$. For $a,b,c>0$, prove that $$2\left (\frac{f(a)+f(b)}{a+b} + \frac{f(b)+f(c)}{b+c}+ \frac{f(c)+f(a)}{c+a} \right) \geq 3\left(\frac{f(a)+f(b)+f(c)}{a+b+c}\right) + \frac{f(a)}{a}+ \frac{f(b)}{b}+ \frac{f(c)}{c}$$
Prove that for all n=3,4,5.... there excist odd x,y such $2^n=x^2 + 7y^2$ .
If a dealer could get his goods for $8\%$ less while keeping his selling price fixed, his profit, based on cost, would be increased to $(x+10)\%$ from his present profit of $x\%$, which is
$\textbf{(A) }12\%\qquad\textbf{(B) }15\%\qquad\textbf{(C) }30\%\qquad\textbf{(D) }50\%\qquad \textbf{(E) }75\%$
Let $S = \{10^n + 1000: n = 0, 1, \dots\}$. Compute the largest positive integer not expressible as the sum of (not necessarily distinct) elements of $S$.
[i]Proposed by Ankan Bhattacharya[/i]
Determine all real solutions to the following equation: \[2^{(2^x)}-3\cdot2^{(2^{x-1}+1)}+8=0.\]
Let $I$ be an incenter of $\triangle ABC$. Denote $D, \ S \neq A$ intersections of $AI$ with $BC, \ O(ABC)$ respectively. Let $K, \ L$ be incenters of $\triangle DSB, \ \triangle DCS$. Let $P$ be a reflection of $I$ with the respect to $KL$. Prove that $BP \perp CP$.
How can you go from the number 11 to 25 by only multiplying with 2 or decreasing with 3 in a minimum number of steps?
A fixed circle $k$ and collinear points $E,F$ and $G$ are given such that the points $E$ and $G$ lie outside the circle $k$ and $F$ lies inside the circle $k$. Prove that, if $ABCD$ is an arbitrary quadrilateral inscribed in the circle $k$ such that the points $E,F$ and $G$ lie on lines $AB,AD$ and $DC$ respectively, then the side $BC$ passes through a fixed point collinear with $E,F$ and $G$, independent of the quadrilateral $ABCD$.
Let $A, B, C$ be real numbers in the interval $\left(0,\frac{\pi}{2}\right)$. Let \begin{align*} X &= \frac{\sin A\sin (A-B)\sin (A-C)}{\sin (B+C)} \\ Y &= \frac{\sin B\sin(B-C)\sin (B-A)}{\sin (C+A)} \\ Z &= \frac{\sin C\sin (C-A)\sin (C-B)}{\sin (A+B)} . \end{align*} Prove that $X+Y+Z \geq 0$.
Let $a>0,b>0,c>0$ and $\sqrt{1987+a}+\sqrt{1987+b}=2\sqrt{1987+c}$. Prove that $\frac{1}{2} (a+b )\ge c $.
(a) Triangles $A_1B_1C_1$ and $A_2B_2C_2$ are inscribed into triangle $ABC$ so that $C_1A_1 \perp BC$, $A_1B_1 \perp CA$, $B_1C_1 \perp AB$, $B_2A_2 \perp BC$, $C_2B_2 \perp CA$, $A_2C_2 \perp AB$. Prove that these triangles are equal.
(b) Points $A_1$, $B_1$, $C_1$, $A_2$, $B_2$, $C_2$ lie inside a triangle $ABC$ so that $A_1$ is on segment $AB_1$, $B_1$ is on segment $BC_1$, $C_1$ is on segment $CA_1$, $A_2$ is on segment $AC_2$, $B_2$ is on segment $BA_2$, $C_2$ is on segment $CB_2$, and the angles $BAA_1$, $CBB_2$, $ACC_1$, $CAA_2$, $ABB_2$, $BCC_2$ are equal. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal.
For what positive integers $n$ is it possible to tile an equilateral triangle of side $n$ with trapezoids each of which has sides $1, 1, 1, 2$?
(NB Vassiliev)
Alice and Bob play a game on a Cartesian Coordinate Plane. At the beginning, Alice chooses a lattice point $ \left(x_{0}, y_{0}\right) $ and places a pudding. Then they plays by turns (B goes first) according to the rules
a. If $ A $ places a pudding on $ \left(x,y\right) $ in the last round, then $ B $ can only place a pudding on one of $ \left(x+2, y+1\right), \left(x+2, y-1\right), \left(x-2, y+1\right), \left(x-2, y-1\right) $
b. If $ B $ places a pudding on $ \left(x,y\right) $ in the last round, then $ A $ can only place a pudding on one of $ \left(x+1, y+2\right), \left(x+1, y-2\right), \left(x-1, y+2\right), \left(x-1, y-2\right) $
Furthermore, if there is already a pudding on $ \left(a,b\right) $, then no one can place a pudding on $ \left(c,d\right) $ where $ c \equiv a \pmod{n}, d \equiv b \pmod{n} $.
1. Who has a winning strategy when $ n = 2018 $
1. Who has a winning strategy when $ n = 2019 $
In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.