Difference between revisions of "2015 USAMO Problems"

Revision as of 12:10, 12 May 2015

Day 1

Problem 1

Solve in integers the equation $x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3.$

Problem 2

Quadrilateral $APBQ$ is inscribed in circle $\omega$ with $\angle P = \angle Q = 90^{\circ}$ and $AP = AQ < BP$ . Let $X$ be a variable point on segment $\overline{PQ}$ . Line $AX$ meets $\omega$ again at $S$ (other than $A$ ). Point $T$ lies on arc $AQB$ of $\omega$ such that $\overline{XT}$ is perpendicular to $\overline{AX}$ . Let $M$ denote the midpoint of chord $\overline{ST}$ . As $X$ varies on segment $\overline{PQ}$ , show that $M$ moves along a circle.

Solution

Problem 3

Day 2

Problem 4

Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\leq i, j, k, l\leq n$ , such that $i<j$ and $k<l$ . A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively,j or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively.

Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.

How many different non-equivalent ways can Steve pile the stones on the grid?

Solution

@@ Line 14: / Line 14: @@
 ==Day 2==
 ===Problem 4===
-Find all functions <math>f:\mathbb{Q}\rightarrow\mathbb{Q}</math> such that<cmath>f(x)+f(t)=f(y)+f(z)</cmath>for all rational numbers <math>x<y<z<t</math> that form an arithmetic progression. (<math>\mathbb{Q}</math> is the set of all rational numbers.)
-[[2015 USAMO Problems/Problem 4|Solution]]
-===Problem 5===
-Let <math>ABCD</math> be a cyclic quadrilateral. Prove that there exists a point <math>X</math> on segment <math>\overline{BD}</math> such that <math>\angle BAC=\angle XAD</math> and <math>\angle BCA=\angle XCD</math> if and only if there exists a point <math>Y</math> on segment <math>\overline{AC}</math> such that <math>\angle CBD=\angle YBA</math> and <math>\angle CDB=\angle YDA</math>.
-[[2015 USAMO Problems/Problem 5|Solution]]
-===Problem 6===
 Steve is piling <math>m\geq 1</math> indistinguishable stones on the squares of an <math>n\times n</math> grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions <math>(i, k), (i, l), (j, k), (j, l)</math> for some <math>1\leq i, j, k, l\leq n</math>, such that <math>i<j</math> and <math>k<l</math>. A stone move consists of either removing one stone from each of <math>(i, k)</math> and <math>(j, l)</math> and moving them to <math>(i, l)</math> and <math>(j, k)</math> respectively,j or removing one stone from each of <math>(i, l)</math> and <math>(j, k)</math> and moving them to <math>(i, k)</math> and <math>(j, l)</math> respectively.
@@ Line 31: / Line 23: @@
 How many different non-equivalent ways can Steve pile the stones on the grid?
-[[2015 USAMO Problems/Problem 6|Solution]]
+[[2015 USAMO Problems/Problem 4|Solution]]
+===Problem 5===
+===Problem 6===

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