Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 USAMO
Let $P$ be a given point inside quadrilateral $ABCD$.  Points $Q_1$ and $Q_2$ are located within $ABCD$ such that

$$\angle Q_1BC=\angle ABP,\quad\angle Q_1CB=\angle DCP,\quad\angle Q_2AD=\angle BAP,\quad\angle Q_2DA=\angle CDP.$$ Prove that $\overline{Q_1Q_2}\parallel\overline{AB}$ if and only if $\overline{Q_1Q_2}\parallel\overline{CD}$.
2011 USAMO
Let $A$ be a set with $|A|=225$, meaning that $A$ has 225 elements.  Suppose further that there are eleven subsets $A_1, \ldots, A_{11}$ of $A$ such that $|A_i|=45$ for $1\leq i\leq11$ and $|A_i\cap A_j|=9$ for $1\leq i<j\leq11$.  Prove that $|A_1\cup A_2\cup\ldots\cup A_{11}|\geq 165$, and give an example for which equality holds.
2010 USAMO
Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by

$P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively.  Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.
2010 USAMO
There are $n$ students standing in a circle, one behind the other. The students have heights $h_1<h_2<\dots <h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or less, the two students are permitted to switch places. Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.
2010 USAMO
The 2010 positive numbers $a_1, a_2, \ldots , a_{2010}$ satisfy the inequality $a_ia_j \le i+j$ for all distinct indices $i, j$. Determine, with proof, the largest possible value of the product $a_1a_2\ldots a_{2010}$.
2010 USAMO
Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.
2010 USAMO
Let $q = \frac{3p-5}{2}$ where $p$ is an odd prime, and let$$

S_q = \frac{1}{2\cdot 3 \cdot 4} + \frac{1}{5\cdot 6 \cdot 7} + \cdots + \frac{1}{q(q+1)(q+2)}

$$Prove that if $\frac{1}{p}-2S_q = \frac{m}{n}$ for integers $m$ and $n$, then $m - n$ is divisible by $p$.
2010 USAMO
A blackboard contains 68 pairs of nonzero integers.  Suppose that for each positive integer $k$ at most one of the pairs $(k, k)$ and $(-k, -k)$ is written on the blackboard.  A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0.  The student then scores one point for each of the 68 pairs in which at least one integer is erased.  Determine, with proof, the largest number $N$ of points that the student can guarantee to score regardless of which 68 pairs have been written on the board.
2009 USAMO
Given circles $ \omega_1$ and $ \omega_2$ intersecting at points $ X$ and $ Y$, let $ \ell_1$ be a line through the center of $ \omega_1$ intersecting $ \omega_2$ at points $ P$ and $ Q$ and let $ \ell_2$ be a line through the center of $ \omega_2$ intersecting $ \omega_1$ at points $ R$ and $ S$.  Prove that if $ P, Q, R$ and $ S$ lie on a circle then the center of this circle lies on line $ XY$.
2009 USAMO
Let $n$ be a positive integer.  Determine the size of the largest subset of $\{ -n, -n+1, \dots, n-1, n\}$ which does not contain three elements $a$, $b$, $c$ (not necessarily distinct) satisfying $a+b+c=0$.
2009 USAMO
We define a chessboard polygon to be a polygon whose sides are situated along lines of the form $ x = a$ or $ y = b$, where $ a$ and $ b$ are integers. These lines divide the interior into unit squares, which are shaded alternately grey and white so that adjacent squares have different colors. To tile a chessboard polygon by dominoes is to exactly cover the polygon by non-overlapping $ 1 \times 2$ rectangles. Finally, a tasteful tiling is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a $ 3 \times 4$ rectangle are shown; the first one is tasteful, while the second is not, due to the vertical dominoes in the upper right corner.



[asy]size(300); pathpen = linewidth(2.5);

void chessboard(int a, int b, pair P){ 

 for(int i = 0; i < a; ++i) for(int j = 0; j < b; ++j) 

  if((i+j) % 2 == 1) fill(shift(P.x+i,P.y+j)*unitsquare,rgb(0.6,0.6,0.6)); 

 D(P--P+(a,0)--P+(a,b)--P+(0,b)--cycle);

}

chessboard(2,2,(2.5,0));fill(unitsquare,rgb(0.6,0.6,0.6));fill(shift(1,1)*unitsquare,rgb(0.6,0.6,0.6)); chessboard(4,3,(6,0)); chessboard(4,3,(11,0)); MP("\mathrm{Distasteful\ tilings}",(2.25,3),fontsize(12)); 



/* draw lines */

D((0,0)--(2,0)--(2,2)--(0,2)--cycle); D((1,0)--(1,2)); D((2.5,1)--(4.5,1)); D((7,0)--(7,2)--(6,2)--(10,2)--(9,2)--(9,0)--(9,1)--(7,1)); D((8,2)--(8,3)); D((12,0)--(12,2)--(11,2)--(13,2)); D((13,1)--(15,1)--(14,1)--(14,3)); D((13,0)--(13,3));[/asy] a) Prove that if a chessboard polygon can be tiled by dominoes, then it can be done so tastefully.



b) Prove that such a tasteful tiling is unique.
2009 USAMO
For $ n\geq2$ let $ a_1, a_2, \ldots a_n$ be positive real numbers such that

$$ (a_1 + a_2 + \cdots + a_n)\left(\frac {1}{a_1} + \frac {1}{a_2} + \cdots + \frac {1}{a_n}\right) \leq \left(n + \frac {1}{2}\right)^2.

$$

Prove that $ \max(a_1, a_2, \ldots, a_n)\leq 4\min(a_1, a_2, \ldots, a_n)$.
2009 USAMO
Trapezoid $ ABCD$, with $ \overline{AB}||\overline{CD}$, is inscribed in circle $ \omega$ and point $ G$ lies inside triangle $ BCD$.  Rays $ AG$ and $ BG$ meet $ \omega$ again at points $ P$ and $ Q$, respectively.  Let the line through $ G$ parallel to $ \overline{AB}$ intersects $ \overline{BD}$ and $ \overline{BC}$ at points $ R$ and $ S$, respectively.  Prove that quadrilateral $ PQRS$ is cyclic if and only if $ \overline{BG}$ bisects $ \angle CBD$.
2009 USAMO
Let $s_1, s_2, s_3, \dots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \dots.$  Suppose that $t_1, t_2, t_3, \dots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$.  Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.
2008 USAMO
Prove that for each positive integer $ n$, there are pairwise relatively prime integers $ k_0,k_1,\ldots,k_n$, all strictly greater than $ 1$, such that $ k_0k_1\ldots k_n-1$ is the product of two consecutive integers.
2008 USAMO
Let $ ABC$ be an acute, scalene triangle, and let $ M$, $ N$, and $ P$ be the midpoints of $ \overline{BC}$, $ \overline{CA}$, and $ \overline{AB}$, respectively. Let the perpendicular bisectors of $ \overline{AB}$ and $ \overline{AC}$ intersect ray $ AM$ in points $ D$ and $ E$ respectively, and let lines $ BD$ and $ CE$ intersect in point $ F$, inside of triangle $ ABC$. Prove that points $ A$, $ N$, $ F$, and $ P$ all lie on one circle.
2008 USAMO
Let $n$ be a positive integer. Denote by $S_n$ the set of points $(x, y)$ with integer coordinates such that $$ \left\lvert x\right\rvert + \left\lvert y + \frac{1}{2} \right\rvert < n. $$ A path is a sequence of distinct points $(x_1 , y_1), (x_2, y_2), \ldots, (x_\ell, y_\ell)$ in $S_n$ such that, for $i = 2, \ldots, \ell$, the distance between $(x_i , y_i)$ and $(x_{i-1} , y_{i-1} )$ is $1$ (in other words, the points $(x_i, y_i)$ and $(x_{i-1} , y_{i-1} )$ are neighbors in the lattice of points with integer coordinates). Prove that the points in $S_n$ cannot be partitioned into fewer than $n$ paths (a partition of $S_n$ into $m$ paths is a set $\mathcal{P}$ of $m$ nonempty paths such that each point in $S_n$ appears in exactly one of the $m$ paths in $\mathcal{P}$).
2008 USAMO
Let $ \mathcal{P}$ be a convex polygon with $ n$ sides, $ n\ge3$. Any set of $ n - 3$ diagonals of $ \mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $ \mathcal{P}$ into $ n - 2$ triangles. If $ \mathcal{P}$ is regular and there is a triangulation of $ \mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $ n$.
2008 USAMO
Three nonnegative real numbers $ r_1$, $ r_2$, $ r_3$ are written on a blackboard. These numbers have the property that there exist integers $ a_1$, $ a_2$, $ a_3$, not all zero, satisfying $ a_1r_1 + a_2r_2 + a_3r_3 = 0$. We are permitted to perform the following operation: find two numbers $ x$, $ y$ on the blackboard with $ x \le y$, then erase $ y$ and write $ y - x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $ 0$ on the blackboard.
2008 USAMO
At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $ 2^k$ for some positive integer $ k$).
2007 USAMO
Let $n$ be a positive integer. Define a sequence by setting $a_{1}= n$ and, for each $k > 1$, letting $a_{k}$ be the unique integer in the range $0\leq a_{k}\leq k-1$ for which $a_{1}+a_{2}+...+a_{k}$ is divisible by $k$. For instance, when $n = 9$ the obtained sequence is $9,1,2,0,3,3,3,...$. Prove that for any $n$ the sequence $a_{1},a_{2},...$ eventually becomes constant.
2007 USAMO
A square grid on the Euclidean plane consists of all points $(m,n)$, where $m$ and $n$ are integers. Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family has radius at least $5$?
2007 USAMO
Let $S$ be a set containing $n^{2}+n-1$ elements, for some positive integer $n$. Suppose that the $n$-element subsets of $S$ are partitioned into two classes. Prove that there are at least $n$ pairwise disjoint sets in the same class.
2007 USAMO
An animal with $n$ cells is a connected figure consisting of $n$ equal-sized cells[1].



A dinosaur is an animal with at least $2007$ cells. It is said to be primitive it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.



(1) Animals are also called polyominoes. They can be defined inductively. Two cells are adjacent if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
2007 USAMO
Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.
2007 USAMO
Let $ABC$ be an acute triangle with $\omega,S$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_{A}$ is tangent internally to $S$ at $A$ and tangent externally to $\omega$. Circle $S_{A}$ is tangent internally to $S$ at $A$ and tangent internally to $\omega$. Let $P_{A}$ and $Q_{A}$ denote the centers of $\omega_{A}$ and $S_{A}$, respectively. Define points $P_{B}, Q_{B}, P_{C}, Q_{C}$ analogously. Prove that

$$8P_{A}Q_{A}\cdot P_{B}Q_{B}\cdot P_{C}Q_{C}\leq R^{3}\; , $$

with equality if and only if triangle $ABC$ is equilateral.
2006 USAMO
Let $p$ be a prime number and let $s$ be an integer with $0 < s < p.$ Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and



$$ \left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p}  $$



if and only if $s$ is not a divisor of $p-1$.



Note: For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of x.
2006 USAMO
For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k + 1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $\tfrac{N}{2}.$
2006 USAMO
For integral $m$, let $p(m)$ be the greatest prime divisor of $m.$ By convention, we set $p(\pm 1) = 1$ and $p(0) = \infty.$ Find all polynomials $f$ with integer coefficients such that the sequence

$$ \{p \left( f \left( n^2 \right) \right) - 2n \}_{n \geq 0}  $$ is bounded above. (In particular, this requires $f \left (n^2 \right ) \neq 0$ for $n \geq 0.$)
2006 USAMO
Find all positive integers $n$ such that there are $k \geq 2$ positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying $a_1 + a_2 + \ldots + a_k = a_1 \cdot a_2 \cdots a_k = n.$
2006 USAMO
A mathematical frog jumps along the number line. The frog starts at $1$, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n + 2^{m_n+1}$ where $2^{m_n}$ is the largest power of $2$ that is a factor of $n.$ Show that if $k \geq 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^ik$ is greater than the minimum number of jumps needed to reach $2^i.$
2006 USAMO
Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$, respectively, such that $\frac{AE}{ED} = \frac{BF}{FC}$. Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$, respectively. Prove that the circumcircles of triangles $SAE$, $SBF$, $TCF$, and $TDE$ pass through a common point.
2005 USAMO
Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.
2005 USAMO
Prove that the system \begin{align*}

x^6+x^3+x^3y+y & = 147^{157} \\
x^3+x^3y+y^2+y+z^9 & = 157^{147}

\end{align*} has no solutions in integers $x$, $y$, and $z$.
2005 USAMO
Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on its side $BC$. Construct a point $C_{1}$ in such a way that the convex quadrilateral $APBC_{1}$ is cyclic, $QC_{1}\parallel CA$, and $C_{1}$ and $Q$ lie on opposite sides of line $AB$. Construct a point $B_{1}$ in such a way that the convex quadrilateral $APCB_{1}$ is cyclic, $QB_{1}\parallel BA$, and $B_{1}$ and $Q$ lie on opposite sides of line $AC$.  Prove that the points $B_{1}$, $C_{1}$, $P$, and $Q$ lie on a circle.
2005 USAMO
Legs $L_1, L_2, L_3, L_4$ of a square table each have  length $n$, where $n$ is a positive integer.  For how many ordered  4-tuples $(k_1, k_2, k_3, k_4)$ of nonnegative integers can we cut  a piece of length $k_i$ from the end of leg $L_i \; (i=1,2,3,4)$ and still have a stable table?



(The table is stable if it can be placed so that  all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)
2005 USAMO
Let $n$ be an integer greater than 1. Suppose $2n$ points are given in the plane, no three of which are collinear. Suppose $n$ of the given $2n$ points are colored blue and the other $n$ colored red. A line in the plane is called a balancing line if it passes through one blue and one red point and, for each side of the line, the number of blue points on that side is equal to the number of red points on the same side.

Prove that there exist at least two balancing lines.
2005 USAMO
For $m$ a positive integer, let $s(m)$ be the sum of the digits of $m$. For $n\ge 2$, let $f(n)$ be the minimal $k$ for which there  exists a set $S$ of $n$ positive integers such that  $s\left(\sum_{x\in X} x\right)=k$ for any nonempty subset $X\subset S$.  Prove that there are constants $0<C_1<C_2$ with

$$C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.$$
2004 USAMO
Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that

$$ 

\frac{1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|. 

$$

When does equality hold?
2004 USAMO
Suppose $a_1, \dots, a_n$ are integers whose greatest common divisor is 1. Let $S$ be a set of integers with the following properties:



(a) For $i=1, \dots, n$, $a_i \in S$.

(b) For $i,j = 1, \dots, n$ (not necessarily distinct), $a_i - a_j \in S$.

(c) For any integers $x,y \in S$, if $x+y \in S$, then $x-y \in S$.



Prove that $S$ must be equal to the set of all integers.
2004 USAMO
For what real values of $k>0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but noncongruent, polygons?
2004 USAMO
Alice and Bob play a game on a 6 by 6 grid.  On his or her turn, a player chooses a rational number not yet appearing in the grid and writes it in an empty square of the grid.  Alice goes first and then the players alternate.  When all squares have numbers written in them, in each row, the square with the greatest number in that row is colored black.  Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can't.  (If two squares share a vertex, Alice can draw a line from one to the other that stays in those two squares.)  Find, with proof, a winning strategy for one of the players.
2004 USAMO
Let $a, b, c > 0$. Prove that $(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \geq (a + b + c)^3$.
2004 USAMO
A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\omega$. Suppose that $$

(AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2.

$$ Prove that $ABCD$ is an isosceles trapezoid.
2003 USAMO
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
2003 USAMO
A convex polygon $\mathcal{P}$ in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon $\mathcal{P}$ are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.
2003 USAMO
Let $n \neq 0$. For every sequence of integers $$ A = a_0,a_1,a_2,\dots, a_n $$ satisfying $0 \le a_i \le i$, for $i=0,\dots,n$, define another sequence $$ t(A)= t(a_0), t(a_1), t(a_2), \dots, t(a_n) $$ by setting $t(a_i)$ to be the number of terms in the sequence $A$ that precede the term $a_i$ and are different from $a_i$. Show that, starting from any sequence $A$ as above, fewer than $n$ applications of the  transformation $t$ lead to a sequence $B$ such that $t(B) = B$.
2003 USAMO
Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$.
2003 USAMO
Let $ a$, $ b$, $ c$ be positive real numbers. Prove that

$$ \dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \le 8.

$$
2003 USAMO
At the vertices of a regular hexagon are written six nonnegative integers whose sum is $2003^{2003}$. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.
2002 USAMO
Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \leq N \leq 2^{2002}$. Prove that it is possible to color every subset of $S$ either black or white so that the following conditions hold:



(a) the union of any two white subsets is white;

(b) the union of any two black subsets is black;

(c) there are exactly $N$ white subsets.
2002 USAMO
Let $ABC$ be a triangle such that

$$ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2,  $$

where $s$ and $r$ denote its semiperimeter and its inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisors and determine these integers.
2002 USAMO
Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots.
2002 USAMO
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$ f(x^2 - y^2) = x f(x) - y f(y)  $$ for all pairs of real numbers $x$ and $y$.
2002 USAMO
Let $a,b$ be integers greater than 2. Prove that there exists a positive integer $k$ and a finite sequence $n_1, n_2, \dots, n_k$ of positive integers such that $n_1 = a$, $n_k = b$, and $n_i n_{i+1}$ is divisible by $n_i + n_{i+1}$ for each $i$ ($1 \leq i < k$).
2002 USAMO
I have an $n \times n$ sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let $b(n)$ be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants $c$ and $d$ such that $$ \dfrac{1}{7} n^2 - cn \leq b(n) \leq \dfrac{1}{5} n^2 + dn  $$ for all $n > 0$.
2001 USAMO
Each of eight boxes contains six balls. Each ball has been colored with one of $n$ colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justification, the smallest integer $n$ for which this is possible.
2001 USAMO
Let $ABC$ be a triangle and let $\omega$ be its incircle. Denote by $D_1$ and $E_1$ the points where $\omega$ is tangent to sides $BC$ and $AC$, respectively. Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$, respectively, such that $CD_2=BD_1$ and $CE_2=AE_1$, and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$. Circle $\omega$ intersects segment $AD_2$ at two points, the closer of which to the vertex $A$ is denoted by $Q$. Prove that $AQ=D_2P$.
2001 USAMO
Let $a, b, c \geq 0$ and satisfy  $$ a^2+b^2+c^2 +abc = 4 . $$ Show that $$ 0 \le ab + bc + ca - abc \leq 2. $$
2001 USAMO
Let $P$ be a point in the plane of triangle $ABC$ such that the segments $PA$, $PB$, and $PC$ are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to $PA$. Prove that $\angle BAC$ is acute.
2001 USAMO
Let $S$ be a set of integers (not necessarily positive) such that



(a) there exist $a,b \in S$ with $\gcd(a,b)=\gcd(a-2,b-2)=1$;

(b) if $x$ and $y$ are elements of $S$ (possibly equal), then $x^2-y$ also belongs to $S$.



Prove that $S$ is the set of all integers.
2001 USAMO
Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.
2000 USAMO
Call a real-valued function $ f$ very convex if

$$ \frac {f(x) + f(y)}{2} \ge f\left(\frac {x + y}{2}\right) + |x - y|

$$

holds for all real numbers $ x$ and $ y$. Prove that no very convex function exists.
2000 USAMO
Let $S$ be the set of all triangles $ABC$ for which $$ 5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r},  $$ where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. Prove that all triangles in $S$ are isosceles  and similar to one another.
2000 USAMO
A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.