Navvye/Olympiad-Problems-CSV huggingface dataset:
2009 USA Team Selection Test
Let $ ABP, BCQ, CAR$ be three non-overlapping triangles erected outside of acute triangle $ ABC$. Let $ M$ be the midpoint of segment $ AP$. Given that $ \angle PAB = \angle CQB = 45^\circ$, $ \angle ABP = \angle QBC = 75^\circ$, $ \angle RAC = 105^\circ$, and $ RQ^2 = 6CM^2$, compute $ AC^2/AR^2$.



Zuming Feng.
2009 USA Team Selection Test
Find all pairs of positive integers $ (m,n)$ such that $ mn - 1$ divides $ (n^2 - n + 1)^2$.



Aaron Pixton.
2009 USA Team Selection Test
Let $ N > M > 1$ be fixed integers. There are $ N$ people playing in a chess tournament; each pair of players plays each other once, with no draws. It turns out that for each sequence of $ M + 1$ distinct players $ P_0, P_1, \ldots P_M$ such that $ P_{i - 1}$ beat $ P_i$ for each $ i = 1, \ldots, M$, player $ P_0$ also beat $ P_M$. Prove that the players can be numbered $ 1,2, \ldots, N$ in such a way that, whenever $ a \geq b + M - 1$, player $ a$ beat player $ b$.



Gabriel Carroll.
2009 USA Team Selection Test
Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations

$$ \begin{cases}x^3 = 3x-12y+50, \\ y^3 = 12y+3z-2, \\ z^3 = 27z + 27x. \end{cases}$$



Razvan Gelca.
2009 USA Team Selection Test
Fix a prime number $ p > 5$. Let $ a,b,c$ be integers no two of which have their difference divisible by $ p$. Let $ i,j,k$ be nonnegative integers such that $ i + j + k$ is divisible by $ p - 1$. Suppose that for all integers $ x$, the quantity

$$ (x - a)(x - b)(x - c)[(x - a)^i(x - b)^j(x - c)^k - 1]$$

is divisible by $ p$. Prove that each of $ i,j,k$ must be divisible by $ p - 1$.



Kiran Kedlaya and Peter Shor.
2009 USA Team Selection Test
Prove that for positive real numbers $x$, $y$, $z$, $$ x^3(y^2+z^2)^2 + y^3(z^2+x^2)^2+z^3(x^2+y^2)^2 \geq xyz\left[xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\right].$$ Zarathustra (Zeb) Brady.
2008 USA Team Selection Test
There is a set of $ n$ coins with distinct integer weights $ w_1, w_2, \ldots , w_n$. It is known that if any coin with weight $ w_k$, where $ 1 \leq k \leq n$, is removed from the set, the remaining coins can be split into two groups of the same weight. (The number of coins in the two groups can be different.) Find all $ n$ for which such a set of coins exists.
2008 USA Team Selection Test
Let $ P$, $ Q$, and $ R$ be the points on sides $ BC$, $ CA$, and $ AB$ of an acute triangle $ ABC$ such that triangle $ PQR$ is equilateral and has minimal area among all such equilateral triangles. Prove that the perpendiculars from $ A$ to line $ QR$, from $ B$ to line $ RP$, and from $ C$ to line $ PQ$ are concurrent.
2008 USA Team Selection Test
For a pair $ A = (x_1, y_1)$ and $ B = (x_2, y_2)$ of points on the coordinate plane, let $ d(A,B) = |x_1 - x_2| + |y_1 - y_2|$. We call a pair $ (A,B)$ of (unordered) points harmonic if $ 1 < d(A,B) \leq 2$. Determine the maximum number of harmonic pairs among 100 points in the plane.
2008 USA Team Selection Test
Two sequences of integers, $ a_1, a_2, a_3, \ldots$ and $ b_1, b_2, b_3, \ldots$, satisfy the equation

$$ (a_n - a_{n - 1})(a_n - a_{n - 2}) + (b_n - b_{n - 1})(b_n - b_{n - 2}) = 0

$$

for each integer $ n$ greater than $ 2$. Prove that there is a positive integer $ k$ such that $ a_k = a_{k + 2008}$.
2008 USA Team Selection Test
Determine the smallest positive real number $ k$ with the following property. Let $ ABCD$ be a convex quadrilateral, and let points $ A_1$, $ B_1$, $ C_1$, and $ D_1$ lie on sides $ AB$, $ BC$, $ CD$, and $ DA$, respectively. Consider the areas of triangles $ AA_1D_1$, $ BB_1A_1$, $ CC_1B_1$ and $ DD_1C_1$; let $ S$ be the sum of the two smallest ones, and let $ S_1$ be the area of quadrilateral $ A_1B_1C_1D_1$. Then we always have $ kS_1\ge S$.



Author: Zuming Feng and Oleg Golberg, USA
2008 USA Team Selection Test
Let $ ABC$ be a triangle with $ G$ as its centroid. Let $ P$ be a variable point on segment $ BC$. Points $ Q$ and $ R$ lie on sides $ AC$ and $ AB$ respectively, such that $ PQ \parallel AB$ and $ PR \parallel AC$. Prove that, as $ P$ varies along segment $ BC$, the circumcircle of triangle $ AQR$ passes through a fixed point $ X$ such that $ \angle BAG = \angle CAX$.
2008 USA Team Selection Test
Mr. Fat and Ms. Taf play a game. Mr. Fat chooses a sequence of positive integers $ k_1, k_2, \ldots , k_n$. Ms. Taf must guess this sequence of integers. She is allowed to give Mr. Fat a red card and a blue card, each with an integer written on it. Mr. Fat replaces the number on the red card with $ k_1$ times the number on the red card plus the number on the blue card, and replaces the number on the blue card with the number originally on the red card. He repeats this process with number $ k_2$. (That is, he replaces the number on the red card with $ k_2$ times the number now on the red card plus the number now on the blue card, and replaces the number on the blue card with the number that was just placed on the red card.) He then repeats this process with each of the numbers $ k_3, \ldots k_n$, in this order. After has has gone through the sequence of integers, Mr. Fat then gives the cards back to Ms. Taf. How many times must Ms. Taf submit the red and blue cards in order to be able to determine the sequence of integers $ k_1, k_2, \ldots k_n$?
2008 USA Team Selection Test
Let $ n$ be a positive integer. Given an integer coefficient polynomial $ f(x)$, define its signature modulo $ n$ to be the (ordered) sequence $ f(1), \ldots , f(n)$ modulo $ n$. Of the $ n^n$ such $ n$-term sequences of integers modulo $ n$, how many are the signature of some polynomial $ f(x)$ if

a) $ n$ is a positive integer not divisible by the square of a prime.

b) $ n$ is a positive integer not divisible by the cube of a prime.
2007 USA Team Selection Test
Circles $ \omega_1$ and $ \omega_2$ meet at $ P$ and $ Q$. Segments $ AC$ and $ BD$ are chords of $ \omega_1$ and $ \omega_2$ respectively, such that segment $ AB$ and ray $ CD$ meet at $ P$. Ray $ BD$ and segment $ AC$ meet at $ X$. Point $ Y$ lies on $ \omega_1$ such that $ PY \parallel BD$. Point $ Z$ lies on $ \omega_2$ such that $ PZ \parallel AC$. Prove that points $ Q,X,Y,Z$ are collinear.
2007 USA Team Selection Test
Let $n$ be a positive integer and let $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$ be two nondecreasing sequences of real numbers such that

$$ a_1 + \dots + a_i \le b_1 + \dots + b_i  \text{ for every } i = 1, \dots, n $$

and

$$ a_1 + \dots + a_n = b_1 + \dots + b_n. $$

Suppose that for every real number $m$, the number of pairs $(i,j)$ with $a_i-a_j=m$ equals the numbers of pairs $(k,\ell)$ with $b_k-b_\ell = m$. Prove that $a_i = b_i$ for $i=1,\dots,n$.
2007 USA Team Selection Test
Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos k \theta$ and $ \cos[(k + 1)\theta ]$ are both rational for some positive integer $ k$, show that $ \theta = \pi/6$.
2007 USA Team Selection Test
Determine whether or not there exist positive integers $ a$ and $ b$ such that $ a$ does not divide $ b^n - n$ for all positive integers $ n$.
2007 USA Team Selection Test
Triangle $ ABC$ is inscribed in circle $ \omega$. The tangent lines to $ \omega$ at $ B$ and $ C$ meet at $ T$. Point $ S$ lies on ray $ BC$ such that $ AS \perp AT$. Points $ B_1$ and $ C_1$ lie on ray $ ST$ (with $ C_1$ in between $ B_1$ and $ S$) such that $ B_1T = BT = C_1T$. Prove that triangles $ ABC$ and $ AB_1C_1$ are similar to each other.
2007 USA Team Selection Test
For a polynomial $ P(x)$ with integer coefficients, $ r(2i - 1)$ (for $ i = 1,2,3,\ldots,512$) is the remainder obtained when $ P(2i - 1)$ is divided by $ 1024$. The sequence

$$ (r(1),r(3),\ldots,r(1023))

$$

is called the remainder sequence of $ P(x)$. A remainder sequence is called complete if it is a permutation of $ (1,3,5,\ldots,1023)$. Prove that there are no more than $ 2^{35}$ different complete remainder sequences.
2006 USA Team Selection Test
A communications network consisting of some terminals is called a $3$-connector if among any three terminals, some two of them can directly communicate with each other. A communications network contains a windmill with $n$ blades if there exist $n$ pairs of terminals $\{x_{1},y_{1}\},\{x_{2},y_{2}\},\ldots,\{x_{n},y_{n}\}$ such that each $x_{i}$ can directly communicate with the corresponding $y_{i}$ and there is a hub terminal that can directly communicate with each of the $2n$ terminals $x_{1}, y_{1},\ldots,x_{n}, y_{n}$ . Determine the minimum value of $f (n)$, in terms of $n$, such that a $3$ -connector with $f (n)$ terminals always contains a windmill with $n$ blades.
2006 USA Team Selection Test
In acute triangle $ABC$ , segments $AD; BE$ , and $CF$ are its altitudes, and $H$ is its orthocenter. Circle $\omega$, centered at $O$, passes through $A$ and $H$ and intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. The circumcircle of triangle $OPQ$ is tangent to segment $BC$ at $R$. Prove that $\frac{CR}{BR}=\frac{ED}{FD}.$
2006 USA Team Selection Test
Find the least real number $k$ with the following property: if the real numbers $x$, $y$, and $z$ are not all positive, then $$k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\geq (xyz)^{2}-xyz+1.$$
2006 USA Team Selection Test
Let $n$ be a positive integer. Find, with proof, the least positive integer $d_{n}$ which cannot be expressed in the form $$\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}},$$

where $a_{i}$ and $b_{i}$ are nonnegative integers for each $i.$
2006 USA Team Selection Test
Let $n$ be a given integer with $n$ greater than $7$ , and let $\mathcal{P}$ be a convex polygon with $n$ sides. 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. A triangle in the triangulation of $\mathcal{P}$ is an interior triangle if all of its sides are diagonals of $\mathcal{P}$. Express, in terms of $n$, the number of triangulations of $\mathcal{P}$ with exactly two interior triangles, in closed form.
2006 USA Team Selection Test
Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of triangle $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle{BAP}= \angle{CAQ}$. Segments $BQ$ and $CP$ meet at $R$. Let $O$ be the circumcenter of triangle $BCR$. Prove that $AO \perp PQ.$
2005 USA Team Selection Test
Let $n$ be an integer greater than $1$. For a positive integer $m$, let $S_{m}= \{ 1,2,\ldots, mn\}$. Suppose that there exists a $2n$-element set $T$ such that

(a) each element of $T$ is an $m$-element subset of $S_{m}$;

(b) each pair of elements of $T$ shares at most one common element;

and

(c) each element of $S_{m}$ is contained in exactly two elements of $T$.



Determine the maximum possible value of $m$ in terms of $n$.
2005 USA Team Selection Test
Let $A_{1}A_{2}A_{3}$ be an acute triangle, and let $O$ and $H$ be its circumcenter and orthocenter, respectively. For $1\leq i \leq 3$, points $P_{i}$ and $Q_{i}$ lie on lines $OA_{i}$ and $A_{i+1}A_{i+2}$ (where $A_{i+3}=A_{i}$), respectively, such that $OP_{i}HQ_{i}$ is a parallelogram. Prove that

$$\frac{OQ_{1}}{OP_{1}}+\frac{OQ_{2}}{OP_{2}}+\frac{OQ_{3}}{OP_{3}}\geq 3.$$
2005 USA Team Selection Test
We choose random a unitary polynomial of degree $n$ and coefficients in the set $1,2,...,n!$. Prove that the probability for this polynomial to be special is between $0.71$ and $0.75$, where a polynomial $g$ is called special if for every $k>1$ in the sequence $f(1), f(2), f(3),...$ there are infinitely many numbers relatively prime with $k$.
2005 USA Team Selection Test
Consider the polynomials $$f(x) =\sum_{k=1}^{n}a_{k}x^{k}\quad\text{and}\quad g(x) =\sum_{k=1}^{n}\frac{a_{k}}{2^{k}-1}x^{k},$$where $a_{1},a_{2},\ldots,a_{n}$ are real numbers and $n$ is a positive integer. Show that if 1 and $2^{n+1}$ are zeros of $g$ then $f$ has a positive zero less than $2^{n}$.
2005 USA Team Selection Test
Find all finite sets $S$ of points in the plane with the following property: for any three distinct points $A,B,$ and $C$ in $S,$ there is a fourth point $D$ in $S$ such that  $A,B,C,$ and $D$ are the vertices of a parallelogram (in some order).
2005 USA Team Selection Test
Let $ABC$ be an acute scalene triangle with $O$ as its circumcenter. Point $P$ lies inside triangle $ABC$ with $\angle PAB = \angle PBC$ and $\angle PAC = \angle PCB$. Point $Q$ lies on line $BC$ with $QA = QP$. Prove that $\angle AQP = 2\angle OQB$.
2004 USA Team Selection Test
Suppose $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ are real numbers such that $$ (a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 -1)(b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 - 1) > (a_1 b_1 + a_2 b_2 + \cdots + a_n b_n - 1)^2. $$ Prove that $a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 > 1$ and $b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 > 1$.
2004 USA Team Selection Test
Assume $n$ is a positive integer. Considers sequences $a_0, a_1, \ldots, a_n$ for which $a_i \in \{1, 2, \ldots , n\}$ for all $i$ and $a_n = a_0$.



(a) Suppose $n$ is odd. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i \pmod{n}$ for all $i = 1, 2, \ldots, n$.



(b) Suppose $n$ is an odd prime. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i, 2i \pmod{n}$ for all $i = 1, 2, \ldots, n$.
2004 USA Team Selection Test
Draw a $2004 \times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array?
2004 USA Team Selection Test
Let $ABC$ be a triangle. Choose a point $D$ in its interior. Let $\omega_1$ be a circle passing through $B$ and $D$ and $\omega_2$ be a circle passing through $C$ and $D$ so that the other point of intersection of the two circles lies on $AD$. Let $\omega_1$ and $\omega_2$ intersect side $BC$ at $E$ and $F$, respectively. Denote by $X$ the intersection of $DF$, $AB$ and $Y$ the intersection of $DE, AC$. Show that $XY \parallel BC$.
2004 USA Team Selection Test
Let $A = (0, 0, 0)$ in 3D space. Define the weight of a point as the sum of the absolute values of the coordinates. Call a point a primitive lattice point if all of its coordinates are integers whose gcd is 1. Let square $ABCD$ be an unbalanced primitive integer square if it has integer side length and also, $B$ and $D$ are primitive lattice points with different weights. Prove that there are infinitely many unbalanced primitive integer squares such that the planes containing the squares are not parallel to each other.
2004 USA Team Selection Test
Define the function $f: \mathbb N \cup \{0\} \to \mathbb{Q}$ as follows: $f(0) = 0$ and $$ f(3n+k) = -\frac{3f(n)}{2} + k ,  $$ for $k = 0, 1, 2$. Show that $f$ is one-to-one and determine the range of $f$.
2003 USA Team Selection Test
For a pair of integers $a$ and $b$, with $0 < a < b < 1000$, set $S\subseteq \{ 1, 2, \dots , 2003\}$ is called a skipping set for $(a, b)$ if for any pair of elements $s_1, s_2 \in S$, $|s_1 - s_2|\not\in \{ a, b\}$. Let $f(a, b)$ be the maximum size of a skipping set for $(a, b)$. Determine the maximum and minimum values of $f$.
2003 USA Team Selection Test
Let $ABC$ be a triangle and let $P$ be a point in its interior. Lines $PA$, $PB$, $PC$ intersect sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Prove that

$$ [PAF]+[PBD]+[PCE]=\frac{1}{2}[ABC]  $$

if and only if $P$ lies on at least one of the medians of triangle $ABC$. (Here $[XYZ]$ denotes the area of triangle $XYZ$.)
2003 USA Team Selection Test
Find all ordered triples of primes $(p, q, r)$ such that $$ p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1.  $$ Reid Barton
2003 USA Team Selection Test
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$ f(m+n)f(m-n) = f(m^2)  $$ for $m,n \in \mathbb{N}$.
2003 USA Team Selection Test
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 $X+Y+Z \geq 0$.
2003 USA Team Selection Test
Let $\overline{AH_1}, \overline{BH_2}$, and $\overline{CH_3}$ be the altitudes of an acute scalene triangle $ABC$. The incircle of triangle $ABC$ is tangent to $\overline{BC}, \overline{CA},$ and $\overline{AB}$ at $T_1, T_2,$ and $T_3$, respectively. For $k = 1, 2, 3$, let $P_i$ be the point on line $H_iH_{i+1}$ (where $H_4 = H_1$) such that $H_iT_iP_i$ is an acute isosceles triangle with $H_iT_i = H_iP_i$. Prove that the circumcircles of triangles $T_1P_1T_2$, $T_2P_2T_3$, $T_3P_3T_1$ pass through a common point.
2002 USA Team Selection Test
Let $ ABC$ be a triangle, and $ A$, $ B$, $ C$ its angles. Prove that

$$ \sin\frac{3A}{2}+\sin\frac{3B}{2}+\sin\frac{3C}{2}\leq \cos\frac{A-B}{2}+\cos\frac{B-C}{2}+\cos\frac{C-A}{2}. $$
2002 USA Team Selection Test
Let $p>5$ be a prime number. For any integer $x$, define

$${f_p}(x) = \sum_{k=1}^{p-1} \frac{1}{(px+k)^2}$$

Prove that for any pair of positive integers $x$, $y$, the numerator of $f_p(x) - f_p(y)$, when written as a fraction in lowest terms, is  divisible by $p^3$.
2002 USA Team Selection Test
Let $n$ be an integer greater than 2, and $P_1, P_2, \cdots , P_n$ distinct points in the plane. Let $\mathcal S$ denote the union of all segments $P_1P_2, P_2P_3, \dots , P_{n-1}P_{n}$. Determine if it is always possible to find points $A$ and $B$ in $\mathcal S$ such that $P_1P_n \parallel AB$ (segment $AB$ can lie on line $P_1P_n$) and $P_1P_n = kAB$, where (1) $k = 2.5$; (2) $k = 3$.
2002 USA Team Selection Test
Let $n$ be a positive integer and let $S$ be a set of $2^n+1$ elements. Let $f$ be a function from the set of two-element subsets of $S$ to $\{0, \dots, 2^{n-1}-1\}$. Assume that for any elements $x, y, z$ of $S$, one of $f(\{x,y\}), f(\{y,z\}), f(\{z, x\})$ is equal to the sum of the other two. Show that there exist $a, b, c$ in $S$ such that $f(\{a,b\}), f(\{b,c\}), f(\{c,a\})$ are all equal to 0.
2002 USA Team Selection Test
Consider the family of  nonisosceles triangles $ABC$ satisfying the property $AC^2 + BC^2 = 2 AB^2$. Points $M$ and $D$ lie on side $AB$ such that $AM = BM$ and $\angle ACD = \angle BCD$. Point $E$ is in the plane such that $D$ is the incenter of triangle $CEM$. Prove that exactly one of the ratios

$$ \frac{CE}{EM}, \quad \frac{EM}{MC}, \quad \frac{MC}{CE}  $$

is constant.
2002 USA Team Selection Test
Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$.
2001 USA Team Selection Test
Let $\{ a_n\}_{n \ge 0}$ be a sequence of real numbers such that $a_{n+1} \ge a_n^2 + \frac{1}{5}$ for all $n \ge 0$. Prove that $\sqrt{a_{n+5}} \ge a_{n-5}^2$ for all $n \ge 5$.
2001 USA Team Selection Test
Express  $$ \sum_{k=0}^n (-1)^k (n-k)!(n+k)!  $$ in closed form.
2001 USA Team Selection Test
For a set $S$, let $|S|$ denote the number of elements in $S$. Let $A$ be a set of positive integers with $|A| = 2001$. Prove that there exists a set $B$ such that



(i) $B \subseteq A$;

(ii) $|B| \ge 668$;

(iii) for any $u, v \in B$ (not necessarily distinct), $u+v \not\in B$.
2001 USA Team Selection Test
There are 51 senators in a senate. The senate needs to be divided into $n$ committees so that each senator is on one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does not necessarily hate senator A.) Find the smallest $n$ such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.
2001 USA Team Selection Test
In triangle $ABC$, $\angle B = 2\angle C$. Let $P$ and $Q$ be points on the perpendicular bisector of segment $BC$ such that rays $AP$ and $AQ$ trisect $\angle A$. Prove that $PQ < AB$ if and only if $\angle B$ is obtuse.
2001 USA Team Selection Test
Let $a,b,c$ be positive real numbers such that $$ a+b+c\geq abc. $$ Prove that at least two of the inequalities $$ \frac{2}{a}+\frac{3}{b}+\frac{6}{c}\geq6,\;\;\;\;\;\frac{2}{b}+\frac{3}{c}+\frac{6}{a}\geq6,\;\;\;\;\;\frac{2}{c}+\frac{3}{a}+\frac{6}{b}\geq6 $$ are true.
2001 USA Team Selection Test
Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle ADC = 135^{\circ}$ and $$AC^2\cdot BD^2 = 2\cdot AB\cdot BC\cdot CD\cdot DA.$$Prove that the diagonals of the quadrilateral $ABCD$ are perpendicular.
2001 USA Team Selection Test
Find all pairs of nonnegative integers $(m,n)$ such that $$(m+n-5)^2=9mn.$$
2001 USA Team Selection Test
Let $A$ be a finite set of positive integers.  Prove that there exists a finite set $B$ of positive integers such that $A \subseteq B$ and



$$\prod_{x\in B} x = \sum_{x\in B} x^2.$$
2000 USA Team Selection Test
Let $a, b, c$ be nonnegative real numbers.  Prove that

$$ \frac{a+b+c}{3} - \sqrt[3]{abc} \leq \max\{(\sqrt{a} - \sqrt{b})^2, (\sqrt{b} - \sqrt{c})^2, (\sqrt{c} - \sqrt{a})^2\}.  $$
2000 USA Team Selection Test
Let $ ABCD$ be a cyclic quadrilateral and let $ E$ and $ F$ be the feet of perpendiculars from the intersection of diagonals $ AC$ and $ BD$ to $ AB$ and $ CD$, respectively. Prove that $ EF$ is perpendicular to the line through the midpoints of $ AD$ and $ BC$.
2000 USA Team Selection Test
Let $p$ be a prime number. For integers $r, s$ such that $rs(r^2 - s^2)$ is not divisible by $p$, let $f(r, s)$ denote the number of integers $n \in \{1, 2, \ldots, p - 1\}$ such that $\{rn/p\}$ and $\{sn/p\}$ are either both less than $1/2$ or both greater than $1/2$. Prove that there exists $N > 0$ such that for $p \geq N$ and all $r, s$, $$ \left\lceil \frac{p-1}{3} \right\rceil \le f(r, s) \le \left\lfloor \frac{2(p-1)}{3} \right\rfloor.  $$
2000 USA Team Selection Test
Let $n$ be a positive integer.  Prove that

$$ \binom{n}{0}^{-1} + \binom{n}{1}^{-1} + \cdots + \binom{n}{n}^{-1} = \frac{n+1}{2^{n+1}} \left( \frac{2}{1} + \frac{2^2}{2} + \cdots + \frac{2^{n+1}}{n+1} \right).  $$
2000 USA Team Selection Test
Let $n$ be a positive integer. A $corner$ is a finite set $S$ of ordered $n$-tuples of positive integers such that if $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ are positive integers with $a_k \geq b_k$ for $k = 1, 2, \ldots, n$ and $(a_1, a_2, \ldots, a_n) \in S$, then $(b_1, b_2, \ldots, b_n) \in S$. Prove that among any infinite collection of corners, there exist two corners, one of which is a subset of the other one.
2000 USA Team Selection Test
Let $ ABC$ be a triangle inscribed in a circle of radius $ R$, and let $ P$ be a point in the interior of triangle $ ABC$. Prove that

$$ \frac {PA}{BC^{2}} + \frac {PB}{CA^{2}} + \frac {PC}{AB^{2}}\ge \frac {1}{R}.

$$

Alternative formulation: If $ ABC$ is a triangle with sidelengths $ BC=a$, $ CA=b$, $ AB=c$ and circumradius $ R$, and $ P$ is a point inside the triangle $ ABC$, then prove that



$ \frac {PA}{a^{2}} + \frac {PB}{b^{2}} + \frac {PC}{c^{2}}\ge \frac {1}{R}$.
2021 Vietnam TST
Define the sequence $(a_n)$ as $a_1 = 1$, $a_{2n} = a_n$ and $a_{2n+1} = a_n + 1$ for all $n\geq 1$.



a) Find all positive integers $n$ such that $a_{kn} = a_n$ for all integers $1 \leq k \leq n$.

b) Prove that there exist infinitely many positive integers $m$ such that $a_{km} \geq a_m$ for all positive integers $k$.
2021 Vietnam TST
In a board of $2021 \times 2021$ grids, we pick $k$ unit squares such that every picked square shares vertice(s) with at most $1$ other picked square. Determine the maximum of $k$.
2021 Vietnam TST
Let $ABC$ be a triangle and $N$ be a point that differs from $A,B,C$. Let $A_b$ be the reflection of $A$ through $NB$, and $B_a$ be the reflection of $B$ through $NA$. Similarly, we define $B_c, C_b, A_c, C_a$. Let $m_a$ be the line through $N$ and perpendicular to $B_cC_b$. Define similarly $m_b, m_c$.



a) Assume that $N$ is the orthocenter of $\triangle ABC$, show that the respective reflection of $m_a, m_b, m_c$ through the bisector of angles $\angle BNC, \angle CNA, \angle ANB$ are the same line.

b) Assume that $N$ is the nine-point center of $\triangle ABC$, show that the respective reflection of $m_a, m_b, m_c$ through $BC, CA, AB$ concur.
2021 Vietnam TST
Let $a,b,c$ are non-negative numbers such that

$$2(a^2+b^2+c^2)+3(ab+bc+ca)=5(a+b+c)$$then prove that $4(a^2+b^2+c^2)+2(ab+bc+ca)+7abc\le 25$