id stringlengths 12 12 | source stringlengths 76 76 | problem stringlengths 59 2.04k | solutions listlengths 0 11 |
|---|---|---|---|
USAMO-2009-3 | https://artofproblemsolving.com/wiki/index.php/2009_USAMO_Problems/Problem_3 | 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 ... | [] |
USAMO-2009-4 | https://artofproblemsolving.com/wiki/index.php/2009_USAMO_Problems/Problem_4 | For \(n \ge 2\) let \(a_1\), \(a_2\), ..., \(a_n\) be positive real numbers such that
\[
(a_1+a_2+ ... +a_n)\left( {1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n} \right) \le \left(n+ {1 \over 2} \right) ^2
\]
Prove that \(\text{max}(a_1, a_2, ... ,a_n) \le 4 \text{min}(a_1, a_2, ... , a_n)\). | [
"Assume without loss of generality that \\(a_1 \\geq a_2 \\geq \\cdots \\geq a_n\\). Now we seek to prove that \\(a_1 \\le 4a_n\\).\n\nBy the Cauchy-Schwarz Inequality,\n\n\\[\n\\begin{align*} (a_n+a_2+ a_3 + ... +a_{n-1}+a_1)\\left({1 \\over a_1} + {1 \\over a_2} + ... +{1 \\over a_n}\\right) &\\ge \\left( \\sqrt{... |
USAMO-2009-5 | https://artofproblemsolving.com/wiki/index.php/2009_USAMO_Problems/Problem_5 | 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 \(\overline{AB}\) intersect \(\overline{BD}\) and \(\ove... | [
"We will use directed angles in this solution. Extend \\(QR\\) to \\(T\\) as follows:\n\n\\[\n[asy] import cse5; import graph; import olympiad; dotfactor = 3; unitsize(1.5inch); path circle = Circle(origin, 1); draw(circle); pair A = (-.6, .8), B = (.6, .8), C = (.9, -sqrt(.19)), D = (-.9, -sqrt(.19)), G = bisect... |
USAMO-2009-6 | https://artofproblemsolving.com/wiki/index.php/2009_USAMO_Problems/Problem_6 | Let \(s_1, s_2, s_3, \ldots\) be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that \(s_1 = s_2 = s_3 = \cdots.\) Suppose that \(t_1, t_2, t_3, \ldots\) 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 a... | [
"Suppose the \\(s_i\\) can be represented as \\(\\frac{a_i}{b_i}\\) for every \\(i\\), and suppose \\(t_i\\) can be represented as \\(\\frac{c_i}{d_i}\\). Let's start with only the first two terms in the two sequences, \\(s_1\\) and \\(s_2\\) for sequence \\(s\\) and \\(t_1\\) and \\(t_2\\) for sequence \\(t\\). Th... |
USAMO-2010-1 | https://artofproblemsolving.com/wiki/index.php/2010_USAMO_Problems/Problem_1 | 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... | [
"Angles are directed mod 180 {don't worry}(take so that the solution can be applied to any configuration) Let the intersection of pq and rs is M.\n\n\\(m\\angle ARM=m\\angle ZRS =m \\angle ZYS =m\\angle YZR = m\\angle YZA = m \\angle YXP = m \\angle XYQ =m \\angle XPQ = m \\angle APM\\) This implies APRM is cyclic ... |
USAMO-2010-2 | https://artofproblemsolving.com/wiki/index.php/2010_USAMO_Problems/Problem_2 | There are \(n\) students standing in a circle, one behind the other. The students have heights \(h_1 < h_2 < \ldots < h_n\). 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... | [
"We adopt the usual convention that \\(\\binom{i}{j} = 0\\) unless \\(0 \\le j \\le i\\). With this, the binomial coefficients are defined for all integers via the recursion:\n\n\\[\n\\binom{0}{0} = 1, \\quad \\binom{n}{k} = \\binom{n-1}{k} + \\binom{n-1}{k-1}\n\\]\n\nIt is clear that the circle is oriented and all... |
USAMO-2010-3 | https://artofproblemsolving.com/wiki/index.php/2010_USAMO_Problems/Problem_3 | 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 \(a_1a_2\cdots a_{2010}\). | [
"The largest possible value is\n\n\\[\n\\prod_{i=1}^{1005}(4i-1) = 3\\times 7 \\times \\ldots \\times 4019.\n\\]\n\n## Proof\n\nNo larger value is possible, since for each consecutive pair of elements: \\((a_{2i-1},a_{2i}), 1\\le i \\le 1005\\), the product is at most \\((2i-1) + 2i = 4i - 1\\), and so the product ... |
USAMO-2010-4 | https://artofproblemsolving.com/wiki/index.php/2010_USAMO_Problems/Problem_4 | 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 \(AB, AC, BI, ID, CI, IE\) ... | [
"We know that angle \\(BIC = 135^{\\circ}\\), as the other two angles in triangle \\(BIC\\) add to \\(45^{\\circ}\\). Assume that only \\(AB, AC, BI\\), and \\(CI\\) are integers. Using the Law of Cosines on triangle BIC,\n\n\\[\n[asy] import olympiad; // Scale unitsize(1inch); // Shape real h = 1.75; real w = 2.... |
USAMO-2010-5 | https://artofproblemsolving.com/wiki/index.php/2010_USAMO_Problems/Problem_5 | Let \(q = \dfrac{3p-5}{2}\) where \(p\) is an odd prime, and let
\[
S_q = \frac{1}{2\cdot 3 \cdot 4} + \frac{1}{5\cdot 6 \cdot 7} + \cdots + \frac{1}{q\cdot (q+1) \cdot (q+2)}.
\]
Prove that if \(\dfrac{1}{p}-2S_q = \dfrac{m}{n}\) for integers \(m\) and \(n\), then \(m-n\) is divisible by \(p\). | [
"Since \\(p\\) is an odd prime, \\(p = 2r + 1\\), for a suitable positive integer \\(r\\), and consequently \\(q = 3r - 1\\).\n\nThe partial-fraction decomposition of the general term of \\(S_q\\) is:\n\n\\[\n\\begin{align*} \\frac{1}{(3k-1)3k(3k+1)} &= \\frac{1}{2}\\left(\\frac{1}{3k-1} - \\frac{2}{3k} + \\frac{1}... |
USAMO-2010-6 | https://artofproblemsolving.com/wiki/index.php/2010_USAMO_Problems/Problem_6 | 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 po... | [
"Answer: 43\n\nAttainability: Consider 8 distinct positive numbers. Let there be 5 pairs for each of the numbers including 2 clones of that number. Let there also be 28 pairs that include the negatives of those numbers such that each negative associates with another negative once and exactly once (in graph theoreti... |
USAMO-2011-1 | https://artofproblemsolving.com/wiki/index.php/2011_USAMO_Problems/Problem_1 | Let \(a\), \(b\), \(c\) be positive real numbers such that \(a^2 + b^2 + c^2 + (a + b + c)^2 \le 4\). Prove that
\[
\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.
\] | [
"Since\n\n\\[\n\\begin{align*} (a+b)^2 + (b+c)^2 + (c+a)^2 &= 2(a^2 + b^2 + c^2 + ab + bc + ca) \\\\ \t&= a^2 + b^2 + c^2 + (a + b + c)^2, \\end{align*}\n\\]\n\nit is natural to consider a change of variables:\n\n\\[\n\\begin{align*} \\alpha &= b + c \\\\ \\beta &= c + a \\\\ \\gamma &= a + b \\end{align*}\n\\]\n\n... |
USAMO-2011-2 | https://artofproblemsolving.com/wiki/index.php/2011_USAMO_Problems/Problem_2 | An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer \(m\) from each of the integers at two neighboring vertices and adding 2m to the opposite vertex, which is not adjacent to either of the first two verti... | [
"Let \\(\\mathbb{F}_5\\) be the field of positive residues modulo 5. We label the vertices of the pentagon clockwise with the residues 0, 1, 2, 3 and 4. For each \\(i \\in \\mathbb{F}_5\\) let \\(n_i\\) be the integer at vertex \\(i\\) and let \\(r_i \\in \\mathbb{F}_5\\) be defined as:\n\n\\[\nr_i \\equiv 4n_{i+... |
USAMO-2011-3 | https://artofproblemsolving.com/wiki/index.php/2011_USAMO_Problems/Problem_3 | In hexagon \(ABCDEF\), which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy \(\angle A = 3\angle D\), \(\angle C = 3\angle F\), and \(\angle E = 3\angle B\). Furthermore \(AB=DE\), \(BC=EF\), and \(CD=FA\). Prove that diagonals \(\overline{AD}\), \(\overline{... | [
"Let \\(\\angle D = \\alpha\\), \\(\\angle F = \\gamma\\), and \\(\\angle B = \\beta\\), \\(AB=DE=p\\), \\(BC=EF=q\\), \\(CD=FA=r\\). Define the vectors:\n\n\\[\n\\vec{u} = \\vec{AB} + \\vec{DE}\n\\]\n\n\\[\n\\vec{v} = \\vec{BC} + \\vec{EF}\n\\]\n\n\\[\n\\vec{w} = \\vec{CD} + \\vec{FA}\n\\]\n\nClearly, \\(\\vec{u}+... |
USAMO-2011-4 | https://artofproblemsolving.com/wiki/index.php/2011_USAMO_Problems/Problem_4 | Consider the assertion that for each positive integer \(n \ge 2\), the remainder upon dividing \(2^{2^n}\) by \(2^n-1\) is a power of 4. Either prove the assertion or find (with proof) a counter-example. | [
"We will show that \\(n = 25\\) is a counter-example.\n\nSince \\(\\textstyle 2^n \\equiv 1 \\pmod{2^n - 1}\\), we see that for any integer \\(k\\), \\(\\textstyle 2^{2^n} \\equiv 2^{(2^n - kn)} \\pmod{2^n-1}\\). Let \\(0 \\le m < n\\) be the residue of \\(2^n \\pmod n\\). Note that since \\(\\textstyle m < n\\) an... |
USAMO-2011-5 | https://artofproblemsolving.com/wiki/index.php/2011_USAMO_Problems/Problem_5 | Let \(P\) be a given point inside quadrilateral \(ABCD\). Points \(Q_1\) and \(Q_2\) are located within \(ABCD\) such that \(\angle Q_1 BC = \angle ABP\), \(\angle Q_1 CB = \angle DCP\), \(\angle Q_2 AD = \angle BAP\), \(\angle Q_2 DA = \angle CDP\). Prove that \(\overline{Q_1 Q_2} \parallel \overline{AB}\) if and only... | [
"Lemma. If \\(AB\\) and \\(CD\\) are not parallel, then \\(AB, CD, Q_1 Q_2\\) are concurrent.\n\nProof. Let \\(AB\\) and \\(CD\\) meet at \\(R\\). Notice that with respect to triangle \\(ADR\\), \\(P\\) and \\(Q_2\\) are isogonal conjugates (this can be proven by dropping altitudes from \\(Q_2\\) to \\(AB\\), \\(CD... |
USAMO-2011-6 | https://artofproblemsolving.com/wiki/index.php/2011_USAMO_Problems/Problem_6 | Let \(A\) be a set with \(|A| = 225\), meaning that \(A\) has 225 elements. Suppose further that there are eleven subsets \(A_1\), \(\dots\), \(A_{11}\) of \(A\) such that \(|A_i | = 45\) for \(1 \le i \le 11\) and \(|A_i \cap A_j| = 9\) for \(1 \le i < j \le 11\). Prove that \(|A_1 \cup A_2 \cup \dots \cup A_{11}| \ge... | [
"## Existence\n\nNote that \\(\\textstyle \\binom{11}3 = 165,\\) and so it is natural to consider placing one element in each intersection of three of the 11 sets. Since each pair of sets is in 9 3-way intersections—one with each of the 9 remaining sets—any two sets will have 9 elements in common. Since \\(\\textst... |
USAMO-2012-1 | https://artofproblemsolving.com/wiki/index.php/2012_USAMO_Problems/Problem_1 | Find all integers \(n \ge 3\) such that among any \(n\) positive real numbers \(a_1\), \(a_2\), \(\dots\), \(a_n\) with
\[
\max(a_1, a_2, \dots, a_n) \le n \cdot \min(a_1, a_2, \dots, a_n),
\]
there exist three that are the side lengths of an acute triangle. | [
"Without loss of generality, assume that the set \\(\\{a\\}\\) is ordered from least to greatest so that the bounding condition becomes \\(a_n \\le n \\cdot a_1.\\) Now set \\(b_i \\equiv \\frac{a_i}{a_1},\\) and since a triangle with sidelengths from \\(\\{a\\}\\) will be similar to the corresponding triangle from... |
USAMO-2012-2 | https://artofproblemsolving.com/wiki/index.php/2012_USAMO_Problems/Problem_2 | A circle is divided into 432 congruent arcs by 432 points. The points are colored in four colors such that some 108 points are colored Red, some 108 points are colored Green, some 108 points are colored Blue, and the remaining 108 points are colored Yellow. Prove that one can choose three points of each color in such a... | [
"If you rotate the red points 431 times, they will overlap with blue points \\(108\\times 108\\) times, for an average of \\(\\frac{108\\times 108}{431}\\) per rotation. Note that this average is slightly greater than 27. Therefore at some point 28 red points overlap with blue points. In other words, there exist 28... |
USAMO-2012-3 | https://artofproblemsolving.com/wiki/index.php/2012_USAMO_Problems/Problem_3 | Determine which integers \(n > 1\) have the property that there exists an infinite sequence \(a_1\), \(a_2\), \(a_3\), \(\dots\) of nonzero integers such that the equality
\[
a_k + 2a_{2k} + \dots + na_{nk} = 0
\]
holds for every positive integer \(k\). | [
"For \\(n\\) equal to any odd prime \\(p\\), the sequence \\(\\left\\{a_i = \\left(\\frac{1-n}{2}\\right)^{m_p\\left(i\\right)}\\right\\}\\), where \\(p^{m_p\\left(i\\right)}\\) is the greatest power of \\(p\\) that divides \\(i\\), gives a valid sequence. Therefore, the set of possible values for \\(n\\) is at lea... |
USAMO-2012-4 | https://artofproblemsolving.com/wiki/index.php/2012_USAMO_Problems/Problem_4 | Find all functions \(f : \mathbb{Z}^+ \to \mathbb{Z}^+\) (where \(\mathbb{Z}^+\) is the set of positive integers) such that \(f(n!) = f(n)!\) for all positive integers \(n\) and such that \(m - n\) divides \(f(m) - f(n)\) for all distinct positive integers \(m\), \(n\). | [
"By the first condition we have \\(f(1)=f(1!)=f(1)!\\) and \\(f(2)=f(2!)=f(2)!\\), so \\(f(1)=1\\) or \\(2\\) and similarly for \\(f(2)\\). By the second condition, we have\n\n\\[\nn\\cdot n!=(n+1)!-n! \\mid f(n+1)!-f(n)! \\qquad \\qquad (1)\n\\]\n\nfor all positive integers \\(n\\).\n\nSuppose that for some \\(n \... |
USAMO-2012-5 | https://artofproblemsolving.com/wiki/index.php/2012_USAMO_Problems/Problem_5 | Let \(P\) be a point in the plane of triangle \(ABC\), and \(\gamma\) a line passing through \(P\). Let \(A'\), \(B'\), \(C'\) be the points where the reflections of lines \(PA\), \(PB\), \(PC\) with respect to \(\gamma\) intersect lines \(BC\), \(AC\), \(AB\), respectively. Prove that \(A'\), \(B'\), \(C'\) are collin... | [
"By the sine law on triangle \\(AB'P\\),\n\n\\[\n\\frac{AB'}{\\sin \\angle APB'} = \\frac{AP}{\\sin \\angle AB'P},\n\\]\n\nso\n\n\\[\nAB' = AP \\cdot \\frac{\\sin \\angle APB'}{\\sin \\angle AB'P}.\n\\]\n\n\\[\n[asy] import graph; import geometry; unitsize(0.5 cm); pair[] A, B, C; pair P, R; A[0] = (2,12); B[0] ... |
USAMO-2012-6 | https://artofproblemsolving.com/wiki/index.php/2012_USAMO_Problems/Problem_6 | For integer \(n \ge 2\), let \(x_1\), \(x_2\), \(\dots\), \(x_n\) be real numbers satisfying
\[
x_1 + x_2 + \dots + x_n = 0, \quad \text{and} \quad x_1^2 + x_2^2 + \dots + x_n^2 = 1.
\]
For each subset \(A \subseteq \{1, 2, \dots, n\}\), define
\[
S_A = \sum_{i \in A} x_i.
\]
(If \(A\) is the empty set, then \(S_A ... | [
"For convenience, let \\(N=\\{1,2,\\dots,n\\}\\).\n\nNote that \\(2\\sum_{1\\leq i<j\\leq n} x_ix_j=\\left(\\sum_{i=1}^{n}x_i\\right)^2-\\left(\\sum_{j=1}^{n} x_i^2\\right)=-1\\), so the sum of the \\(x_i\\) taken two at a time is \\(-1/2\\). Now consider the following sum:\n\n\\[\n\\sum_{A\\subseteq N}S_A^2=2^{n-1... |
USAMO-2013-1 | https://artofproblemsolving.com/wiki/index.php/2013_USAMO_Problems/Problem_1 | In triangle \(ABC\), points \(P,Q,R\) lie on sides \(BC,CA,AB\) respectively. Let \(\omega_A\), \(\omega_B\), \(\omega_C\) denote the circumcircles of triangles \(AQR\), \(BRP\), \(CPQ\), respectively. Given the fact that segment \(AP\) intersects \(\omega_A\), \(\omega_B\), \(\omega_C\) again at \(X,Y,Z\) respectively... | [
"\\[\n[asy] /* DRAGON 0.0.9.6 Homemade Script by v_Enhance. */ import olympiad; import cse5; size(11cm); real lsf=0.8000; real lisf=2011.0; defaultpen(fontsize(10pt)); /* Initialize Objects */ pair A = (-1.0, 3.0); pair B = (-3.0, -3.0); pair C = (4.0, -3.0); pair P = (-0.6698198198198195, -3.0); pair Q = (1.140646... |
USAMO-2013-2 | https://artofproblemsolving.com/wiki/index.php/2013_USAMO_Problems/Problem_2 | For a positive integer \(n\geq 3\) plot \(n\) equally spaced points around a circle. Label one of them \(A\), and place a marker at \(A\). One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of \(2n\) distinct moves available; two from each ... | [
"We label the points in clockwise order as \\(1,2,3,\\ldots,n\\), where point \\(A\\) is the same as point \\(1\\). We start and end at point \\(1\\), and we must cross over it, either by visiting it again, or else by making the move from point \\(n\\) to point \\(2\\). We interpret each of these cases in terms of ... |
USAMO-2013-3 | https://artofproblemsolving.com/wiki/index.php/2013_USAMO_Problems/Problem_3 | Let \(n\) be a positive integer. There are \(\tfrac{n(n+1)}{2}\) marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing \(n\) marks. Initially, each mark has the black side up. An operation is to choose a line parallel to the sides of the triangle, and fli... | [
"Solution by inconsistent\n\nThis problem is headache-inducing... but also fun. Ouch.\n\nLet \\(n = 4k + b\\) where \\(b \\in {1, 2, 3, 4}\\). Then I claim the answer is \\(6k+1, 6k+2, 6k+3, 6k+6\\) in these four cases. First, note that the base cases \\(n = 1, 2, 3\\) are trivial by checking to be \\(1, 2, 3\\). N... |
USAMO-2013-4 | https://artofproblemsolving.com/wiki/index.php/2013_USAMO_Problems/Problem_4 | Find all real numbers \(x,y,z\geq 1\) satisfying
\[
\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.
\] | [
"The key Lemma is:\n\n\\[\n\\sqrt{a-1}+\\sqrt{b-1} \\le \\sqrt{ab}\n\\]\n\nfor all \\(a,b \\ge 1\\). Equality holds when \\((a-1)(b-1)=1\\).\n\nThis is proven easily.\n\n\\[\n\\sqrt{a-1}+\\sqrt{b-1} = \\sqrt{a-1}\\sqrt{1}+\\sqrt{1}\\sqrt{b-1} \\le \\sqrt{(a-1+1)(b-1+1)} = \\sqrt{ab}\n\\]\n\nby Cauchy.\n\nEquality t... |
USAMO-2013-5 | https://artofproblemsolving.com/wiki/index.php/2013_USAMO_Problems/Problem_5 | Given positive integers \(m\) and \(n\), prove that there is a positive integer \(c\) such that the numbers \(cm\) and \(cn\) have the same number of occurrences of each non-zero digit when written in base ten. | [
"This solution is adopted from the official solution. Both the problem and the solution were suggested by Richard Stong.\n\nWithout Loss of Generality, suppose \\(m \\geq n \\geq 1\\). By prime factorization of \\(n\\), we can find a positive integer \\(c_1\\) such that \\(c_1n=10^s n_1\\) where \\(n_1\\) is relati... |
USAMO-2013-6 | https://artofproblemsolving.com/wiki/index.php/2013_USAMO_Problems/Problem_6 | Let \(ABC\) be a triangle. Find all points \(P\) on segment \(BC\) satisfying the following property: If \(X\) and \(Y\) are the intersections of line \(PA\) with the common external tangent lines of the circumcircles of triangles \(PAB\) and \(PAC\), then
\[
\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=... | [
"Let circle \\(PAB\\) (i.e. the circumcircle of \\(PAB\\)), \\(PAC\\) be \\(\\omega_1, \\omega_2\\) with radii \\(r_1\\), \\(r_2\\) and centers \\(O_1, O_2\\), respectively, and \\(d\\) be the distance between their centers.\n\nLemma. \\(XY = \\frac{r_1 + r_2}{d} \\sqrt{d^2 - (r_1 - r_2)^2}.\\)\n\nProof. Let the ex... |
USAMO-2014-1 | https://artofproblemsolving.com/wiki/index.php/2014_USAMO_Problems/Problem_1 | Let \(a,b,c,d\) be real numbers such that \(b-d \ge 5\) and all zeros \(x_1, x_2, x_3,\) and \(x_4\) of the polynomial \(P(x)=x^4+ax^3+bx^2+cx+d\) are real. Find the smallest value the product \((x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)\) can take. | [
"Using the hint we turn the equation into \\(\\prod_{k=1} ^4 (x_k-i)(x_k+i) \\implies P(i)P(-i) \\implies (b-d-1)^2 + (a-c)^2 \\implies \\boxed{16}\\). This minimum is achieved when all the \\(x_i\\) are equal to \\(1\\).\n\nA more detailed version goes as follows:\n\nObserve that \\(x^2+1=x^2-(-1)=x^2-i^2=(x-i)(x+... |
USAMO-2014-2 | https://artofproblemsolving.com/wiki/index.php/2014_USAMO_Problems/Problem_2 | Let \(\mathbb{Z}\) be the set of integers. Find all functions \(f : \mathbb{Z} \rightarrow \mathbb{Z}\) such that
\[
xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))
\]
for all \(x, y \in \mathbb{Z}\) with \(x \neq 0\). | [
"Note: This solution is kind of rough. I didn't want to put my 7-page solution all over again. It would be nice if someone could edit in the details of the expansions.\n\nLemma 1: \\(f(0) = 0\\). Proof: Assume the opposite for a contradiction. Plug in \\(x = 2f(0)\\) (because we assumed that \\(f(0) \\neq 0\\)), \\... |
USAMO-2014-3 | https://artofproblemsolving.com/wiki/index.php/2014_USAMO_Problems/Problem_3 | Prove that there exists an infinite set of points
\[
\ldots,\,\,\,\,P_{-3},\,\,\,\,P_{-2},\,\,\,\,P_{-1},\,\,\,\,P_0,\,\,\,\,P_1,\,\,\,\,P_2,\,\,\,\,P_3,\,\,\,\,\ldots
\]
in the plane with the following property: For any three distinct integers \(a,b,\) and \(c\), points \(P_a\), \(P_b\), and \(P_c\) are collinear if... | [
"Consider an elliptic curve with a generator \\(g\\), such that \\(g\\) is not a root of \\(0\\). By repeatedly adding \\(g\\) to itself under the standard group operation, with can build \\(g, 2g, 3g, \\ldots\\) as well as \\(-g, -2g, -3g, \\ldots\\). If we let\n\n\\[\nP_k = (3k-2014)g\n\\]\n\nthen we can observe ... |
USAMO-2014-4 | https://artofproblemsolving.com/wiki/index.php/2014_USAMO_Problems/Problem_4 | Let \(k\) be a positive integer. Two players \(A\) and \(B\) play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with \(A\) moving first. In his move, \(A\) may choose two adjacent hexagons in the grid which are empty and place a counter i... | [
"We claim that the minimum \\(k\\) such that A cannot create a \\(k\\) in a row is \\(\\boxed{6}\\).\n\nIt is easy to verify that player A can create a 5 in a row.\n\nLet A place two counters anywhere, and B take off one of them. Then, A should create a \"triangle of hexagons\" by placing two adjacent counters also... |
USAMO-2014-5 | https://artofproblemsolving.com/wiki/index.php/2014_USAMO_Problems/Problem_5 | Let \(ABC\) be a triangle with orthocenter \(H\) and let \(P\) be the second intersection of the circumcircle of triangle \(AHC\) with the internal bisector of the angle \(\angle BAC\). Let \(X\) be the circumcenter of triangle \(APB\) and \(Y\) the orthocenter of triangle \(APC\). Prove that the length of segment \(XY... | [
"Let \\(O_1\\) be the center of \\((AHPC)\\), \\(O\\) be the center of \\((ABC)\\). Note that \\((O_1)\\) is the reflection of \\((O)\\) across \\(AC\\), so \\(AO=AO_1\\). Additionally\n\n\\[\n\\angle AYC=180-\\angle APC=180-\\angle AHC=\\angle B\n\\]\n\nso \\(Y\\) lies on \\((O)\\). Now since \\(XO,OO_1,XO_1\\) ar... |
USAMO-2014-6 | https://artofproblemsolving.com/wiki/index.php/2014_USAMO_Problems/Problem_6 | Prove that there is a constant \(c>0\) with the following property: If \(a, b, n\) are positive integers such that \(\gcd(a+i, b+j)>1\) for all \(i, j\in\{0, 1, \ldots n\}\), then
\[
\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.
\] | [
"The following solution is due to Gabriel Dospinescu and v_Enhance (also known as Evan Chen).\n\nLet \\(N = n+1\\) and assume \\(N\\) is (very) large. We construct an \\(N \\times N\\) with cells \\((i,j)\\) where \\(0 \\le i, j \\le n\\) and in each cell place a prime \\(p\\) dividing \\(\\gcd (a+i, b+j)\\).\n\nTh... |
USAMO-2015-1 | https://artofproblemsolving.com/wiki/index.php/2015_USAMO_Problems/Problem_1 | Solve in integers the equation
\[
x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3.
\] | [
"We first notice that both sides must be integers, so \\(\\frac{x+y}{3}\\) must be an integer.\n\nWe can therefore perform the substitution \\(x+y = 3t\\) where \\(t\\) is an integer.\n\nThen:\n\n\\[\n(3t)^2 - xy = (t+1)^3\n\\]\n\n\\[\n9t^2 + x (x - 3t) = t^3 + 3t^2 + 3t + 1\n\\]\n\n\\[\n4x^2 - 12xt + 9t^2 = 4t^3 -... |
USAMO-2015-2 | https://artofproblemsolving.com/wiki/index.php/2015_USAMO_Problems/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 \(\overline{XT}\) i... | [
"We will use coordinate geometry.\n\nWithout loss of generality, let the circle be the unit circle centered at the origin,\n\n\\[\nA=(1,0) P=(1-a,b), Q=(1-a,-b)\n\\]\n\n, where \\((1-a)^2+b^2=1\\).\n\nLet angle \\(\\angle XAB=A\\), which is an acute angle, \\(\\tan{A}=t\\), then \\(X=(1-a,at)\\).\n\nAngle \\(\\angl... |
USAMO-2015-3 | https://artofproblemsolving.com/wiki/index.php/2015_USAMO_Problems/Problem_3 | Let \(S = \{1, 2, ..., n\}\), where \(n \ge 1\). Each of the \(2^n\) subsets of \(S\) is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set \(T \subseteq S\), we then write \(f(T)\) for the number of subsets of T that are blue.
Determine the number of colori... | [
"Define function: \\(C(T)=1\\) if the set T is colored blue, and \\(C(T)=0\\) if \\(T\\) is colored red. Define the \\(\\text{Core} =\\text{intersection of all } T \\text{ where } C(T)=1\\).\n\nThe empty set is denoted as \\(\\varnothing\\), \\(\\cap\\) denotes intersection, and \\(\\cup\\) denotes union. Let \\(S_... |
USAMO-2015-4 | https://artofproblemsolving.com/wiki/index.php/2015_USAMO_Problems/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... | [
"Let the number of stones in row \\(i\\) be \\(r_i\\) and let the number of stones in column \\(i\\) be \\(c_i\\). Since there are \\(m\\) stones, we must have \\(\\sum_{i=1}^n r_i=\\sum_{i=1}^n c_i=m\\)\n\nLemma 1: If any \\(2\\) pilings are equivalent, then \\(r_i\\) and \\(c_i\\) are the same in both pilings \\(... |
USAMO-2015-5 | https://artofproblemsolving.com/wiki/index.php/2015_USAMO_Problems/Problem_5 | Let \(a, b, c, d, e\) be distinct positive integers such that \(a^4 + b^4 = c^4 + d^4 = e^5\). Show that \(ac + bd\) is a composite number. | [
"Note: This solution is definitely not what the folks at MAA intended, but it works!\n\nLook at the statement \\(a^4+b^4=e^5\\). This can be viewed as a special case of Beal's Conjecture (http://en.wikipedia.org/wiki/Beal%27s_conjecture), stating that the equation \\(A^x+B^y=C^z\\) has no solutions over positive in... |
USAMO-2015-6 | https://artofproblemsolving.com/wiki/index.php/2015_USAMO_Problems/Problem_6 | Consider \(0<\lambda<1\), and let \(A\) be a multiset of positive integers. Let \(A_n=\{a\in A: a\leq n\}\). Assume that for every \(n\in\mathbb{N}\), the set \(A_n\) contains at most \(n\lambda\)numbers. Show that there are infinitely many \(n\in\mathbb{N}\) for which the sum of the elements in \(A_n\) is at most \(\f... | [
"Proposed by mengmeng142857.\n\nWe prove this by contradiction. Suppose that the number of times that an integer \\(a\\) appears in \\(A\\) is \\(k_a\\). Let the sum of the elements in \\(A_n\\) be \\(S_n=\\sum_{i=1}^{n}i\\cdot k_i\\). If there are only finitely many \\(n\\in\\mathbb{N}\\) such that \\(S_n\\leq\\fr... |
USAMO-2016-1 | https://artofproblemsolving.com/wiki/index.php/2016_USAMO_Problems/Problem_1 | Let \(X_1, X_2, \ldots, X_{100}\) be a sequence of mutually distinct nonempty subsets of a set \(S\). Any two sets \(X_i\) and \(X_{i+1}\) are disjoint and their union is not the whole set \(S\), that is, \(X_i\cap X_{i+1}=\emptyset\) and \(X_i\cup X_{i+1}\neq S\), for all \(i\in\{1, \ldots, 99\}\). Find the smallest p... | [
"The answer is that \\(|S| \\ge 8\\).\n\nFirst, we provide a inductive construction for \\(S = \\left\\{ 1, \\dots, 8 \\right\\}\\). Actually, for \\(n \\ge 4\\) we will provide a construction for \\(S = \\left\\{ 1, \\dots, n \\right\\}\\) which has \\(2^{n-1} + 1\\) elements in a line. (This is sufficient, since ... |
USAMO-2016-2 | https://artofproblemsolving.com/wiki/index.php/2016_USAMO_Problems/Problem_2 | Prove that for any positive integer \(k,\)
\[
\left(k^2\right)!\cdot\prod_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}
\]
is an integer. | [
"Define \\(v_p(N)\\) for all rational numbers \\(N\\) and primes \\(p\\), where if \\(N=\\frac{x}{y}\\), then \\(v_p(N)=v_p(x)-v_p(y)\\), and \\(v_p(x)\\) is the greatest power of \\(p\\) that divides \\(x\\) for integer \\(x\\). Note that the expression(that we're trying to prove is an integer) is clearly rational... |
USAMO-2016-3 | https://artofproblemsolving.com/wiki/index.php/2016_USAMO_Problems/Problem_3 | Let \(\triangle ABC\) be an acute triangle, and let \(I_B, I_C,\) and \(O\) denote its \(B\)-excenter, \(C\)-excenter, and circumcenter, respectively. Points \(E\) and \(Y\) are selected on \(\overline{AC}\) such that \(\angle ABY = \angle CBY\) and \(\overline{BE}\perp\overline{AC}.\) Similarly, points \(F\) and \(Z\)... | [
"This problem can be proved in the following two steps.\n\n1. Let \\(I_A\\) be the \\(A\\)-excenter, then \\(I_A,O,\\) and \\(P\\) are colinear. This can be proved by the Trigonometric Form of Ceva's Theorem for \\(\\triangle I_AI_BI_C.\\)\n\n2. Show that \\(I_AY^2-I_AZ^2=OY^2-OZ^2,\\) which implies \\(\\overline{O... |
USAMO-2016-4 | https://artofproblemsolving.com/wiki/index.php/2016_USAMO_Problems/Problem_4 | Find all functions \(f:\mathbb{R}\rightarrow \mathbb{R}\) such that for all real numbers \(x\) and \(y\),
\[
(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.
\] | [
"Step 1: Set \\(x = y = 0\\) to obtain \\(f(0) = 0.\\)\n\nStep 2: Set \\(x = 0\\) to obtain \\(f(y)f(-y) = f(y)^2.\\)\n\n\\(\\indent\\) In particular, if \\(f(y) \\ne 0\\) then \\(f(y) = f(-y).\\)\n\n\\(\\indent\\) In addition, replacing \\(y \\to -t\\), it follows that \\(f(t) = 0 \\implies f(-t) = 0\\) for all \\... |
USAMO-2016-5 | https://artofproblemsolving.com/wiki/index.php/2016_USAMO_Problems/Problem_5 | An equilateral pentagon \(AMNPQ\) is inscribed in triangle \(ABC\) such that \(M\in\overline{AB},\) \(Q\in\overline{AC},\) and \(N, P\in\overline{BC}.\) Let \(S\) be the intersection of lines \(MN\) and \(PQ.\) Denote by \(\ell\) the angle bisector of \(\angle MSQ.\)
Prove that \(\overline{OI}\) is parallel to \(\ell,... | [
"Let \\(D\\) be the intersection of line \\(AI\\) and the circumcircle of \\(\\Delta ABC\\) (other than \\(A\\)), then \\(OD\\perp BC\\). Let \\(R\\) be the point such that \\(NPQR\\) is a rhombus. It follows that \\(OD\\perp QR\\).\n\nSince \\(AM=AQ\\), \\(AI\\perp MQ\\), or \\(DI\\perp MQ\\). It follows that \\(\... |
USAMO-2016-6 | https://artofproblemsolving.com/wiki/index.php/2016_USAMO_Problems/Problem_6 | Integers \(n\) and \(k\) are given, with \(n\ge k\ge 2.\) You play the following game against an evil wizard.
The wizard has \(2n\) cards; for each \(i = 1, ..., n,\) there are two cards labeled \(i.\) Initially, the wizard places all cards face down in a row, in unknown order.
You may repeatedly make moves of the fo... | [
"## Case I:\n\nWe first prove that the game is winnable whenever \\(n > k\\) by demonstrating a winning strategy in this case.\n\nOn the \\(i\\)th move, choose the \\(k\\) cards in positions \\(i\\) through \\(i+k-1.\\) Assuming that you do not win on any earlier move, repeat this for \\(1\\le i \\le 2n-k+1.\\)\n\n... |
USAMO-2017-1 | https://artofproblemsolving.com/wiki/index.php/2017_USAMO_Problems/Problem_1 | Prove that there are infinitely many distinct pairs \((a,b)\) of relatively prime positive integers \(a>1\) and \(b>1\) such that \(a^b+b^a\) is divisible by \(a+b\). | [
"Let \\(n=a+b\\). Since \\(gcd(a,b)=1\\), we know \\(gcd(a,n)=1\\). We can rewrite the condition as\n\n\\[\na^{n-a}+(n-a)^a \\equiv 0 \\mod{n}\n\\]\n\n\\[\na^{n-a}\\equiv-(-a)^a \\mod{n}\n\\]\n\nAssume \\(a\\) is odd. Since we need to prove an infinite number of pairs exist, it suffices to show that infinitely many... |
USAMO-2017-2 | https://artofproblemsolving.com/wiki/index.php/2017_USAMO_Problems/Problem_2 | Let \(m_1, m_2, \ldots, m_n\) be a collection of \(n\) positive integers, not necessarily distinct. For any sequence of integers \(A = (a_1, \ldots, a_n)\) and any permutation \(w = w_1, \ldots, w_n\) of \(m_1, \ldots, m_n\), define an \(A\)-inversion of \(w\) to be a pair of entries \(w_i, w_j\) with \(i < j\) for whi... | [] |
USAMO-2017-3 | https://artofproblemsolving.com/wiki/index.php/2017_USAMO_Problems/Problem_3 | Let \(ABC\) be a scalene triangle with circumcircle \(\Omega\) and incenter \(I.\) Ray \(AI\) meets \(BC\) at \(D\) and \(\Omega\) again at \(M;\) the circle with diameter \(DM\) cuts \(\Omega\) again at \(K.\) Lines \(MK\) and \(BC\) meet at \(S,\) and \(N\) is the midpoint of \(IS.\) The circumcircles of \(\triangle ... | [
"Let \\(X\\) be the point on circle \\(\\Omega\\) opposite \\(M\\). This means \\(\\angle MAX = 90^\\circ, BC \\perp XM.\\)\n\n\\(\\angle XKM = \\angle DKM = 90^\\circ \\implies\\) the points \\(X, D,\\) and \\(K\\) are collinear.\n\nLet \\(D' = BC \\cap XM \\implies DD' \\perp XM \\implies\\)\n\n\\(S\\) is the ort... |
USAMO-2017-4 | https://artofproblemsolving.com/wiki/index.php/2017_USAMO_Problems/Problem_4 | Let \(P_1, \ldots, P_{2n}\) be \(2n\) distinct points on the unit circle \(x^2 + y^2 = 1\) other than \((1,0)\). Each point is colored either red or blue, with exactly \(n\) of them red and exactly \(n\) of them blue. Let \(R_1, \ldots, R_n\) be any ordering of the red points. Let \(B_1\) be the nearest blue point to \... | [
"I define a sequence to be, starting at \\((1,0)\\) and tracing the circle counterclockwise, and writing the color of the points in that order - either R or B. For example, possible sequences include \\(RB\\), \\(RBBR\\), \\(BBRRRB\\), \\(BRBRRBBR\\), etc. Note that choosing an \\(R_1\\) is equivalent to choosing a... |
USAMO-2017-5 | https://artofproblemsolving.com/wiki/index.php/2017_USAMO_Problems/Problem_5 | Let \(\mathbf{Z}\) denote the set of all integers. Find all real numbers \(c > 0\) such that there exists a labeling of the lattice points \(( x, y ) \in \mathbf{Z}^2\) with positive integers for which: only finitely many distinct labels occur, and for each label \(i\), the distance between any two points labeled \(i\)... | [
"See page 11 of this PDF: https://web.evanchen.cc/exams/USAMO-2017-notes.pdf",
"For \\(c\\le 1,\\) we can label every lattice point \\(1.\\) For \\(c\\le \\sqrt[4]{2},\\) we can make a \"checkerboard\" labeling, i.e. label \\((x, y)\\) with \\(1\\) if \\(x+y\\) is even and \\(2\\) if \\(x+y\\) is odd. One can eas... |
USAMO-2017-6 | https://artofproblemsolving.com/wiki/index.php/2017_USAMO_Problems/Problem_6 | Find the minimum possible value of
\[
\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4},
\]
given that \(a,b,c,d,\) are nonnegative real numbers such that \(a+b+c+d=4\). | [] |
USAMO-2018-1 | https://artofproblemsolving.com/wiki/index.php/2018_USAMO_Problems/Problem_1 | Let \(a,b,c\) be positive real numbers such that \(a+b+c=4\sqrt[3]{abc}\). Prove that
\[
2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.
\] | [
"WLOG let \\(a \\leq b \\leq c\\). Add \\(2(ab+bc+ca)\\) to both sides of the inequality and factor to get:\n\n\\[\n4(a(a+b+c)+bc) \\geq (a+b+c)^2\n\\]\n\n\\[\n\\frac{4a\\sqrt[3]{abc}+bc}{2} \\geq 2\\sqrt[3]{a^2b^2c^2}\n\\]\n\nThe last inequality is true by AM-GM. Since all these steps are reversible, the proof is ... |
USAMO-2018-2 | https://artofproblemsolving.com/wiki/index.php/2018_USAMO_Problems/Problem_2 | Find all functions \(f:(0,\infty) \to (0,\infty)\) such that
\[
f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{z}\right) + f\left(z+\frac{1}{x}\right) = 1
\]
for all \(x,y,z >0\) with \(xyz =1.\) | [
"Obviously, the output of \\(f\\) lies in the interval \\((0,1)\\). Define \\(g:(0,1)\\to(0,1)\\) as \\(g(x)=f\\left(\\frac1x-1\\right)\\). Then for any \\(a,b,c\\in(0,1)\\) such that \\(a+b+c=1\\), we have \\(g(a)=f\\left(\\frac1a-1\\right)=f\\left(\\frac{1-a}a\\right)=f\\left(\\frac{b+c}a\\right)\\). We can trans... |
USAMO-2018-3 | https://artofproblemsolving.com/wiki/index.php/2018_USAMO_Problems/Problem_3 | For a given integer \(n\ge 2,\) let \(\{a_1,a_2,…,a_m\}\) be the set of positive integers less than \(n\) that are relatively prime to \(n.\) Prove that if every prime that divides \(m\) also divides \(n,\) then \(a_1^k+a_2^k + \dots + a_m^k\) is divisible by \(m\) for every positive integer \(k.\) | [
"https://maa.org/sites/default/files/pdf/AMC/usamo/2018/2018USAMO.pdf The integer m in the statement of the problem is ϕ(n), where ϕ is the Euler totient function. Throughout our proof we write p s || m, if s is the greatest power of p that divides m. We begin with the following lemma: Lemma 1. If p is a prime and ... |
USAMO-2018-4 | https://artofproblemsolving.com/wiki/index.php/2018_USAMO_Problems/Problem_4 | Let \(p\) be a prime, and let \(a_1, \dots, a_p\) be integers. Show that there exists an integer \(k\) such that the numbers
\[
a_1 + k, a_2 + 2k, \dots, a_p + pk
\]
produce at least \(\tfrac{1}{2} p\) distinct remainders upon division by \(p\). | [
"\\(\\textbf{Lemma: }\\) For fixed \\(i\\neq j,\\) where \\(i, j\\in\\{1, 2, ..., p\\},\\) the statement \\(a_i + ik\\equiv a_j + jk\\text{ (mod } p\\text{)}\\) holds for exactly one \\(k\\in {1, 2, ..., p}.\\)\n\n\\(\\textbf{Proof: }\\) Notice that the left side minus the right side is congruent to \\((a_i - a_j) ... |
USAMO-2018-5 | https://artofproblemsolving.com/wiki/index.php/2018_USAMO_Problems/Problem_5 | In convex cyclic quadrilateral \(ABCD,\) we know that lines \(AC\) and \(BD\) intersect at \(E,\) lines \(AB\) and \(CD\) intersect at \(F,\) and lines \(BC\) and \(DA\) intersect at \(G.\) Suppose that the circumcircle of \(\triangle ABE\) intersects line \(CB\) at \(B\) and \(P\), and the circumcircle of \(\triangle ... | [
"\\[\n[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(13cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle =... |
USAMO-2018-6 | https://artofproblemsolving.com/wiki/index.php/2018_USAMO_Problems/Problem_6 | Let \(a_n\) be the number of permutations \((x_1, x_2, \dots, x_n)\) of the numbers \((1,2,\dots, n)\) such that the \(n\) ratios \(\frac{x_k}{k}\) for \(1\le k\le n\) are all distinct. Prove that \(a_n\) is odd for all \(n\ge 1.\) | [
"Write out the mapping of each \\(k\\) to \\(x_k\\) as follows:\n\n\\[\n1\\ \\ 2\\ \\ 3\\ \\ \\dots \\ \\ n\n\\]\n\n\\[\nx_1\\ x_2\\ x_3\\ \\dots \\ x_n\n\\]\n\nNow, consider what happens when the two rows are swapped (and the top-bottom pairs are reordered so that the top reads (1,2,3,...,n)). This will result in ... |
USAMO-2019-1 | https://artofproblemsolving.com/wiki/index.php/2019_USAMO_Problems/Problem_1 | Let \(\mathbb{N}\) be the set of positive integers. A function \(f:\mathbb{N}\to\mathbb{N}\) satisfies the equation
\[
\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}
\]
for all positive integers \(n\). Given this information, determine all possible values of \(f(1000)\). | [
"Let \\(f^r(x)\\) denote the result when \\(f\\) is applied to \\(f^{r-1}(x)\\), where \\(f^1(x)=f(x)\\). \\(\\hfill \\break \\hfill \\break\\) If \\(f(p)=f(q)\\), then \\(f^2(p)=f^2(q)\\) and \\(f^{f(p)}(p)=f^{f(q)}(q)\\)\n\n\\[\n\\implies p^2=f^2(p)\\cdot f^{f(p)}(p)=f^2(q)\\cdot f^{f(q)}(q)=q^2\n\\]\n\n\\[\n\\im... |
USAMO-2019-2 | https://artofproblemsolving.com/wiki/index.php/2019_USAMO_Problems/Problem_2 | Let \(ABCD\) be a cyclic quadrilateral satisfying \(AD^2 + BC^2 = AB^2\). The diagonals of \(ABCD\) intersect at \(E\). Let \(P\) be a point on side \(\overline{AB}\) satisfying \(\angle APD = \angle BPC\). Show that line \(PE\) bisects \(\overline{CD}\). | [
"Let \\(PE \\cap DC = M\\). Also, let \\(N\\) be the midpoint of \\(AB\\). Note that only one point \\(P\\) satisfies the given angle condition. With this in mind, construct \\(P'\\) with the following properties:\n\n(1) \\(AP' \\cdot AB = AD^2\\)\n\n(2) \\(BP' \\cdot AB = CD^2\\)\n\nClaim: \\(P = P'\\)\n\nProof: T... |
USAMO-2019-3 | https://artofproblemsolving.com/wiki/index.php/2019_USAMO_Problems/Problem_3 | Let \(K\) be the set of all positive integers that do not contain the digit \(7\) in their base-\(10\) representation. Find all polynomials \(f\) with nonnegative integer coefficients such that \(f(n)\in K\) whenever \(n\in K\). | [
"I claim the only such polynomials are of the form \\(f(n)=k\\) where \\(k\\in K\\), or \\(f(n)=an+b\\) where \\(a\\) is a power of 10, \\(b\\in K\\), and \\(b<a\\). Obviously, these polynomials satisfy the conditions. We now prove that no other polynomial works. That is, we prove that for any other polynomial \\(f... |
USAMO-2019-4 | https://artofproblemsolving.com/wiki/index.php/2019_USAMO_Problems/Problem_4 | Let \(n\) be a nonnegative integer. Determine the number of ways that one can choose \((n+1)^2\) sets \(S_{i,j}\subseteq\{1,2,\ldots,2n\}\), for integers \(i,j\) with \(0\leq i,j\leq n\), such that:
\(\bullet\) for all \(0\leq i,j\leq n\), the set \(S_{i,j}\) has \(i+j\) elements; and
\(\bullet\) \(S_{i,j}\subseteq S... | [
"Note that there are \\((2n)!\\) ways to choose \\(S_{1, 0}, S_{2, 0}... S_{n, 0}, S_{n, 1}, S_{n, 2}... S_{n, n}\\), because there are \\(2n\\) ways to choose which number \\(S_{1, 0}\\) is, \\(2n-1\\) ways to choose which number to append to make \\(S_{2, 0}\\), \\(2n-2\\) ways to choose which number to append to... |
USAMO-2019-5 | https://artofproblemsolving.com/wiki/index.php/2019_USAMO_Problems/Problem_5 | Two rational numbers \(\frac{m}{n}\) and \(\frac{n}{m}\) are written on a blackboard, where \(m\) and \(n\) are relatively prime positive integers. At any point, Evan may pick two of the numbers \(x\) and \(y\) written on the board and write either their arithmetic mean \(\frac{x+y}{2}\) or their harmonic mean \(\frac{... | [
"We claim that all odd \\(m, n\\) work if \\(m+n\\) is a positive power of 2.\n\nProof: We first prove that \\(m+n=2^k\\) works. By weighted averages we have that \\(\\frac{n(\\frac{m}{n})+(2^k-n)\\frac{n}{m}}{2^k}=\\frac{m+n}{2^k}=1\\) can be written, so the solution set does indeed work. We will now prove these a... |
USAMO-2019-6 | https://artofproblemsolving.com/wiki/index.php/2019_USAMO_Problems/Problem_6 | Find all polynomials \(P\) with real coefficients such that
\[
\frac{P(x)}{yz}+\frac{P(y)}{zx}+\frac{P(z)}{xy}=P(x-y)+P(y-z)+P(z-x)
\]
holds for all nonzero real numbers \(x,y,z\) satisfying \(2xyz=x+y+z\). | [
"If \\(P(x)=c\\) for a constant \\(c,\\) then \\(\\dfrac{c(x+y+z)}{xyz}=3c\\). We have \\(2c=3c.\\) Therefore \\(c=0.\\)\n\nNow consider the case of non-constant polynomials. First we have\n\n\\[\nxP(x)+yP(y)+zP(z)=xyz(P(x-y)+P(y-z)+P(z-x))\n\\]\n\nfor all nonzero real numbers \\(x,y,z\\) satisfying \\(2xyz=x+y+z\\... |
USAMO-2020-1 | https://artofproblemsolving.com/wiki/index.php/2020_USAMO_Problems/Problem_1 | Let \(ABC\) be a fixed acute triangle inscribed in a circle \(\omega\) with center \(O\). A variable point \(X\) is chosen on minor arc \(AB\) of \(\omega\), and segments \(CX\) and \(AB\) meet at \(D\). Denote by \(O_1\) and \(O_2\) the circumcenters of triangles \(ADX\) and \(BDX\), respectively. Determine all points... | [
"Let \\(E\\) be midpoint \\(AD.\\) Let \\(F\\) be midpoint \\(BD \\implies\\)\n\n\\[\nEF = ED + FD = \\frac {AD}{2} + \\frac {BD}{2} = \\frac {AB}{2}.\n\\]\n\n\\(E\\) and \\(F\\) are the bases of perpendiculars dropped from \\(O_1\\) and \\(O_2,\\) respectively.\n\nTherefore \\(O_1O_2 \\ge EF = \\frac {AB}{2}.\\)\n... |
USAMO-2020-2 | https://artofproblemsolving.com/wiki/index.php/2020_USAMO_Problems/Problem_2 | An empty \(2020 \times 2020 \times 2020\) cube is given, and a \(2020 \times 2020\) grid of square unit cells is drawn on each of its six faces. A beam is a \(1 \times 1 \times 2020\) rectangular prism. Several beams are placed inside the cube subject to the following conditions:
- The two \(1 \times 1\) faces of each... | [
"Take one vertex of the cube as origin and establish 3D coordinates along the cube's edges.\n\nDefine a beam as \\(x-dir\\) if its long edge is parallel to x-axis. Similarly for \\(y-dir\\) and \\(z-dir\\).\n\nDefine a beam's location as (direction, (\\(1 \\times 1\\) face's location in 2D coordinate).\n\nFor examp... |
USAMO-2020-3 | https://artofproblemsolving.com/wiki/index.php/2020_USAMO_Problems/Problem_3 | Let \(p\) be an odd prime. An integer \(x\) is called a quadratic non-residue if \(p\) does not divide \(x - t^2\) for any integer \(t\).
Denote by \(A\) the set of all integers \(a\) such that \(1 \le a < p\), and both \(a\) and \(4 - a\) are quadratic non-residues. Calculate the remainder when the product of the ele... | [
"We will be using finite field theory and the Frobenius endomorphism freely. Also we will just give a sketch and leave out details like it's out of fashion.\n\n[b]Case 1[/b]: \\(p\\equiv 3\\pmod{4}\\). Work in \\(\\mathbb{F}_{p^2} = \\mathbb{F}_p[i]\\). Suppose \\(a\\in A\\). Then, \\(a=-x^2\\) for some \\(x\\in\\m... |
USAMO-2020-4 | https://artofproblemsolving.com/wiki/index.php/2020_USAMO_Problems/Problem_4 | Suppose that \((a_1, b_1), (a_2, b_2), \ldots , (a_{100}, b_{100})\) are distinct ordered pairs of nonnegative integers. Let \(N\) denote the number of pairs of integers \((i, j)\) satisfying \(1 \le i < j \le 100\) and \(|a_ib_j - a_j b_i|=1\). Determine the largest possible value of \(N\) over all possible choices of... | [
"Let's start off with just \\((a_1, b_1), (a_2, b_2)\\) and suppose that it satisfies the given condition. We could use \\((1, 1), (1, 2)\\) for example. We should maximize the number of conditions that the third pair satisfies. We find out that the third pair should equal \\((a_1+a_2, b_1+b_2)\\):\n\nWe know this ... |
USAMO-2020-5 | https://artofproblemsolving.com/wiki/index.php/2020_USAMO_Problems/Problem_5 | A finite set \(S\) of points in the coordinate plane is called overdetermined if \(|S| \ge 2\) and there exists a nonzero polynomial \(P(t)\), with real coefficients and of degree at most \(|S| - 2\), satisfying \(P(x) = y\) for every point \((x, y) \in S\).
For each integer \(n \ge 2\), find the largest integer \(k\)... | [
"The answer is \\(2^{n-1}-n\\). To construct this, have \\(n-1\\) points on a horizontal line and \\(1\\) point not on it. Then, any subset that does not include the last point is overdetermined.\\\\\n\nFor the bound, the main idea of the problem is the following claim.\\\\\n\nClaim: If \\(S\\) and \\(T\\) are over... |
USAMO-2020-6 | https://artofproblemsolving.com/wiki/index.php/2020_USAMO_Problems/Problem_6 | Let \(n \ge 2\) be an integer. Let \(x_1 \ge x_2 \ge \cdots \ge x_n\) and \(y_1 \ge y_2 \ge \cdots \ge y_n\) be \(2n\) real numbers such that
\begin{align*} 0 &= x_1 + x_2 + \cdots + x_n = y_1 + y_2 + \cdots + y_n \\ \text{and }1 &= x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2. \end{align*}
Prove that
\[
\sum... | [] |
USAMO-2021-1 | https://artofproblemsolving.com/wiki/index.php/2021_USAMO_Problems/Problem_1 | Rectangles \(BCC_1B_2,\) \(CAA_1C_2,\) and \(ABB_1A_2\) are erected outside an acute triangle \(ABC.\) Suppose that
\[
\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.
\]
Prove that lines \(B_1C_2,\) \(C_1A_2,\) and \(A_1B_2\) are concurrent. | [
"Let \\(D\\) be the second point of intersection of the circles \\(AB_1B\\) and \\(AA_1C.\\) Then:\n\n\\[\n\\begin{align*} \\angle ADB &= 180^\\circ – \\angle AB_1B,&\\angle ADC &= 180^\\circ – \\angle AA_1C\\\\ \\angle BDC &= 360^\\circ – \\angle ADB – \\angle ADC\\\\ &= 360^\\circ – (180^\\circ – \\angle AB_1B) –... |
USAMO-2021-2 | https://artofproblemsolving.com/wiki/index.php/2021_USAMO_Problems/Problem_2 | The Planar National Park is an undirected 3-regular planar graph (i.e. all vertices have degree 3). That is: every trail has its two endpoints at two different junctions whereas each junction is the endpoint of exactly three trails. Trails only intersect at junctions (in particular, trails only meet at endpoints). Fina... | [] |
USAMO-2021-3 | https://artofproblemsolving.com/wiki/index.php/2021_USAMO_Problems/Problem_3 | Let \(n \geq 2\) be an integer. An \(n \times n\) board is initially empty. Each minute, you may perform one of three moves:
- If there is an L-shaped tromino region of three cells without stones on the board (see figure; rotations not allowed), you may place a stone in each of those cells.
- If all cells in a column ... | [
"Label the bottom right cell as \\(\\omega^0\\) where \\(\\omega = e^{i2\\pi/3}\\). Then label each cell with \\(\\omega^d\\) where \\(d\\) is the Manhattan distance of the current cell from the bottom right cell.\n\nNo matter where an L-shaped tromino is placed, the sum of the labels in the cells is,\n\n\\[\n\\ome... |
USAMO-2021-4 | https://artofproblemsolving.com/wiki/index.php/2021_USAMO_Problems/Problem_4 | A finite set \(S\) of positive integers has the property that, for each \(s\in S\), and each positive integer divisor \(d\) of \(s\), there exists a unique element \(t \in S\) satisfying \(\gcd(s,t)=d\) (the elements \(s\) and \(t\) could be equal).
Given this information, find all possible values for the number of el... | [] |
USAMO-2021-5 | https://artofproblemsolving.com/wiki/index.php/2021_USAMO_Problems/Problem_5 | Let \(n \geq 4\) be an integer. Find all positive real solutions to the following system of \(2n\) equations:
\begin{align*} a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\ a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\ a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7} ... | [] |
USAMO-2021-6 | https://artofproblemsolving.com/wiki/index.php/2021_USAMO_Problems/Problem_6 | Let \(ABCDEF\) be a convex hexagon satisfying \(\overline{AB} \parallel \overline{DE}\), \(\overline{BC} \parallel \overline{EF}\), \(\overline{CD} \parallel \overline{FA}\), and
\[
AB \cdot DE = BC \cdot EF = CD \cdot FA.
\]
Let \(X\), \(Y\), and \(Z\) be the midpoints of \(\overline{AD}\), \(\overline{BE}\), and \(... | [
"Let \\(M_1\\), \\(M_2\\), and \\(M_3\\) be the midpoints of \\(CE\\), \\(AE\\), \\(AC\\) and \\(N_1\\), \\(N_2\\), and \\(N_3\\) be the midpoints of \\(DF\\), \\(BF\\), and \\(BD\\). Also, let \\(H\\) be the orthocenter of \\(XYZ\\). Note that we can use parallel sides to see that \\(X\\), \\(Z\\), and \\(M_3\\) a... |
USAMO-2022-1 | https://artofproblemsolving.com/wiki/index.php/2022_USAMO_Problems/Problem_1 | Let \(a\) and \(b\) be positive integers. The cells of an \((a+b+1)\times (a+b+1)\) grid are colored amber and bronze such that there are at least \(a^2+ab-b\) amber cells and at least \(b^2+ab-a\) bronze cells. Prove that it is possible to choose \(a\) amber cells and \(b\) bronze cells such that no two of the \(a+b\)... | [
"We proceed with induction. Base case: \\(a=b=1\\). We fill in a 3x3 grid that has at least one bronze and one amber cell. We choose one bronze and one amber cell. Note that if the given bronze and amber cells are not in the same row or column, we are done. However, if they are, consider a cell that is not in the s... |
USAMO-2022-2 | https://artofproblemsolving.com/wiki/index.php/2022_USAMO_Problems/Problem_2 | Let \(b\geq2\) and \(w\geq2\) be fixed integers, and \(n=b+w\). Given are \(2b\) identical black rods and \(2w\) identical white rods, each of side length \(1\).
We assemble a regular \(2n\)-gon using these rods so that parallel sides are the same color. Then, a convex \(2b\)-gon \(B\) is formed by translating the bla... | [
"First notice that the black rods and the white rods form polygons iff in the original \\(2n\\)-gon, if a side is a color \\(x\\), then the side that is parallel to that side in the original \\(2n\\)-gon is also the color \\(x\\).\n\nWe can prove that the difference in areas is only affected by the values of \\(b\\... |
USAMO-2022-3 | https://artofproblemsolving.com/wiki/index.php/2022_USAMO_Problems/Problem_3 | Let \(\mathbb{R}_{>0}\) be the set of all positive real numbers. Find all functions \(f:\mathbb{R}_{>0} \to \mathbb{R}_{>0}\) such that for all \(x,y\in \mathbb{R}_{>0}\) we have
\[
f(x) = f(f(f(x)) + y) + f(xf(y)) f(x+y).
\] | [
"[WIP]"
] |
USAMO-2022-4 | https://artofproblemsolving.com/wiki/index.php/2022_USAMO_Problems/Problem_4 | Find all pairs of primes \((p, q)\) for which \(p-q\) and \(pq-q\) are both perfect squares. | [
"Since \\(q(p-1)\\) is a perfect square and \\(q\\) is prime, we should have \\(p - 1 = qb^2\\) for some positive integer \\(b\\). Let \\(a^2 = p - q\\). Therefore, \\(q = p - a^2\\), and substituting that into the \\(p - 1 = qb^2\\) and solving for \\(p\\) gives\n\n\\[\np = \\frac{a^2b^2 - 1}{b^2 - 1} = \\frac{(ab... |
USAMO-2022-5 | https://artofproblemsolving.com/wiki/index.php/2022_USAMO_Problems/Problem_5 | A function \(f: \mathbb{R}\to \mathbb{R}\) is \(\textit{essentially increasing}\) if \(f(s)\leq f(t)\) holds whenever \(s\leq t\) are real numbers such that \(f(s)\neq 0\) and \(f(t)\neq 0\).
Find the smallest integer \(k\) such that for any 2022 real numbers \(x_1,x_2,\ldots , x_{2022},\) there exist \(k\) essentiall... | [
"Coming soon."
] |
USAMO-2022-6 | https://artofproblemsolving.com/wiki/index.php/2022_USAMO_Problems/Problem_6 | There are \(2022\) users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.)
Starting now, Mathbook will only allow a new friendship to be formed between two users if they have at least two friends in common. What is the minimum number o... | [
"To answer this question, we need to consider how friendships on Mathbook are formed. If two users have at least two friends in common, then they will be able to become friends with each other. This means that the minimum number of friendships that must already exist in order for every user to eventually become fri... |
USAMO-2023-1 | https://artofproblemsolving.com/wiki/index.php/2023_USAMO_Problems/Problem_1 | In an acute triangle \(ABC\), let \(M\) be the midpoint of \(\overline{BC}\). Let \(P\) be the foot of the perpendicular from \(C\) to \(AM\). Suppose that the circumcircle of triangle \(ABP\) intersects line \(BC\) at two distinct points \(B\) and \(Q\). Let \(N\) be the midpoint of \(\overline{AQ}\). Prove that \(NB=... | [
"Let \\(X\\) be the foot from \\(A\\) to \\(\\overline{BC}\\). By definition, \\(\\angle AXM = \\angle MPC = 90^{\\circ}\\). Thus, \\(\\triangle AXM \\sim \\triangle CPM\\), and \\(\\triangle BMP \\sim \\triangle AMQ\\).\n\nFrom this, we have \\(\\frac{MP}{MX} = \\frac{MC}{MA} = \\frac{MP}{MQ}\\), as \\(MC=MB\\). T... |
USAMO-2023-2 | https://artofproblemsolving.com/wiki/index.php/2023_USAMO_Problems/Problem_2 | Let \(\mathbb{R}^{+}\) be the set of positive real numbers. Find all functions \(f:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) such that, for all \(x, y \in \mathbb{R}^{+}\),
\[
f(xy + f(x)) = xf(y) + 2
\] | [
"Make the following substitutions to the equation:\n\n1. \\((x, 1) \\rightarrow f(x + f(x)) = xf(1) + 2\\)\n\n2. \\((1, x + f(x)) \\rightarrow f(x + f(x) + f(1)) = f(x + f(x)) + 2 = xf(1) + 4\\)\n\n3. \\((x, 1 + \\frac{f(1)}{x}) \\rightarrow f(x + f(x) + f(1)) = xf\\biggl(1 + \\frac{f(1)}{x}\\biggr) + 2\\)\n\nIt th... |
USAMO-2023-3 | https://artofproblemsolving.com/wiki/index.php/2023_USAMO_Problems/Problem_3 | Consider an \(n\)-by-\(n\) board of unit squares for some odd positive integer \(n\). We say that a collection \(C\) of identical dominoes is a maximal grid-aligned configuration on the board if \(C\) consists of \((n^2-1)/2\) dominoes where each domino covers exactly two neighboring squares and the dominoes don't over... | [
"We claim the answer is \\((\\frac{n+1}{2})^2\\).\n\nFirst, consider a checkerboard tiling of the board with 4 colors: R, G, B, Y. Number each column from \\(1\\) to \\(n\\) from left to right and each row from \\(1\\) to \\(n\\) from top to bottom. We color a tile R if its row and column are odd, a tile G is its r... |
USAMO-2023-4 | https://artofproblemsolving.com/wiki/index.php/2023_USAMO_Problems/Problem_4 | A positive integer \(a\) is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer \(n\) on the board with \(n+a\), and on Bob's turn he must replace some even integer \(n\) on the board with \(n/2\). Alice goes first and they a... | [
"The contrapositive of the claim is somewhat easier to conceptualize: If it is not guaranteed that the game will end (i.e. the game could potentially last forever), then Bob is not able to force the game to end (i.e. Alice can force it to last forever). So we want to prove that, if the game can potentially last ind... |
USAMO-2023-5 | https://artofproblemsolving.com/wiki/index.php/2023_USAMO_Problems/Problem_5 | Let \(n\geq3\) be an integer. We say that an arrangement of the numbers \(1\), \(2\), \(\dots\), \(n^2\) in a \(n \times n\) table is row-valid if the numbers in each row can be permuted to form an arithmetic progression, and column-valid if the numbers in each column can be permuted to form an arithmetic progression. ... | [
"The answer is all \\(\\boxed{\\text{prime } n}\\).\n\n## Proof that primes work\n\nSuppose \\(n=p\\) is prime. Then, let the arithmetic progressions in the \\(i\\)th row have least term \\(a_i\\) and common difference \\(d_i\\). For each cell with integer \\(k\\), assign a monomial \\(x^k\\). The sum of the monomi... |
USAMO-2023-6 | https://artofproblemsolving.com/wiki/index.php/2023_USAMO_Problems/Problem_6 | Let ABC be a triangle with incenter \(I\) and excenters \(I_a\), \(I_b\), \(I_c\) opposite \(A\), \(B\), and \(C\), respectively. Given an arbitrary point \(D\) on the circumcircle of \(\triangle ABC\) that does not lie on any of the lines \(II_{a}\), \(I_{b}I_{c}\), or \(BC\), suppose the circumcircles of \(\triangle ... | [
"\\[\n[asy] size(500); pair A,B,C,D,E,F,G,H,I,J,K,IA,IB,IC,P,Q; B=(0,0); C=(8,0); A=intersectionpoint(Circle(B,6),Circle(C,9)); I=incenter(A,B,C); path c=circumcircle(A,B,C); J=intersectionpoint(I--(4*I-3*A),c); IA=2*J-I; IB=2*intersectionpoint(I--(4*I-3*B),c)-I; IC=2*intersectionpoint(I--(4*I-3*C),c)-I; K=intersec... |
USAMO-2024-1 | https://artofproblemsolving.com/wiki/index.php/2024_USAMO_Problems/Problem_1 | Find all integers \(n \geq 3\) such that the following property holds: if we list the divisors of \(n !\) in increasing order as \(1=d_1<d_2<\cdots<d_k=n!\), then we have
\[
d_2-d_1 \leq d_3-d_2 \leq \cdots \leq d_k-d_{k-1} .
\] | [
"We can start by verifying that \\(n=3\\) and \\(n=4\\) work by listing out the factors of \\(3!\\) and \\(4!\\). We can also see that \\(n=5\\) does not work because the terms \\(15, 20\\), and \\(24\\) are consecutive factors of \\(5!\\). Also, \\(n=6\\) does not work because the terms \\(6, 8\\), and \\(9\\) app... |
USAMO-2024-2 | https://artofproblemsolving.com/wiki/index.php/2024_USAMO_Problems/Problem_2 | Let \(S_1, S_2, \ldots, S_{100}\) be finite sets of integers whose intersection is not empty. For each non-empty \(T \subseteq\left\{S_1, S_2, \ldots, S_{100}\right\}\), the size of the intersection of the sets in \(T\) is a multiple of the number of sets in \(T\). What is the least possible number of elements that are... | [
"Let's determine the smallest possible number of elements that appear in at least 50 of the sets.\n\nFirst, we establish some notation. For any subset \\(T \\subseteq \\{S_1, S_2, \\ldots, S_{100}\\}\\), let \\(\\cap T\\) denote the intersection of all sets in \\(T\\). By the problem statement, \\(|\\cap T|\\) is d... |
USAMO-2024-3 | https://artofproblemsolving.com/wiki/index.php/2024_USAMO_Problems/Problem_3 | Let \(m\) be a positive integer. A triangulation of a polygon is \(m\)-balanced if its triangles can be colored with \(m\) colors in such a way that the sum of the areas of all triangles of the same color is the same for each of the \(m\) colors. Find all positive integers \(n\) for which there exists an \(m\)-balanced... | [
"The answer is if and only if \\(m\\) is a proper divisor of \\(n\\).\n\nWe represent the vertices of the regular \\(n\\)-gon using complex numbers. Let \\(\\omega = \\exp(2\\pi i/n)\\) be a primitive \\(n\\)-th root of unity, and label the vertices as \\(1, \\omega, \\omega^2, \\ldots, \\omega^{n-1}\\).\n\nKey Lem... |
USAMO-2024-4 | https://artofproblemsolving.com/wiki/index.php/2024_USAMO_Problems/Problem_4 | Let \(m\) and \(n\) be positive integers. A circular necklace contains \(m n\) beads, each either red or blue. It turned out that no matter how the necklace was cut into \(m\) blocks of \(n\) consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair... | [
"We need to determine all possible positive integer pairs \\((m, n)\\) such that there exists a circular necklace of \\(mn\\) beads, each colored red or blue, satisfying the following condition:\n\nNo matter how the necklace is cut into \\(m\\) blocks of \\(n\\) consecutive beads, each block has a distinct number o... |
USAMO-2024-5 | https://artofproblemsolving.com/wiki/index.php/2024_USAMO_Problems/Problem_5 | Point \(D\) is selected inside acute triangle \(ABC\) so that \(\angle DAC=\angle ACB\) and \(\angle BDC=90^\circ+\angle BAC\). Point \(E\) is chosen on ray \(BD\) so that \(AE=EC\). Let \(M\) be the midpoint of \(BC\). Show that line \(AB\) is tangent to the circumcircle of triangle \(BEM\). | [
"Let \\(\\angle DBT = \\alpha\\) and \\(\\angle BEM = \\beta\\). Extend AD intersects BC at point T, then TC = TA, TE is perpendicular to AC\n\nThus, AB is the tangent of the circle BEM\n\nThen the question is equivalent as the \\(\\angle ABT\\) is the auxillary angle of \\(\\angle BEM\\).\n\nontinued"
] |
USAMO-2024-6 | https://artofproblemsolving.com/wiki/index.php/2024_USAMO_Problems/Problem_6 | Continued
Let \(n>2\) be an integer and let \(\ell \in\{1,2, \ldots, n\}\). A collection \(A_1, \ldots, A_k\) of (not necessarily distinct) subsets of \(\{1,2, \ldots, n\}\) is called \(\ell\)-large if \(\left|A_i\right| \geq \ell\) for all \(1 \leq i \leq k\). Find, in terms of \(n\) and \(\ell\), the largest real nu... | [] |
USAMO-2025-1 | https://artofproblemsolving.com/wiki/index.php/2025_USAMO_Problems/Problem_1 | Let \(k\) and \(d\) be positive integers. Prove that there exists a positive integer \(N\) such that for every odd integer \(n>N\), the digits in the base-\(2n\) representation of \(n^k\) are all greater than \(d\). | [
"We define a remainder operation \\(\\,a \\bmod b\\,\\) to be the remainder when \\(a\\) is divided by \\(b\\). Also, let \\(\\lfloor x\\rfloor\\) be the usual floor function.\n\nBase-\\((2n)\\) Representation:\n\n\\[\nn^k \\;=\\; a_{k-1}\\,(2n)^{k-1} \\;+\\; a_{k-2}\\,(2n)^{k-2} \\;+\\;\\dots\\;+\\;a_1\\,(2n)\\;+\... |
USAMO-2025-2 | https://artofproblemsolving.com/wiki/index.php/2025_USAMO_Problems/Problem_2 | Let \(n\) and \(k\) be positive integers with \(k<n\). Let \(P(x)\) be a polynomial of degree \(n\) with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers \(a_0, a_1, \dots, a_k\) such that the polynomial \(a_kx^k+\cdots+a_1x+a_0\) divides \(P(x)\), the product \(a_0a_1\... | [
"We proceed by contradiction. Assume that all roots of \\(P(x)\\) are real. Let the distinct roots be \\(r_1, r_2, \\ldots, r_n\\), all nonzero since the constant term is nonzero.\n\nConsider any subset of \\(k\\) roots \\(\\{r_{i_1}, r_{i_2}, \\ldots, r_{i_k}\\}\\) and form the polynomial:\n\n\\[\nQ(x) = \\prod_{j... |
USAMO-2025-3 | https://artofproblemsolving.com/wiki/index.php/2025_USAMO_Problems/Problem_3 | Alice the architect and Bob the builder play a game. First, Alice chooses two points \(P\) and \(Q\) in the plane and a subset \(\mathcal{S}\) of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one u... | [
"https://artofproblemsolving.com/wiki/index.php/File:2025_USAMO_PROBLEM_3.jpeg\n\nThe answer is that Alice wins. Let's define a Bob-set V to be a set of points in the plane with no three collinear and with all distances at least 1. The point of the problem is to prove the following fact. Claim — Given a Bob-set V ⊆... |
USAMO-2025-4 | https://artofproblemsolving.com/wiki/index.php/2025_USAMO_Problems/Problem_4 | Let \(H\) be the orthocenter of acute triangle \(ABC\), let \(F\) be the foot of the altitude from \(C\) to \(AB\), and let \(P\) be the reflection of \(H\) across \(BC\). Suppose that the circumcircle of triangle \(FAP\) intersects line \(BC\) at two distinct points \(X\) and \(Y\). Prove that \(C\) is the midpoint of... | [
"Let AP intersects BC at D. Extend FC to the point E on the circumcircle \\(\\omega\\) of \\(FAP\\). Since \\(H\\) is the orthocenter of \\(\\Delta ABC\\), we know that \\(HD = DP\\) or \\(HP = 2HD\\), and \\(AH \\cdot HD = CH \\cdot HF\\). Next we use the power of H in \\(\\omega\\): \\(AH \\cdot HP = CH \\cdot HE... |
USAMO-2025-5 | https://artofproblemsolving.com/wiki/index.php/2025_USAMO_Problems/Problem_5 | Determine, with proof, all positive integers \(k\) such that
\[
\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k
\]
is an integer for every positive integer \(n.\) | [
"https://artofproblemsolving.com/wiki/index.php/File:2025_USAMO_PROBLEM_5_1.jpg\n\nhttps://artofproblemsolving.com/wiki/index.php/File:2025_USAMO_PROBLEM_5_2.jpg",
"Let\n\n\\[\nX=\\{(A_1,\\dots,A_k):A_j\\subseteq\\{1,\\dots,n\\},\\;|A_1|=\\cdots=|A_k|\\}.\n\\]\n\nThen\n\n\\[\n|X|=\\sum_{i=0}^n\\binom{n}{i}^k.\n\\... |
USAMO-2025-6 | https://artofproblemsolving.com/wiki/index.php/2025_USAMO_Problems/Problem_6 | Let \(m\) and \(n\) be positive integers with \(m\geq n\). There are \(m\) cupcakes of different flavors arranged around a circle and \(n\) people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person \(P\), it i... | [
"https://artofproblemsolving.com/wiki/index.php/File:2025_USAMO_PROBLEM_6.jpeg\n\nArbitrarily pick any one person — call her Pip — and her n arcs. The initial idea is to try to apply Hall's marriage lemma to match the n people with Pip's arcs (such that each such person is happy with their matched arc). To that end... |
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