id stringlengths 10 10 | source stringlengths 74 74 | problem stringlengths 57 1.29k | solutions listlengths 0 9 |
|---|---|---|---|
IMO-2009-5 | https://artofproblemsolving.com/wiki/index.php/2009_IMO_Problems/Problem_5 | Determine all functions \(f\) from the set of positive integers to the set of positive integers such that, for all positive integers \(a\) and \(b\), there exists a non-degenerate triangle with sides of lengths
\[
a,f(b)
\]
and
\[
f(b+f(a)-1).
\]
(A triangle is non-degenerate if its vertices are not collinear.) | [
"Answer: The only such function is \\(f(x)=x\\).\n\nIt is easy to see that this function satisfy the condition. We are going to proof that this is the only such function.\n\nWe start with\n\nLemma. If 1, \\(a\\), \\(b\\) are sides of a non-degenerate triangle then \\(a=b\\).\n\nProof. In this case \\(a<b+1\\), ther... |
IMO-2009-6 | https://artofproblemsolving.com/wiki/index.php/2009_IMO_Problems/Problem_6 | Let \(a_1,a_2,\ldots,a_n\) be distinct positive integers and let \(M\) be a set of \(n-1\) positive integers not containing \(s=a_1+a_2+\ldots+a_n\). A grasshopper is to jump along the real axis, starting at the point \(0\) and making \(n\) jumps to the right with lengths \(a_1,a_2,\ldots,a_n\) in some order. Prove tha... | [
"We will use strong induction on \\(n\\). When \\(n = 1\\), there are no elements in \\(M\\), so the one jump can be made without landing on a point in \\(M\\). When \\(n = 2\\), we consider two cases. If \\(a_1\\) is not in \\(M\\), then the order \\(a_1, a_2\\) will work. If \\(a_1\\) is in \\(M\\), then \\(a_2\\... |
IMO-2010-1 | https://artofproblemsolving.com/wiki/index.php/2010_IMO_Problems/Problem_1 | Find all functions \(f:\mathbb{R}\rightarrow\mathbb{R}\) such that for all \(x,y\in\mathbb{R}\) the following equality holds
\[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor
\]
where \(\left\lfloor a\right\rfloor\) is greatest integer not greater than \(a.\) | [
"Put \\(x=y=0\\). Then \\(f(0)=0\\) or \\(\\lfloor f(0) \\rfloor=1\\).\n\n\\(\\bullet\\) If \\(\\lfloor f(0) \\rfloor=1\\), putting \\(y=0\\) we get \\(f(x)=f(0)\\), that is f is constant. Substituing in the original equation we find \\(f(x)=0, \\ \\forall x \\in \\mathbb{R}\\) or \\(f(x)=a, \\ \\forall x \\in \\ma... |
IMO-2010-2 | https://artofproblemsolving.com/wiki/index.php/2010_IMO_Problems/Problem_2 | Given a triangle \(ABC\), with \(I\) as its incenter and \(\Gamma\) as its circumcircle, \(AI\) intersects \(\Gamma\) again at \(D\). Let \(E\) be a point on arc \(BDC\), and \(F\) a point on the segment \(BC\), such that \(\angle BAF=\angle CAE< \frac12\angle BAC\). If \(G\) is the midpoint of \(IF\), prove that the i... | [
"Note that it suffices to prove alternatively that if \\(EI\\) meets the circle again at \\(J\\) and \\(JD\\) meets \\(IF\\) at \\(G\\), then \\(G\\) is the midpoint of \\(IF\\). Let \\(JD\\) meet \\(BC\\) at \\(K\\).\n\nObservation 1. D is the midpoint of arc \\(BDC\\) because it lies on angle bisector \\(AI\\).\n... |
IMO-2010-3 | https://artofproblemsolving.com/wiki/index.php/2010_IMO_Problems/Problem_3 | Find all functions \(g:\mathbb{N}\rightarrow\mathbb{N}\) such that \(\left(g(m)+n\right)\left(g(n)+m\right)\) is a perfect square for all \(m,n\in\mathbb{N}.\) | [
"Suppose such function \\(g\\) exist then:\n\nLemma 1) \\(g(m) \\ne g(m+1)\\)\n\nbut \\(\\left(g(m)+m\\right)^2<\\left(g(m+1)+m\\right)\\left(g(m)+m+1\\right)<\\left(g(m)+m+1\\right)^2\\).\n\nLemma 2) \\(|g(m)-g(m+1)| = 1\\) (we have show that it can't be 0)\n\nThen there must exist a prime number \\(p\\) such that... |
IMO-2010-4 | https://artofproblemsolving.com/wiki/index.php/2010_IMO_Problems/Problem_4 | Let \(P\) be a point interior to triangle \(ABC\) (with \(CA \neq CB\)). The lines \(AP\), \(BP\) and \(CP\) meet again its circumcircle \(\Gamma\) at \(K\), \(L\), respectively \(M\). The tangent line at \(C\) to \(\Gamma\) meets the line \(AB\) at \(S\). Show that from \(SC = SP\) follows \(MK = ML\). | [
"Without loss of generality, suppose that \\(AS > BS\\). By Power of a Point, \\(SP^2 = SC^2 = SB \\cdot SA\\), so \\(\\overline{SP}\\) is tangent to the circumcircle of \\(\\triangle ABP\\). Thus, \\(\\angle KPS = 180 - \\angle SPA = \\widehat{AP}/2 = \\angle ABP\\). It follows that after some angle-chasing,\n\n\\... |
IMO-2010-5 | https://artofproblemsolving.com/wiki/index.php/2010_IMO_Problems/Problem_5 | Each of the six boxes \(B_1\), \(B_2\), \(B_3\), \(B_4\), \(B_5\), \(B_6\) initially contains one coin. The following operations are allowed
Type 1) Choose a non-empty box \(B_j\), \(1\leq j \leq 5\), remove one coin from \(B_j\) and add two coins to \(B_{j+1}\);
Type 2) Choose a non-empty box \(B_k\), \(1\leq k \leq... | [
"Let the notation \\([a_1,a_2,a_3,a_4,a_5,a_6]\\) be the configuration in which the \\(x\\)-th box has \\(a_x\\) coin,\n\nLet \\(T=2010^{2010^{2010}}\\).\n\nOur starting configuration is \\([1,1,1,1,1,1]\\)\n\nCompound move 1: \\([a,0]\\rightarrow[0,2a]\\), this is just repeated type 1 move on all \\(a\\) coins.\n\... |
IMO-2010-6 | https://artofproblemsolving.com/wiki/index.php/2010_IMO_Problems/Problem_6 | Let \(a_1, a_2, a_3\) be a sequence of positive real numbers, and \(s\) be a positive integer, such that
\[
a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.
\]
Prove there exist positive integers \(\ell \leq s\) and \(N\), such that
\[
a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for a... | [
"So for solving This Problem, we need to take a assumption that,\n\n- \\(n = t_{1}+...+t_{s}\\)\n- \\(t_{i} \\text{is divisibe by i for every i}\\)\n\n - \\(\\text{there exist indices i} \\not\\ge \\text{j with i+j} \\not\\le \\text{s,}t_{i}\\not\\le\\text{i and }t_{j} \\ge \\text{j}\\)\n - \\(\\text{there exists... |
IMO-2011-1 | https://artofproblemsolving.com/wiki/index.php/2011_IMO_Problems/Problem_1 | Given any set \(A = \{a_1, a_2, a_3, a_4\}\) of four distinct positive integers, we denote the sum \(a_1 +a_2 +a_3 +a_4\) by \(s_A\). Let \(n_A\) denote the number of pairs \((i, j)\) with \(1 \leq i < j \leq 4\) for which \(a_i +a_j\) divides \(s_A\). Find all sets \(A\) of four distinct positive integers which achiev... | [
"Firstly, if we order \\(a_1 \\ge a_2 \\ge a_3 \\ge a_4\\), we see \\(2(a_3 + a_4) \\ge (a_1+a_2)+(a_3+a_4) = s_A \\geq 0\\), so \\((a_3, a_4)\\) isn't a couple that satisfies the conditions of the problem. Also, \\(2(a_4 + a_2) = (a_4 + a_4) + (a_2 + a_2) \\ge (a_4+a_3)+(a_2+a_1) = s_A \\ge 0\\), so again \\((a_2,... |
IMO-2011-2 | https://artofproblemsolving.com/wiki/index.php/2011_IMO_Problems/Problem_2 | Let \(\mathcal{S}\) be a finite set of at least two points in the plane. Assume that no three points of \(\mathcal S\) are collinear. A windmill is a process that starts with a line \(\ell\) going through a single point \(P \in \mathcal S\). The line rotates clockwise about the pivot \(P\) until the first time that the... | [
"Choose a coordinate system so that all points in \\(\\mathcal S\\) have distinct x-coordinates. Number the points \\(P_i=(x_i,y_i)\\) of \\(\\mathcal S\\) by increasing x-coordinates: \\(x_1<x_2<\\ldots<x_N\\).\n\nIn order to divide the set \\(\\mathcal S\\) into two halves, define \\(n\\) so that \\(N=2n+1+d\\) w... |
IMO-2011-3 | https://artofproblemsolving.com/wiki/index.php/2011_IMO_Problems/Problem_3 | Let \(f: \mathbb R \to \mathbb R\) be a real-valued function defined on the set of real numbers that satisfies
\[
f(x + y) \le yf(x) + f(f(x))
\]
for all real numbers \(x\) and \(y\). Prove that \(f(x) = 0\) for all \(x \le 0\). | [
"Let \\(P(x,y)\\) be the given assertion. Comparing \\(P(x,f(y)-x)\\) and \\(P(y,f(x)-y)\\) yields,\n\n\\[\nxf(x)+yf(y)\\leq 2f(x)f(y).\n\\]\n\n\\(y\\mapsto 2f(x)\\implies xf(x)\\leq 0. \\qquad (*)\\)\n\n\\[\n\\textbf{Claim: }f(k)\\leq 0~~\\forall k.\n\\]\n\n\\(Proof.\\) Suppose \\(\\exists k:f(k)>0,\\) then\n\n\\[... |
IMO-2011-4 | https://artofproblemsolving.com/wiki/index.php/2011_IMO_Problems/Problem_4 | Let \(n > 0\) be an integer. We are given a balance and \(n\) weights of weight \(2^0,2^1, \cdots ,2^{n-1}\). We are to place each of the \(n\) weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been ... | [
"Call our answer \\(W(n)\\). We proceed to prove \\(W(n)=(2n-1)!!\\).\n\nIt is evident \\(W(1)=1\\).\n\nNow, the key observation is that smaller weights can never add up to the weight of a larger weight, ie which side is heavier is determined completely by the heaviest weight currently placed. It follows, therefore... |
IMO-2011-5 | https://artofproblemsolving.com/wiki/index.php/2011_IMO_Problems/Problem_5 | Let \(f\) be a function from the set of integers to the set of positive integers. Suppose that, for any two integers \(m\) and \(n\), the difference \(f(m) - f(n)\) is divisible by \(f(m - n)\). Prove that, for all integers \(m\) and \(n\) with \(f(m) \leq f(n)\), the number \(f(n)\) is divisible by \(f(m)\). | [
"Solution 1 Let \\(f\\) be a function from the set of integers to the set of positive integers. Suppose that, for any two integers \\(m\\) and \\(n\\), the difference \\(f(m) - f(n)\\) is divisible by \\(f(m - n)\\). Prove that, for all integers \\(m\\) and \\(n\\) with \\(f(m) \\leq f(n)\\), the number \\(f(n)\\) ... |
IMO-2011-6 | https://artofproblemsolving.com/wiki/index.php/2011_IMO_Problems/Problem_6 | Let \(ABC\) be an acute triangle with circumcircle \(\Gamma\). Let \(\ell\) be a tangent line to \(\Gamma\), and let \(\ell_a, \ell_b\) and \(\ell_c\) be the lines obtained by reflecting \(\ell\) in the lines \(BC\), \(CA\) and \(AB\), respectively. Show that the circumcircle of the triangle determined by the lines \(\... | [
"Without loss of generality, let \\(\\Gamma\\) be the unit circle and let \\(\\ell\\) be the line \\(y=1\\).\n\nDenote the coordinates of \\(A,B,C\\) by \\((x_a, y_a)\\) and similarly for B and C.\n\nWe get \\(x_a^2+y_a^2=1\\), ...\n\nThe equation for line \\(AB\\) is \\(\\frac{x-x_a}{y-y_a}=\\frac{x-x_b}{y-y_b}\\)... |
IMO-2012-1 | https://artofproblemsolving.com/wiki/index.php/2012_IMO_Problems/Problem_1 | Given triangle \(ABC\) the point \(J\) is the centre of the excircle opposite the vertex \(A.\) This excircle is tangent to the side \(BC\) at \(M\), and to the lines \(AB\) and \(AC\) at \(K\) and \(L\), respectively. The lines \(LM\) and \(BJ\) meet at \(F\), and the lines \(KM\) and \(CJ\) meet at \(G.\) Let \(S\) b... | [
"First, \\(BK = BM\\) because \\(BK\\) and \\(BM\\) are both tangents from \\(B\\) to the excircle \\(J\\). Then \\(BJ \\bot KM\\). Call the \\(X\\) the intersection between \\(BJ\\) and \\(KM\\). Similarly, let the intersection between the perpendicular line segments \\(CJ\\) and \\(LM\\) be \\(Y\\). We have \\(\\... |
IMO-2012-2 | https://artofproblemsolving.com/wiki/index.php/2012_IMO_Problems/Problem_2 | Let \({{a}_{2}}, {{a}_{3}}, \cdots, {{a}_{n}}\) be positive real numbers that satisfy \({{a}_{2}}\cdot {{a}_{3}}\cdots {{a}_{n}}=1\) . Prove that
\[
\left(a_2+1\right)^2\cdot \left(a_3+1\right)^3\cdots \left(a_n+1\right)^n\gneq n^n
\] | [
"The inequality between arithmetic and geometric mean implies\n\n\\[\n{{\\left( {{a}_{k}}+1 \\right)}^{k}}={{\\left( {{a}_{k}}+\\frac{1}{k-1}+\\frac{1}{k-1}+\\cdots +\\frac{1}{k-1} \\right)}^{k}}\\ge {{k}^{k}}\\cdot {{a}_{k}}\\cdot \\frac{1}{{{\\left( k-1 \\right)}^{k-1}}}=\\frac{{{k}^{k}}}{{{\\left( k-1 \\right)}^... |
IMO-2012-3 | https://artofproblemsolving.com/wiki/index.php/2012_IMO_Problems/Problem_3 | The liar’s guessing game is a game played between two players A and B. The rules of the game depend on two positive integers \(k\) and \(n\) which are known to both players.
At the start of the game the player A chooses integers \(x\) and \(N\) with \(1 \le x \le N\). Player A keeps \(x\) secret, and truthfully tells ... | [
"It suffices to show that there is a winning strategy for \\(n = 2^k-1\\), as a winning strategy for any \\(n \\ge 2^k\\) easily gives a winning strategy for \\(n = 2^k-1\\).\n\nPlayer B first divides all integers \\(1\\) through \\(N\\) into \\(2^k\\) nonempty sets \\(T_s\\), where \\(s\\) is some binary string of... |
IMO-2012-4 | https://artofproblemsolving.com/wiki/index.php/2012_IMO_Problems/Problem_4 | Find all functions \(f: \mathbb{Z} \to \mathbb{Z}\) such that, for all integers \(a, b,\) and \(c\) that satisfy \(a + b+ c = 0\), the following equality holds:
\[
f(a)^2 + f(b)^2 + f(c)^2 = 2f(a)f(b) + 2f(b)f(c) + 2f(c)f(a).
\]
(Here \(\mathbb{Z}\) denotes the set of integers.) | [
"Consider \\(a = b = c = 0.\\) Then \\(f(0)^2 + f(0)^2 + f(0)^2 = 2f(0)f(0) + 2f(0)f(0) + 2f(0)f(0) \\Rightarrow 3f(0)^2 = 6f(0)^2 \\Rightarrow\\)\n\n\\[\nf(0) = 0.\n\\]\n\nNow we look at \\(b = -a, c = 0.\\) \\(f(a)^2 + f(-a)^2 + f(0)^2 = 2f(a)f(-a) + 2f(-a)f(0) + 2f(0)f(a) \\Rightarrow\\) \\(f(a)^2 + f(-a)^2 = 2f... |
IMO-2012-5 | https://artofproblemsolving.com/wiki/index.php/2012_IMO_Problems/Problem_5 | Let \(ABC\) be a triangle with \(\angle BCA=90^{\circ}\), and let \(D\) be the foot of the altitude from \(C\). Let \(X\) be a point in the interior of the segment \(CD\). Let \(K\) be the point on the segment \(AX\) such that \(BK=BC\). Similarly, let \(L\) be the point on the segment \(BX\) such that \(AL=AC\). Let \... | [
"Let \\(\\Gamma\\), \\(\\Gamma'\\), \\(\\Gamma''\\) be the circumcircle of triangle \\(ABC\\), the circle with its center as \\(A\\) and radius as \\(AC\\), and the circle with its center as \\(B\\) and radius as \\(BC\\), Respectively. Since the center of \\(\\Gamma\\) lies on \\(BC\\), the three circles above are... |
IMO-2012-6 | https://artofproblemsolving.com/wiki/index.php/2012_IMO_Problems/Problem_6 | Find all positive integers \(n\) for which there exist non-negative integers \(a_1, a_2, \ldots, a_n\) such that
\[
\frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} = \frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}} = 1.
\] | [] |
IMO-2013-1 | https://artofproblemsolving.com/wiki/index.php/2013_IMO_Problems/Problem_1 | Prove that for any pair of positive integers \(k\) and \(n\), there exist \(k\) positive integers \(m_1,m_2,...,m_k\) (not necessarily different) such that
\(1+\frac{2^k-1}{n}=(1+\frac{1}{m_1})(1+\frac{1}{m_2})...(1+\frac{1}{m_k})\). | [
"We prove the claim by induction on \\(k\\).\n\nBase case: If \\(k = 1\\) then \\(1 +\\frac{2^1-1}{n} = 1 + \\frac{1}{n}\\), so the claim is true for all positive integers \\(n\\).\n\nInductive hypothesis: Suppose that for some \\(m \\in \\mathbb{Z}^{+}\\) the claim is true for \\(k = m\\), for all \\(n \\in \\math... |
IMO-2013-2 | https://artofproblemsolving.com/wiki/index.php/2013_IMO_Problems/Problem_2 | A configuration of \(4027\) points in the plane is called Colombian if it consists of \(2013\) red points and \(2014\) blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configurati... | [
"We can start off by imagining the points in their worst configuration. With some trials, we find \\(2013\\) lines to be the answer to the worst cases. We can assume the answer is \\(2013\\). We will now prove it.\n\nWe will first prove that the sufficient number of lines required for a good arrangement for a confi... |
IMO-2013-3 | https://artofproblemsolving.com/wiki/index.php/2013_IMO_Problems/Problem_3 | Let the excircle of triangle \(ABC\) opposite the vertex \(A\) be tangent to the side \(BC\) at the point \(A_1\). Define the points \(B_1\) on \(CA\) and \(C_1\) on \(AB\) analogously, using the excircles opposite \(B\) and \(C\), respectively. Suppose that the circumcentre of triangle \(A_1B_1C_1\) lies on the circum... | [
"\\[\n[asy] unitsize(2.5cm); void b() { pointpen=black; pathpen=rgb(0.4,0.6,0.8); pointfontpen=fontsize(10); pen dd=linetype(\"4 8\"); /* Define the excenter */ pair excenter(pair A=(0,0), pair B=(0,0), pair C=(0,0)) { return extension(A,bisectorpoint(C,A,B),B,rotate(90,B)*bisectorpoint(A,B,C)); } /* Draw points *... |
IMO-2013-4 | https://artofproblemsolving.com/wiki/index.php/2013_IMO_Problems/Problem_4 | Let \(ABC\) be an acute triangle with orthocenter \(H\), and let \(W\) be a point on the side \(BC\), lying strictly between \(B\) and \(C\). The points \(M\) and \(N\) are the feet of the altitudes from \(B\) and \(C\), respectively. Denote by \(\omega_1\) is the circumcircle of \(BWN\), and let \(X\) be the point on ... | [
"\\[\n[asy] //Original diagram by suli, August 2014. Feel free to make edits, but please leave this comment in place. import olympiad; import math; unitsize(10); pair A = (15,25), B = (0,0), C = (20,0), W = (12,0); pair H = orthocenter(A, B, C); pair N = extension(C,H, A,B); pair M = extension(B,H, A,C); pair L = e... |
IMO-2013-5 | https://artofproblemsolving.com/wiki/index.php/2013_IMO_Problems/Problem_5 | Let \(\mathbb Q_{>0}\) be the set of all positive rational numbers. Let \(f:\mathbb Q_{>0}\to\mathbb R\) be a function satisfying the following three conditions:
(i) for all \(x,y\in\mathbb Q_{>0}\), we have \(f(x)f(y)\geq f(xy)\); (ii) for all \(x,y\in\mathbb Q_{>0}\), we have \(f(x+y)\geq f(x)+f(y)\); (iii) there ex... | [] |
IMO-2013-6 | https://artofproblemsolving.com/wiki/index.php/2013_IMO_Problems/Problem_6 | Let \(n \ge 3\) be an integer, and consider a circle with \(n + 1\) equally spaced points marked on it. Consider all labellings of these points with the numbers \(0, 1, ... , n\) such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotat... | [] |
IMO-2014-1 | https://artofproblemsolving.com/wiki/index.php/2014_IMO_Problems/Problem_1 | Let \(a_0<a_1<a_2<\cdots \quad\) be an infinite sequence of positive integers, Prove that there exists a unique integer \(n\ge1\) such that
\[
a_n<\frac{a_0+a_1+\cdots + a_n}{n}\le a_{n+1}.
\] | [
"Define \\(f(n) = a_0 + a_1 + \\dots + a_n - n a_{n+1}\\). (In particular, \\(f(0) = a_0.\\)) Notice that because \\(a_{n+2} \\ge a_{n+1}\\), we have\n\n\\[\na_0 + a_1 + \\dots + a_n - n a_{n+1} > a_0 + a_1 + \\dots + a_n + a_{n+1} - (n+1) a_{n+2}.\n\\]\n\nThus, \\(f(n) > f(n+1)\\); i.e., \\(f\\) is monotonic decre... |
IMO-2014-2 | https://artofproblemsolving.com/wiki/index.php/2014_IMO_Problems/Problem_2 | Let \(n\ge2\) be an integer. Consider an \(n\times n\) chessboard consisting of \(n^2\) unit squares. A configuration of \(n\) rooks on this board is \(\textit{peaceful}\) if every row and every column contains exactly one rook. Find the greatest positive integer \(k\) such that, for each peaceful configuration of \(n\... | [
"We claim the answer is \\(k = \\lceil \\sqrt{n}\\rceil - 1\\), where \\(\\lceil n\\rceil\\) is the ceiling function of \\(n\\); i.e., the least integer greater than or equal to \\(n\\). Notice that \\(\\lceil n\\rceil < n + 1\\).\n\nFirst, we shall show that each \\(n \\times n\\) chessboard with a peaceful config... |
IMO-2014-3 | https://artofproblemsolving.com/wiki/index.php/2014_IMO_Problems/Problem_3 | Convex quadrilateral \(ABCD\) has \(\angle{ABC}=\angle{CDA}=90^{\circ}\). Point \(H\) is the foot of the perpendicular from \(A\) to \(BD\). Points \(S\) and \(T\) lie on sides \(AB\) and \(AD\), respectively, such that \(H\) lies inside \(\triangle{SCT}\) and
\[
\angle{CHS}-\angle{CSB}=90^{\circ},\quad \angle{THC}-\a... | [
"\\[\n[asy] import cse5; import graph; import olympiad; dotfactor = 3; unitsize(1.5inch); path circle = Circle(origin, 1); // draw(circle); pair A = (0,1), C=(0,-1); pair Oo = (0,-0.05); pair Bb = rotate(-8,Oo)*(2,-0.05), Dd =rotate(-8, Oo)*(-2,-0.05); pair B = IP(Dd--Bb, circle, 1); pair D = IP(Dd--Bb, circle, 0... |
IMO-2014-4 | https://artofproblemsolving.com/wiki/index.php/2014_IMO_Problems/Problem_4 | Points \(P\) and \(Q\) lie on side \(BC\) of acute-angled \(\triangle{ABC}\) so that \(\angle{PAB}=\angle{BCA}\) and \(\angle{CAQ}=\angle{ABC}\). Points \(M\) and \(N\) lie on lines \(AP\) and \(AQ\), respectively, such that \(P\) is the midpoint of \(AM\), and \(Q\) is the midpoint of \(AN\). Prove that lines \(BM\) a... | [
"\\[\n[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(10.60000000000002cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */... |
IMO-2014-5 | https://artofproblemsolving.com/wiki/index.php/2014_IMO_Problems/Problem_5 | For each positive integer \(n\), the Bank of Cape Town issues coins of denomination \(\tfrac{1}{n}\). Given a finite collection of such coins (of not necessarily different denominations) with total value at most \(99+\tfrac{1}{2}\), prove that it is possible to split this collection into \(100\) or fewer groups, such t... | [
"The bound is not tight. We'll prove the result for at most \\(k - \\frac{k}{2k+1}\\) with \\(k\\) groups.\n\nFirst, perform the following optimizations. - If any coin of size \\(\\frac{1}{2m}\\) appears twice, then replace it with a single coin of size \\(\\frac{1}{m}\\). - If any coin of size \\(\\frac{1}{2m+1}\\... |
IMO-2014-6 | https://artofproblemsolving.com/wiki/index.php/2014_IMO_Problems/Problem_6 | A set of lines in the plane is in \(\textit{general position}\) if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its \(\textit{finite regions}\). Prove that for all sufficiently large \(n\), in ... | [
"Call the \\(\\binom{n}{2}\\) intersections, well, points. Then each line will have \\(n-1\\) points. We call 2 points (on a line) neighbors if there are no other points on the line segment joining those 2. Then each finite region has to be a convex polygon whose any pair of neighboring vertices are, well, neighbor... |
IMO-2015-1 | https://artofproblemsolving.com/wiki/index.php/2015_IMO_Problems/Problem_1 | We say that a finite set \(\mathcal{S}\) in the plane is balanced if, for any two different points \(A\), \(B\) in \(\mathcal{S}\), there is a point \(C\) in \(\mathcal{S}\) such that \(AC=BC\). We say that \(\mathcal{S}\) is centre-free if for any three points \(A\), \(B\), \(C\) in \(\mathcal{S}\), there is no point ... | [
"Part (a): We explicitly construct the sets \\(\\mathcal{S}\\). For odd \\(n\\), \\(\\mathcal{S}\\) can be taken to be the vertices of regular polygons \\(P_n\\) with \\(n\\) sides: given any two vertices \\(A\\) and \\(B\\), one of the two open half-spaces into which \\(AB\\) divides \\(P_n\\) contains an odd numb... |
IMO-2015-2 | https://artofproblemsolving.com/wiki/index.php/2015_IMO_Problems/Problem_2 | Determine all triples of positive integers \((a,b,c)\) such that each of the numbers
\[
ab-c,\; bc-a,\; ca-b
\]
is a power of 2.
(A power of 2 is an integer of the form \(2^n\) where \(n\) is a non-negative integer ). | [
"The solutions for \\((a,b,c)\\) are \\((2,2,2)\\), \\((2,2,3)\\), \\((2,6,11)\\), \\((3,5,7)\\), and permutations of these triples.\n\nThroughout the proof, we assume \\(a \\leq b \\leq c\\), so that \\(ab-c = 2^m\\), \\(ca-b = 2^n\\), \\(bc-a=2^p\\), with \\(m \\leq n \\leq p\\). Note that \\(a>1\\) since otherwi... |
IMO-2015-3 | https://artofproblemsolving.com/wiki/index.php/2015_IMO_Problems/Problem_3 | Let \(ABC\) be an acute triangle with \(AB>AC\). Let \(\Gamma\) be its circumcircle, \(H\) its orthocenter, and \(F\) the foot of the altitude from \(A\). Let \(M\) be the midpoint of \(BC\). Let \(Q\) be the point on \(\Gamma\) such that \(\angle HKQ=90^\circ\). Assume that the points \(A\), \(B\), \(C\), \(K\), and \... | [
"\\[\nup\n\\]\n\nWe know that there is a negative inversion which is at \\(H\\) and swaps the nine-point circle and \\(\\Gamma\\). And this maps:\n\n\\(A \\longleftrightarrow F\\). Also, let \\(M \\longleftrightarrow Q`\\). Of course \\(\\triangle HFM \\sim \\triangle HQ'A\\) so \\(\\angle HQ'A = 90\\). Hence, \\(Q... |
IMO-2015-4 | https://artofproblemsolving.com/wiki/index.php/2015_IMO_Problems/Problem_4 | Triangle \(ABC\) has circumcircle \(\Omega\) and circumcenter \(O\). A circle \(\Gamma\) with center \(A\) intersects the segment \(BC\) at points \(D\) and \(E\), such that \(B\), \(D\), \(E\), and \(C\) are all different and lie on line \(BC\) in this order. Let \(F\) and \(G\) be the points of intersection of \(\Gam... | [
"Lemma (On three chords). If two lines pass through different endpoints of two circles' common chord, then the other two chords cut by these lines on the circles are parallel. Proof The second and the third chords are anti-parallel to the first (common) chord with respect to the given lines, so they are parallel to... |
IMO-2015-5 | https://artofproblemsolving.com/wiki/index.php/2015_IMO_Problems/Problem_5 | Let \(\mathbb{R}\) be the set of real numbers. Determine all functions \(f\):\(\mathbb{R}\rightarrow\mathbb{R}\) satisfying the equation
\[
f(x+f(x+y))+f(xy) = x+f(x+y)+yf(x)
\]
for all real numbers \(x\) and \(y\).
Proposed by Dorlir Ahmeti, Albania | [
"\\(f(x+f(x+y)) + f(xy) = x + f(x+y) + yf(x)\\) for all real numbers \\(x\\) and \\(y\\).\n\n(1) Put \\(x=y=0\\) in the equation, We get\\(f(0 + f(0)) + f(0) = 0 + f(0) + 0\\) or \\(f(f(0)) = 0\\) Let \\(f(0) = k\\), then \\(f(k) = 0\\)\n\n(2) Put \\(x=0, y=k\\) in the equation, We get \\(f(0 + f(k)) + f(0) = 0 + f... |
IMO-2015-6 | https://artofproblemsolving.com/wiki/index.php/2015_IMO_Problems/Problem_6 | The sequence \(a_1,a_2,\dots\) of integers satisfies the conditions: (i) \(1\le a_j\le2015\) for all \(j\ge1\), (ii) \(k+a_k\neq \ell+a_\ell\) for all \(1\le k<\ell\). Prove that there exist two positive integers \(b\) and \(N\) for which
\[
\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2
\]
for all integers \(m\... | [
"We can prove the more general statement. Theorem Let \\(T\\) be a non-negative integer parameter. If given a sequence \\(a_1,a_2,\\dots\\) that satisfies the conditions: (i) \\(1 \\le a_j \\le T+1\\) for all \\(j \\le 1;\\) (ii) \\(k+a_k \\ne \\ell+a_\\ell\\) for all \\(1 \\le k<\\ell,\\) then there exist two inte... |
IMO-2016-1 | https://artofproblemsolving.com/wiki/index.php/2016_IMO_Problems/Problem_1 | Triangle \(BCF\) has a right angle at \(B\). Let \(A\) be the point on line \(CF\) such that \(FA=FB\) and \(F\) lies between \(A\) and \(C\). Point \(D\) is chosen so that \(DA=DC\) and \(AC\) is the bisector of \(\angle{DAB}\). Point \(E\) is chosen so that \(EA=ED\) and \(AD\) is the bisector of \(\angle{EAC}\). Let... | [
"\\[\n2016IMOQ1Solution.jpg\n\\]\n\nThe Problem shows that \\(\\angle DAC = \\angle DCA = \\angle CAD\\), it follows that \\(AB \\parallel CD\\). Extend \\(DC\\) to intersect \\(AB\\) at \\(G\\), we get \\(\\angle GFA = \\angle GFB = \\angle CFD\\). Making triangles \\(\\triangle CDF\\) and \\(\\triangle AGF\\) sim... |
IMO-2016-2 | https://artofproblemsolving.com/wiki/index.php/2016_IMO_Problems/Problem_2 | Find all integers \(n\) for which each cell of \(n \times n\) table can be filled with one of the letters \(I,M\) and \(O\) in such a way that:
Note. The rows and columns of an \(n \times n\) table are each labelled \(1\) to \(n\) in a natural order. Thus each cell corresponds to a pair of positive integer \((i,j)\) w... | [
"Here is a solution using counting in two ways.\n\nIt's obvious that \\(3 \\mid n\\). We consider all the squares indexed \\((3k+2,3l+2)\\) and call it [i]good[/i] square. Let \\(a\\) be number of [i]good[/i] squares that are filled with \\(I\\). We can see that every good square lies on both type of diagonals. So ... |
IMO-2016-3 | https://artofproblemsolving.com/wiki/index.php/2016_IMO_Problems/Problem_3 | Let \(P = A_1A_2 \cdots A_k\) be a convex polygon in the plane. The vertices \(A_1,A_2,\dots, A_k\) have integral coordinates and lie on a circle. Let \(S\) be the area of \(P\). An odd positive integer \(n\) is given such that the squares of the side lengths of \(P\) are integers divisible by \(n\). Prove that \(2S\) ... | [
"Note that \\(2S\\) is always an integer for any lattice polygon, so it remains to show that it is divisible by \\(n\\). It clearly suffices to prove the problem for when \\(n=p^m\\) is a prime power. We proceed using induction on \\(k\\), with the base case of \\(k=3\\) settled by Heron's formula: If \\(a,b,c\\) a... |
IMO-2016-4 | https://artofproblemsolving.com/wiki/index.php/2016_IMO_Problems/Problem_4 | A set of positive integers is called fragrant if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let \(P(n)=n^2+n+1\). What is the least possible positive integer value of \(b\) such that there exists a non-negative integer \(a\) for which... | [
"Consider \\(P(x)\\) and \\(P(x+y)\\). We note that in order to \\(p \\mid P(x)\\) and \\(p \\mid P(x+y)-P(x)\\) we must have \\(p \\mid x^2+x+1\\) and \\(p \\mid y(2x+y+1)\\). It is obvious that \\(p \\equiv 1 \\pmod{6}\\) since \\(p \\mid n^2+n+1 \\mid (2n+1)^2+3\\) or \\(\\left( \\tfrac{-3}{p} \\right)=1\\).\n\n... |
IMO-2016-5 | https://artofproblemsolving.com/wiki/index.php/2016_IMO_Problems/Problem_5 | The equation
\[
(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)
\]
is written on the board, with \(2016\) linear factors on each side. What is the least possible value of \(k\) for which it is possible to erase exactly \(k\) of these \(4032\) linear factors so that at least one factor remains on each side and the ... | [] |
IMO-2016-6 | https://artofproblemsolving.com/wiki/index.php/2016_IMO_Problems/Problem_6 | There are \(n\ge 2\) line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands \(n-1\) times. Every time he claps,each frog will immediately jump forwa... | [
"Beautiful problem. Despite being placed a bit higher in both the Shortlist and the actual exam, kudos to the proposer for involving combi, geo and NT in one single problem. Anyway, here's my solution: Color each of the \\(2n\\) endpoints [color=#00f]blue[/color] and all the \\(\\binom{n}{2}\\) intersection points ... |
IMO-2017-1 | https://artofproblemsolving.com/wiki/index.php/2017_IMO_Problems/Problem_1 | For each integer \(a_0 > 1\), define the sequence \(a_0, a_1, a_2, \ldots\) for \(n \geq 0\) as
\[
a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases}
\]
Determine all values of \(a_0\) such that there exists a number \(A\) such that \(a_n = A\... | [
"First we observe the following:\n\nWhen we start with \\(a_0=3\\), we get \\(a_1=6\\), \\(a_2=9\\), \\(a_3=3\\) and the pattern \\(3,6,9\\) repeats.\n\nWhen we start with \\(a_0=6\\), we get \\(a_1=9\\), \\(a_2=3\\), \\(a_3=6\\) and the pattern \\(3,6,9\\) repeats.\n\nWhen we start with \\(a_0=9\\), we get \\(a_1=... |
IMO-2017-2 | https://artofproblemsolving.com/wiki/index.php/2017_IMO_Problems/Problem_2 | Let \(\mathbb{R}\) be the set of real numbers , determine all functions \(f:\mathbb{R}\rightarrow\mathbb{R}\) such that for any real numbers \(x\) and \(y\)
\[
f(f(x)f(y)) + f(x+y)=f(xy)
\] | [
"Looking at the equation one can deduce that the functions that will work will be linear. That is, a polynomial of at most a degree of 1.\n\nThus, \\(f\\) is in the form \\(f(x)=mx+b\\)\n\nTherefore,\n\n\\[\nf((mx+b)(my+b))+m(x+y)+b=mxy+b\n\\]\n\n\\[\nf(m^2xy+mb(x+y)+b^2)+m(x+y)+b=mxy+b\n\\]\n\n\\[\nm(m^2xy+mb(x+y)... |
IMO-2017-3 | https://artofproblemsolving.com/wiki/index.php/2017_IMO_Problems/Problem_3 | A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, \(A_0\), and the hunter's starting point, \(B_0\), are the same. After \(n-1\) rounds of the game, the rabbit is at point \(A_{n-1}\) and the hunter is at point \(B_{n-1}\). In the nth round of the game, three things occur... | [
"Answer: No. There is no such strategy for the hunter. The rabbit will always “Win”\n\nProof: Suppose on the contrary that the answer is Yes. Therefore, there exists a strategy for the hunter to always “win” no matter how the rabbit moved or how the radar pinged. We will show that with bad luck from radar pings, th... |
IMO-2017-4 | https://artofproblemsolving.com/wiki/index.php/2017_IMO_Problems/Problem_4 | Let \(R\) and \(S\) be different points on a circle \(\Omega\) such that \(RS\) is not a diameter. Let \(\ell\) be the tangent line to \(\Omega\) at \(R\). Point \(T\) is such that \(S\) is the midpoint of the line segment \(RT\). Point \(J\) is chosen on the shorter arc \(RS\) of \(\Omega\) so that the circumcircle \(... | [
"We construct inversion which maps \\(KT\\) into the circle \\(\\omega_1\\) and \\(\\Gamma\\) into \\(\\Gamma.\\) Than we prove that \\(\\omega_1\\) is tangent to \\(\\Gamma.\\)\n\nQuadrangle \\(RJSK\\) is cyclic \\(\\implies \\angle RSJ = \\angle RKJ.\\)\n\nQuadrangle \\(AJST\\) is cyclic \\(\\implies \\angle RSJ ... |
IMO-2017-5 | https://artofproblemsolving.com/wiki/index.php/2017_IMO_Problems/Problem_5 | An integer \(N \ge 2\) is given. A collection of \(N(N + 1)\) soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove \(N(N - 1)\) players from this row leaving a new row of \(2N\) players in which the following \(N\) conditions hold:
(\(1\)) no one stands between the two talles... | [
"The answer is \\(\\boxed{2016}\\). Clearly, if we erase less than \\(2016\\) terms, then some term will appear on both sides by the pigeonhole principle, thereby causing a real root. Now, we show how we can erase \\(2016\\) terms and have no real roots.\n\nLet \\(f(x)=(x-1)(x-4)\\) and \\(g(x)=(x-2)(x-3)\\). It is... |
IMO-2017-6 | https://artofproblemsolving.com/wiki/index.php/2017_IMO_Problems/Problem_6 | An ordered pair \((x, y)\) of integers is a primitive point if the greatest common divisor of \(x\) and \(y\) is \(1\). Given a finite set \(S\) of primitive points, prove that there exist a positive integer \(n\) and integers \(a_0, a_1, \ldots , a_n\) such that, for each \((x, y)\) in \(S\), we have:
\[
a_0x^n + a_1... | [
"The proof goes by induction in the number \\(n=|S|\\) of points of the set. The base case is trivial by Bézout's Theorem. Write \\(S =\\{(a_1,b_1)., \\ldots, (a_n,b_n), (a_{n_1},b_{n+1})\\}\\) and let \\(g(x,y)\\) be a homogeneous polynomial of degree \\(\\ell\\) such that \\(g(a_i,b_i)=1\\) for \\(1 \\leq i \\leq... |
IMO-2018-1 | https://artofproblemsolving.com/wiki/index.php/2018_IMO_Problems/Problem_1 | Let \(\Gamma\) be the circumcircle of acute triangle \(ABC\). Points \(D\) and \(E\) are on segments \(AB\) and \(AC\) respectively such that \(AD = AE\). The perpendicular bisectors of \(BD\) and \(CE\) intersect minor arcs \(AB\) and \(AC\) of \(\Gamma\) at points \(F\) and \(G\) respectively. Prove that lines \(DE\)... | [
"http://wiki-images.artofproblemsolving.com/5/5d/FB_IMG_1531446409131.jpg\n\nThe diagram is certainly not to scale, but the argument is sound (I believe) and involves re-ordering the construction as specified in the original problem so that an identical state of affairs results, yet in so doing differently it is ma... |
IMO-2018-2 | https://artofproblemsolving.com/wiki/index.php/2018_IMO_Problems/Problem_2 | Find all numbers \(n \ge 3\) for which there exists real numbers \(a_1, a_2, ..., a_{n+2}\) satisfying \(a_{n+1} = a_1, a_{n+2} = a_2\) and
\[
a_{i}a_{i+1} + 1 = a_{i+2}
\]
for \(i = 1, 2, ..., n.\) | [
"We find at least one series of real numbers for \\(n = 3,\\) for each \\(n = 3k\\) and we prove that if \\(n = 3k \\pm 1,\\) then the series does not exist.\n\nCase 1\n\nLet \\(n = 3.\\) We get system of equations\n\n\\[\n\\begin{cases} a_1 a_2 + 1 = a_3 \\\\a_2 a_3 + 1 = a_1 \\\\a_3 a_1 + 1 = a_2 \\end{cases}\n\\... |
IMO-2018-3 | https://artofproblemsolving.com/wiki/index.php/2018_IMO_Problems/Problem_3 | An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from \... | [
"Trivially it is required that every positive integer from \\(1\\) to \\(1+2+3+\\cdots+2018\\) appears exactly once.\n\nLet \\(M_n\\) denote the maximum number in the \\(n\\)th row and let \\(m_n\\) denote the minimum number in the \\(n\\)th row.\n\nNow assume \\(n\\leq 2017\\) and consider the numbers directly bel... |
IMO-2018-4 | https://artofproblemsolving.com/wiki/index.php/2018_IMO_Problems/Problem_4 | A site is any point \((x, y)\) in the plane such that \(x\) and \(y\) are both positive integers less than or equal to 20. Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance betw... | [
"The maximal K is 100. Amy can reach at least 100 by playing only on sites for which x+y is even. There are 200 such sites, none are of distance \\(\\sqrt{5}\\) from each other and Ben can occupy at most half of them. On the other hand Ben can prevent Amy from reaching more than 100 using the following strategy: Pi... |
IMO-2018-5 | https://artofproblemsolving.com/wiki/index.php/2018_IMO_Problems/Problem_5 | Let \(a_1, a_2, \dots\) be an infinite sequence of positive integers. Suppose that there is an integer\(N > 1\) such that, for each \(n \geq N\), the number \(\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots +\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}\) is an integer. Prove that there is a positive integer \(M\) such that \(a_m = a_{m+1... | [
"The condition implies that the difference \\(S(n) = \\frac{a_{n+1}}{a_1} - \\frac{a_n}{a_1} + \\frac{a_n}{a_{n+1}}\\) is an integer for all \\(n > N\\). We proceed by \\(p\\)-adic valuation only henceforth.\n\n[b][color=red]Claim:[/color][/b] If \\(p \\nmid a_1\\), then \\(\\nu_p(a_{n+1}) \\le \\nu_p(a_n)\\) for \... |
IMO-2018-6 | https://artofproblemsolving.com/wiki/index.php/2018_IMO_Problems/Problem_6 | A convex quadrilateral \(ABCD\) satisfies \(AB\cdot CD=BC \cdot DA.\) Point \(X\) lies inside \(ABCD\) so that \(\angle XAB = \angle XCD\) and \(\angle XBC = \angle XDA.\) Prove that \(\angle BXA + \angle DXC = 180^{\circ}\) | [
"We want to find the point \\(X.\\) Let \\(E\\) and \\(F\\) be the intersection points of \\(AB\\) and \\(CD,\\) and \\(BC\\) and \\(DA,\\) respectively. The poinx \\(X\\) is inside \\(ABCD,\\) so points \\(E,A,X,C\\) follow in this order.\n\n\\(\\angle XAB = \\angle XCD \\implies \\angle XAE + \\angle XCE = 180^\\... |
IMO-2019-1 | https://artofproblemsolving.com/wiki/index.php/2019_IMO_Problems/Problem_1 | Let \(\mathbb{Z}\) be the set of integers. Determine all functions \(f : \mathbb{Z} \to \mathbb{Z}\) such that, for all integers \(a\) and \(b\),
\[
f(2a) + 2f(b) = f(f(a + b)).
\] | [
"The only solutions are \\(f(x)=0, 2x+c.\\) For some integer \\(c.\\)\n\nObviously these work. We prove these are the only linear solutions. Plug \\(a=0\\) and \\(b=0\\) separately to get that \\(f(2x)=2f(x)-f(0).\\) Plug \\((0, a+b)\\) to see \\(f(0)+f(a+b)=f(a)+f(b),\\) and subtracting \\(2f(0)\\) from both sides... |
IMO-2019-2 | https://artofproblemsolving.com/wiki/index.php/2019_IMO_Problems/Problem_2 | In triangle \(ABC\), point \(A_1\) lies on side \(BC\) and point \(B_1\) lies on side \(AC\). Let \(P\) and \(Q\) be points on segments \(AA_1\) and \(BB_1\), respectively, such that \(PQ\) is parallel to \(AB\). Let \(P_1\) be a point on line \(PB_1\), such that \(B_1\) lies strictly between \(P\) and \(P_1\), and \(\... | [
"The essence of the proof is to build a circle through the points \\(P, Q,\\) and two additional points \\(A_0\\) and \\(B_0,\\) then we prove that the points \\(P_1\\) and \\(Q_1\\) lie on the same circle.\n\nWe assume that the intersection point of \\(AP\\) and \\(BQ\\) lies on the segment \\(PA_1.\\) If it lies ... |
IMO-2019-3 | https://artofproblemsolving.com/wiki/index.php/2019_IMO_Problems/Problem_3 | A social network has \(2019\) users, some pairs of whom are friends. Whenever user \(A\) is friends with user \(B\), user \(B\) is also friends with user \(A\). Events of the following kind may happen repeatedly, one at a time: Three users \(A\), \(B\), and \(C\) such that \(A\) is friends with both \(B\) and \(C\), bu... | [
"Let \\(G\\) be a graph with \\(2019\\) vertices representing the users and edges corresponding to their friendships.\n\nWe have the following properties:\n\n1. \\(G\\) is connected, since two non-adjacent vertices with no common friends will have at least \\(1009+1009=2018\\) vertices neighboring them, which is mo... |
IMO-2019-4 | https://artofproblemsolving.com/wiki/index.php/2019_IMO_Problems/Problem_4 | Find all pairs \((k,n)\) of positive integers such that
\[
k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).
\] | [
"\\[\nLHS\n\\]\n\n\\(k! = 1\\) (when \\(k = 1\\)), \\(2\\) (when \\(k = 2\\)), \\(6\\) (when \\(k = 3\\))\n\n\\(RHS = 1\\)(when \\(n = 1\\)), \\(6\\) (when \\(n = 2\\))\n\nHence, \\((1,1)\\), \\((3,2)\\) satisfy\n\nFor \\(k = 2: RHS\\) is strictly increasing, and will never satisfy \\(k\\) = 2 for integer n since \... |
IMO-2019-5 | https://artofproblemsolving.com/wiki/index.php/2019_IMO_Problems/Problem_5 | The Bank of Bath issues coins with an \(H\) on one side and a \(T\) on the other. Harry has \(n\) of these coins arranged in a line from left to right. He repeatedly performs the following operation:
If there are exactly \(k > 0\) coins showing \(H\), then he turns over the \(k^{th}\) coin from the left; otherwise, al... | [
"Don't worry, this is way simpler than it's length may suggest :)\n\nClaim: The expected value of \\(L(C)\\) is \\(\\frac{n(n+1)}{4}\\).\n\nWe prove parts A and B simultaneously using induction.\n\nBase case: \\(n=1\\)\n\n\\(L(T)=0\\) and \\(L(H)=1\\), and these are the only possibilities, so clearly the process te... |
IMO-2019-6 | https://artofproblemsolving.com/wiki/index.php/2019_IMO_Problems/Problem_6 | Let \(I\) be the incenter of acute triangle \(ABC\) with \(AB \neq AC\). The incircle \(\omega\) of \(ABC\) is tangent to sides \(BC\), \(CA\), and \(AB\) at \(D\), \(E\), and \(F\), respectively. The line through \(D\) perpendicular to \(EF\) meets \(\omega\) again at \(R\). Line \(AR\) meets ω again at \(P\). The cir... | [
"Step 1\n\nWe find an auxiliary point \\(S.\\)\n\nLet \\(G\\) be the antipode of \\(D\\) on \\(\\omega, GD = 2R,\\) where \\(R\\) is radius \\(\\omega.\\)\n\nWe define \\(A' = PG \\cap AI.\\)\n\n\\(RD||AI, PRGD\\) is cyclic \\(\\implies \\angle IAP = \\angle DRP = \\angle DGP.\\)\n\n\\(RD||AI, RD \\perp RG, RI=GI \... |
IMO-2020-1 | https://artofproblemsolving.com/wiki/index.php/2020_IMO_Problems/Problem_1 | Consider the convex quadrilateral \(ABCD\). The point \(P\) is in the interior of \(ABCD\). The following ratio equalities hold:
\[
\angle PAD : \angle PBA : \angle DPA = 1 : 2 : 3 = \angle CBP : \angle BAP : \angle BPC.
\]
Prove that the following three lines meet in a point: the internal bisectors of angles \(\angl... | [
"Let the perpendicular bisector of \\(AP,BP\\) meet at point \\(O\\), those two lines meet at \\(AD,BC\\) at \\(N,M\\) respectively.\n\nAs the problem states, denote that \\(\\angle{PBC}=\\alpha, \\angle{BAP}=2\\alpha, \\angle {BPC}=3\\alpha\\). We can express another triple with \\(\\beta\\) as well. Since the per... |
IMO-2020-2 | https://artofproblemsolving.com/wiki/index.php/2020_IMO_Problems/Problem_2 | The real numbers \(a\), \(b\), \(c\), \(d\) are such that \(a \geq b \geq c \geq d > 0\) and \(a + b + c + d = 1\). Prove that
\[
(a + 2b + 3c + 4d) a^a b^b c^c d^d < 1.
\] | [
"Using Weighted AM-GM we get\n\n\\[\n\\frac{a\\cdot a +b\\cdot b +c\\cdot c +d\\cdot d}{a+b+c+d} \\ge \\sqrt[a+b+c+d]{a^a b^b c^c d^d}\n\\]\n\n\\[\n\\implies a^a b^b c^c d^d \\le a^2 +b^2 +c^2 +d^2\n\\]\n\nSo,\n\n\\[\n(a+2b+3c+4d) a^ab^bc^cd^d \\le (a+2b+3c+4d)(a^2+b^2+c^2+d^2)\n\\]\n\nNow notice that\n\nSo, we get... |
IMO-2020-3 | https://artofproblemsolving.com/wiki/index.php/2020_IMO_Problems/Problem_3 | There are \(4n\) pebbles of weights \(1, 2, 3, . . . , 4n\). Each pebble is colored in one of \(n\) colors and there are four pebbles of each color. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied:
- The total weights of both piles are the same.
- Each pile c... | [] |
IMO-2020-4 | https://artofproblemsolving.com/wiki/index.php/2020_IMO_Problems/Problem_4 | There is an integer \(n > 1\). There are \(n^2\) stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, \(A\) and \(B\), operates \(k\) cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The \(k\) cable cars of \... | [] |
IMO-2020-5 | https://artofproblemsolving.com/wiki/index.php/2020_IMO_Problems/Problem_5 | A deck of \(n > 1\) cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards.
For which \(n\) does it follow that the numbers on the cards are all... | [
"Claim : For all n > 1, all numbers must be equal Contradiction: Let us assume this is not true and for a certain n, there are k distinct positive integers which can be written in ascending order as follows :\n\nzk > zk-1 > zk-2 > … > z1\n\nSince zk is the largest of the numbers, it has to be greater than 1. This i... |
IMO-2020-6 | https://artofproblemsolving.com/wiki/index.php/2020_IMO_Problems/Problem_6 | Prove that there exists a positive constant \(c\) such that the following statement is true:
Consider an integer \(n > 1\), and a set \(S\) of n points in the plane such that the distance between any two different points in \(S\) is at least \(1\). It follows that there is a line \(\ell\) separating \(S\) such that th... | [
"For any unit vector \\(v\\), let \\(a_v=\\min_{p\\in S} p \\cdot v\\) and \\(b_v = \\max_{p\\in S} p\\cdot v\\). If \\(b_v - a_v\\geq n^{2/3}\\) then we can find a line \\(\\ell\\) perpendicular to \\(v\\) such that \\(\\ell\\) separates \\(S\\), and any point in \\(S\\) is at least \\(\\Omega(n^{2/3}/n) = \\Omega... |
IMO-2021-1 | https://artofproblemsolving.com/wiki/index.php/2021_IMO_Problems/Problem_1 | Let \(n \geq 100\) be an integer. Ivan writes the numbers \(n, n+1, \ldots, 2 n\) each on different cards. He then shuffles these \(n+1\) cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square. | [
"If we can guarantee that there exist \\(3\\) cards such that every pair of them sum to a perfect square, then we can guarantee that one of the piles contains \\(2\\) cards that sum to a perfect square. Assume the perfect squares \\(p^2\\), \\(q^2\\), and \\(r^2\\) satisfy the following system of equations:\n\n\\[\... |
IMO-2021-2 | https://artofproblemsolving.com/wiki/index.php/2021_IMO_Problems/Problem_2 | Show that the inequality
\[
\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|} \le \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}
\]
holds for all real numbers \(x_1,x_2,\dots,x_n\). | [
"then,\n\n\\[\n\\sum \\sum x_i^2+x_j^2-2x_ix_j \\leq \\sum \\sum x_i^2+x_j^2+2x_ix_j \\to \\sum \\sum 4x_ix_j\\geq 0,\n\\]\n\ntherefore we have to prove that\n\n\\[\n\\sum \\sum a_ia_j\\geq 0\n\\]\n\nfor every list \\(x_i\\), and we can describe this to\n\n\\[\n\\sum \\sum a_ia_j=\\sum a_i^2 + \\sum\\sum a_ia_j(i\\... |
IMO-2021-3 | https://artofproblemsolving.com/wiki/index.php/2021_IMO_Problems/Problem_3 | Let \(D\) be an interior point of the acute triangle \(ABC\) with \(AB > AC\) so that \(\angle DAB= \angle CAD\). The point \(E\) on the segment \(AC\) satisfies \(\angle ADE= \angle BCD\), the point \(F\) on the segment \(AB\) satisfies \(\angle FDA= \angle DBC\), and the point \(X\) on the line \(AC\) satisfies \(CX=... | [
"We prove that circles \\(ACD, EXD\\) and \\(\\Omega_0\\) centered at \\(P\\) (the intersection point \\(BC\\) and \\(EF)\\) have a common chord.\n\nLet \\(P\\) be the intersection point of the tangent to the circle \\(\\omega_2 = BDC\\) at the point \\(D\\) and the line \\(BC, A'\\) is inverse to \\(A\\) with resp... |
IMO-2021-4 | https://artofproblemsolving.com/wiki/index.php/2021_IMO_Problems/Problem_4 | Let \(\Gamma\) be a circle with centre \(I\), and \(ABCD\) a convex quadrilateral such that each of the segments \(AB, BC, CD\) and \(DA\) is tangent to \(\Gamma\). Let \(\Omega\) be the circumcircle of the triangle \(AIC\). The extension of \(BA\) beyond \(A\) meets \(\Omega\) at \(X\), and the extension of \(BC\) bey... | [
"Let \\(O\\) be the centre of \\(\\Omega\\).\n\nFor \\(AB=BC\\) the result follows simply. By Pitot's Theorem we have\n\n\\[\nAB + CD = BC + AD\n\\]\n\nso that, \\(AD = CD.\\) The configuration becomes symmetric about \\(OI\\) and the result follows immediately.\n\nNow assume WLOG \\(AB < BC\\). Then \\(T\\) lies b... |
IMO-2021-5 | https://artofproblemsolving.com/wiki/index.php/2021_IMO_Problems/Problem_5 | Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention... | [
"We will start by introducing some notation.\n\n- Let the holes be denoted by \\(H_1, H_2, \\dots, H_{2021}\\). The index \\(i\\) in \\(H_i\\) is considered modulo \\(2021\\)\n- Let the nuts be denoted by \\(N_1, N_2, \\dots, N_{2021}\\) and define \\(N_i < N_j\\) when \\(i < j\\).\n- Let a nut \\(n\\) in hole \\(H... |
IMO-2021-6 | https://artofproblemsolving.com/wiki/index.php/2021_IMO_Problems/Problem_6 | Let \(m \ge 2\) be an integer, \(A\) be a finite set of (not necessarily positive) integers, and \(B_1, B_2, B_3 , \ldots, B_m\) be subsets of \(A\). Assume that for each \(k = 1, 2,...,m\) the sum of the elements of \(B_k\) is \(m^k\). Prove that \(A\) contains at least \(m/2\) elements. | [] |
IMO-2022-1 | https://artofproblemsolving.com/wiki/index.php/2022_IMO_Problems/Problem_1 | The Bank of Oslo issues two types of coin: aluminium (denoted A) and bronze (denoted B). Marianne has \(n\) aluminium coins and \(n\) bronze coins, arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer \(k\le 2n\), Marianne re... | [
"We call a chain basic when it is the largest possible for the coins it consists of. Let \\(A=[i,j]\\) be the basic chain with the \\(i\\)-th and \\(j\\)-th coins being the first and last, respectively.\n\nClaim:\n\n\\[\nk \\notin \\{1, 2, \\ldots, n-1\\} \\cup \\{\\lceil \\frac{3n}{2} \\rceil + 1, \\ldots, 2n\\}.\... |
IMO-2022-2 | https://artofproblemsolving.com/wiki/index.php/2022_IMO_Problems/Problem_2 | Let \(\mathbb{R}^+\) denote the set of positive real numbers. Find all functions \(f : \mathbb{R}^+ \to \mathbb{R}^+\) such that for each \(x \in \mathbb{R}^+\), there is exactly one \(y \in \mathbb{R}^+\) satisfying
\[
xf (y) + yf (x) \le 2
\]
. | [
"The unique solution is the function \\( f(x) = \\frac{1}{x} \\) for every \\( x \\in \\mathbb{R}^+ \\). This function clearly satisfies the required property since the expression \\( xf(y) + yf(x) = \\frac{x}{y} + \\frac{y}{x} \\) is greater than 2 for every \\( y \\neq x \\) (directly from AM-GM) and equal to 2 (... |
IMO-2022-3 | https://artofproblemsolving.com/wiki/index.php/2022_IMO_Problems/Problem_3 | Let \(k\) be a positive integer and let \(S\) be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of \(S\) around a circle such that the product of any two neighbours is of the form \(x^2 + x + k\) for some positive integer \(x\). | [] |
IMO-2022-4 | https://artofproblemsolving.com/wiki/index.php/2022_IMO_Problems/Problem_4 | Let \(ABCDE\) be a convex pentagon such that \(BC = DE\). Assume that there is a point \(T\) inside \(ABCDE\) with \(TB = TD\), \(TC = TE\) and \(\angle ABT = \angle TEA\). Let line \(AB\) intersect lines \(CD\) and \(CT\) at points \(P\) and \(Q\), respectively. Assume that the points \(P, B, A, Q\) occur on their lin... | [
"\\[\nTB = TD, TC = TE, BC = DE \\implies\n\\]\n\n\\[\n\\triangle TBC = \\triangle TDE \\implies \\angle BTC = \\angle DTE.\n\\]\n\n\\[\n\\angle BTQ = 180^\\circ - \\angle BTC = 180^\\circ - \\angle DTE = \\angle STE\n\\]\n\n\\[\n\\angle ABT = \\angle AET \\implies \\triangle TQB \\sim \\triangle TSE \\implies\n\\... |
IMO-2022-5 | https://artofproblemsolving.com/wiki/index.php/2022_IMO_Problems/Problem_5 | Find all triples \((a,b,p)\) of positive integers with \(p\) prime and
\[
a^p = b! + p
\] | [
"Case 1: \\(b < p\\)\n\n- Since \\(b!\\) is indivisible by \\(p\\), then \\(a\\) must also be indivisible by \\(p\\).\n\n- If \\(a \\le b\\), then \\(a^p-b!\\) is divisible by \\(a\\), so \\(a\\) must be a divisor of \\(p\\), but \\(a=1\\) obviously has no solutions and we ruled out \\(a=p\\) already. For \\(a > b\... |
IMO-2022-6 | https://artofproblemsolving.com/wiki/index.php/2022_IMO_Problems/Problem_6 | Let \(n\) be a positive integer. A Nordic square is an \(n \times n\) board containing all the integers from \(1\) to \(n^2\) so that each cell contains exactly one number. Two different cells are considered adjacent if they share an edge. Every cell that is adjacent only to cells containing larger numbers is called a ... | [
"The minimum total number of uphill paths in a Nordic square is \\(2n(n-1)+1\\)\n\nProof: For every pair of adjacent cells in the grid, there is at least one pathway that begins at the higher valued cell, steps to the lower valued cell, and then continues stepping to lower adjacent cells until (by finite descent) a... |
IMO-2023-1 | https://artofproblemsolving.com/wiki/index.php/2023_IMO_Problems/Problem_1 | Determine all composite integers \(n>1\) that satisfy the following property: if \(d_1,d_2,\dots,d_k\) are all the positive divisors of \(n\) with \(1=d_1<d_2<\dots<d_k=n\), then \(d_i\) divides \(d_{i+1}+d_{i+2}\) for every \(1\le i \le k-2\). | [
"If \\(n\\) has at least \\(2\\) prime divisors, WLOG let \\(p<q\\) be the smallest two of these primes. Then the ordered tuple of divisors is of the form \\((1,\\, p,\\, p^2 \\dots,\\, p^a,\\, q \\dots,\\, n)\\) for some integer \\(a\\geq 1\\).\n\nTo prove this claim, note that \\(p\\) is the smallest prime that d... |
IMO-2023-2 | https://artofproblemsolving.com/wiki/index.php/2023_IMO_Problems/Problem_2 | Let \(ABC\) be an acute-angled triangle with \(AB < AC\). Let \(\Omega\) be the circumcircle of \(ABC\). Let \(S\) be the midpoint of the arc \(CB\) of \(\Omega\) containing \(A\). The perpendicular from \(A\) to \(BC\) meets \(BS\) at \(D\) and meets \(\Omega\) again at \(E \neq A\). The line through \(D\) parallel to... | [
"Denote the point diametrically opposite to a point \\(S\\) through \\(S' \\implies AS'\\) is the internal angle bisector of \\(\\angle BAC\\).\n\nDenote the crosspoint of \\(BS\\) and \\(AS'\\) through \\(H, \\angle ABS = \\varphi.\\)\n\n\\[\nAE \\perp BC, SS' \\perp BC \\implies \\overset{\\Large\\frown} {AS} = \... |
IMO-2023-3 | https://artofproblemsolving.com/wiki/index.php/2023_IMO_Problems/Problem_3 | For each integer \(k \geqslant 2\), determine all infinite sequences of positive integers \(a_1, a_2, \ldots\) for which there exists a polynomial \(P\) of the form \(P(x)=x^k+c_{k-1} x^{k-1}+\cdots+c_1 x+c_0\), where \(c_0, c_1, \ldots, c_{k-1}\) are non-negative integers, such that
\[
P\left(a_n\right)=a_{n+1} a_{n+... | [
"Let \\(f(n)\\) and \\(g(j)\\) be functions of positive integers \\(n\\) and \\(j\\) respectively.\n\nLet \\(a_{n}=a_{1}+f(n)\\), then \\(a_{n+1}=a_{1}+f(n+1)\\), and \\(a_{n+k}=a_{1}+f(n+k)\\)\n\nLet \\(P=\\prod_{j=1}^{k}\\left ( a_{n+j} \\right ) = \\prod_{j=1}^{k}\\left ( a_{n}+g(j)) \\right )\\)\n\nIf we want t... |
IMO-2023-4 | https://artofproblemsolving.com/wiki/index.php/2023_IMO_Problems/Problem_4 | Let \(x_1, x_2, \cdots , x_{2023}\) be pairwise different positive real numbers such that
\[
a_n = \sqrt{(x_1+x_2+ \text{···} +x_n)(\frac1{x_1} + \frac1{x_2} + \text{···} +\frac1{x_n})}
\]
is an integer for every \(n = 1,2,\cdots,2023\). Prove that \(a_{2023} \ge 3034\). | [
"We solve for \\(a_{n+2}\\) in terms of \\(a_n\\) and \\(x.\\) \\(a_{n+2}^2 \\\\ = (\\sum^{n+2}_{k=1}x_k)(\\sum^{n+2}_{k=1}\\frac1{x_k}) \\\\ = (x_{n+1}+x_{n+2}+\\sum^{n}_{k=1}x_k)(\\frac{1}{x_{n+1}}+\\frac{1}{x_{n+2}}+\\sum^{n}_{k=1}\\frac1{x_k}) \\\\ = \\frac{x_{n+1}}{x_{n+1}} + \\frac{x_{n+1}}{x_{n+2}} + \\frac{... |
IMO-2023-5 | https://artofproblemsolving.com/wiki/index.php/2023_IMO_Problems/Problem_5 | Let \(n\) be a positive integer. A Japanese triangle consists of \(1 + 2 + \dots + n\) circles arranged in an equilateral triangular shape such that for each \(i = 1\), \(2\), \(\dots\), \(n\), the \(i^{th}\) row contains exactly \(i\) circles, exactly one of which is coloured red. A ninja path in a Japanese triangle i... | [] |
IMO-2023-6 | https://artofproblemsolving.com/wiki/index.php/2023_IMO_Problems/Problem_6 | Let \(ABC\) be an equilateral triangle. Let \(A_1,B_1,C_1\) be interior points of \(ABC\) such that \(BA_1=A_1C\), \(CB_1=B_1A\), \(AC_1=C_1B\), and
\[
\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ
\]
Let \(BC_1\) and \(CB_1\) meet at \(A_2,\) let \(CA_1\) and \(AC_1\) meet at \(B_2,\) and let \(AB_1\) and \(BA_1\... | [] |
IMO-2024-1 | https://artofproblemsolving.com/wiki/index.php/2024_IMO_Problems/Problem_1 | Determine all real numbers \(\alpha\) such that, for every positive integer \(n\), the integer
\[
\lfloor \alpha \rfloor + \lfloor 2\alpha \rfloor + \dots +\lfloor n\alpha \rfloor
\]
is a multiple of \(n\). (Note that \(\lfloor z \rfloor\) denotes the greatest integer less than or equal to \(z\). For example, \(\lflo... | [
"To solve the problem, we need to find all real numbers \\( \\alpha \\) such that, for every positive integer \\( n \\), the integer\n\n\\[\nS_n(\\alpha) = \\lfloor \\alpha \\rfloor + \\lfloor 2\\alpha \\rfloor + \\dots + \\lfloor n\\alpha \\rfloor\n\\]\n\nis divisible by \\( n \\), i.e., \\( S_n(\\alpha) \\equiv 0... |
IMO-2024-2 | https://artofproblemsolving.com/wiki/index.php/2024_IMO_Problems/Problem_2 | Find all positive integer pairs \((a,b),\) such that there exists positive integer \(g,N,\)
\[
\gcd (a^n+b,b^n+a)=g
\]
holds for all integer \(n\ge N\). | [
"We will determine all pairs \\((a,b)\\) of positive integers such that \\(\\gcd(a^n+b,b^n+a)=g\\) for all \\(n \\geq N\\).\n\nFirst, \\((1,1)\\) works with \\(g=2\\). Now for any solution \\((a,b)\\):\n\n\\begin{lemma} \\(g = \\gcd(a,b)\\) or \\(g = 2\\gcd(a,b)\\). \\end{lemma}\n\n\\begin{proof} Since \\(g\\) divi... |
IMO-2024-3 | https://artofproblemsolving.com/wiki/index.php/2024_IMO_Problems/Problem_3 | Let \(a_1, a_2, a_3, \dots\) be an infinite sequence of positive integers, and let \(N\) be a positive integer. Suppose that, for each \(n > N\), \(a_n\) is equal to the number of times \(a_{n-1}\) appears in the list \(a_1, a_2, \dots, a_{n-1}\).
Prove that at least one of the sequence \(a_1, a_3, a_5, \dots\) and \(... | [] |
IMO-2024-4 | https://artofproblemsolving.com/wiki/index.php/2024_IMO_Problems/Problem_4 | Let \(ABC\) be a triangle with \(AB < AC < BC\). Let the incentre and incircle of triangle \(ABC\) be \(I\) and \(\omega\), respectively. Let \(X\) be the point on line \(BC\) different from \(C\) such that the line through \(X\) parallel to \(AC\) is tangent to \(\omega\). Similarly, let \(Y\) be the point on line \(B... | [
"Part 1: Derive tangent values \\(\\angle AIL\\) and \\(\\angle AIK\\) with trig values of angles \\(\\frac{A}{2}\\), \\(\\frac{B}{2}\\), \\(\\frac{C}{2}\\)\n\nPart 2: Derive tangent values \\(\\angle XPM\\) and \\(\\angle YPM\\) with side lengths \\(AB\\), \\(BC\\), \\(CA\\), where \\(M\\) is the midpoint of \\(BC... |
IMO-2024-5 | https://artofproblemsolving.com/wiki/index.php/2024_IMO_Problems/Problem_5 | Turbo the snail plays a game on a board with 2024 rows and 2023 columns. There are hidden monsters in 2022 of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the last row, and that each column contains at most o... | [] |
IMO-2024-6 | https://artofproblemsolving.com/wiki/index.php/2024_IMO_Problems/Problem_6 | Let \(\mathbb{Q}\) be the set of rational numbers. A function \(f: \mathbb{Q} \to \mathbb{Q}\) is called \(\emph{aquaesulian}\) if the following property holds: for every \(x,y \in \mathbb{Q}\),
\[
f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y).
\]
Show that there exists an integer \(c\) such that fo... | [] |
IMO-2025-1 | https://artofproblemsolving.com/wiki/index.php/2025_IMO_Problems/Problem_1 | A line in the plane is called sunny if it is not parallel to any of the \(x\)–axis, the \(y\)–axis, and the line \(x+y=0\).
Let \(n\ge3\) be a given integer. Determine all nonnegative integers \(k\) such that there exist \(n\) distinct lines in the plane satisfying both of the following:
- for all positive integers \... | [
"Consider a valid construction for \\(k \\ge 4\\).\n\n\\[\n\\text{Claim: One of the } n \\text{ lines must be } x=1, y=1, \\text{ or } x+y=n.\n\\]\n\nProof: Assume for the sake of contradiction not. Then, the following holds:\n\n\\[\n\\quad \\text{1. } x=1 \\text{ is not in our lines.}\n\\]\n\nOtherwise, two points... |
IMO-2025-2 | https://artofproblemsolving.com/wiki/index.php/2025_IMO_Problems/Problem_2 | Let \(\Omega\) and \(\Gamma\) be circles with centers \(M\) and \(N\), respectively, such that the radius of \(\Omega\) is less than the radius of \(\Gamma\). Suppose circles \(\Omega\) and \(\Gamma\) intersect at two distinct points \(A\) and \(B\). Line \(MN\) intersects \(\Omega\) at \(C\) and \(\Gamma\) at \(D\), s... | [
"Throughout the solution, we define\n\n\\[\n\\alpha := \\angle DCA = \\angle BCD \\implies \\angle PAD = \\angle CAB = 90^\\circ - \\alpha\n\\]\n\\[\n\\beta := \\angle ADC = \\angle CDB \\implies \\angle CAP = \\angle BAD = 90^\\circ - \\beta.\n\\]\n\nIgnore the points \\(H, M, N\\) for now and focus on the remaini... |
IMO-2025-3 | https://artofproblemsolving.com/wiki/index.php/2025_IMO_Problems/Problem_3 | Let \(\mathbb{N}\) denote the set of positive integers. A function \(f: \mathbb{N} \rightarrow \mathbb{N}\) is said to be bonza if
\[
f(a)
\]
divides
\[
b^{a} - f(b)^{f(a)}
\]
for all positive integers \(a\) and \(b\). Determine the smallest real constant \(c\) such that \(f(n) \leq cn\) for all bonza functions \(f... | [
"The answer is \\(c = 4\\).\n\nLet \\(P(a,b)\\) denote the given statement \\(f(a) \\mid b^a - f(b)^{f(a)}\\).\n\n---\n\n**Claim —** We have \\(f(n) \\mid n^n\\) for all \\(n\\).\n\n*Proof.* Take \\(P(n,n)\\). \\(\\square\\)\n\n---\n\n**Claim —** Unless \\(f = \\mathrm{id}\\), we have \\(f(p) = 1\\) for all odd pri... |
IMO-2025-4 | https://artofproblemsolving.com/wiki/index.php/2025_IMO_Problems/Problem_4 | A proper divisor of a positive integer \(N\) is a positive divisor of \(N\) other than \(N\) itself.
The infinite sequence \(a_1,a_2,\dots\) consists of positive integers, each of which has at least three proper divisors. For each \(n\ge1\), the integer \(a_{n+1}\) is the sum of the three largest proper divisors of \(... | [
"The answer is \\(a_1 = 12^e \\cdot 6 \\cdot \\ell\\) for any \\(e, \\ell \\geq 0\\) with \\(\\gcd(\\ell,10)=1\\).\n\nLet \\(S\\) denote the set of positive integers with at least three divisors. For \\(x \\in S\\), let \\(\\psi(x)\\) denote the sum of the three largest ones, so that \\(\\psi(a_n)=a_{n+1}\\).\n\n--... |
IMO-2025-5 | https://artofproblemsolving.com/wiki/index.php/2025_IMO_Problems/Problem_5 | Alice and Bazza are playing the inekoalaty game, a two‑player game whose rules depend on a positive real number \(\lambda\) which is known to both players. On the \(n\)th turn of the game (starting with \(n=1\)) the following happens:
- If \(n\) is odd, Alice chooses a nonnegative real number \(x_n\) such that \(x_1 +... | [
"The answer is that Alice has a winning strategy for \\(\\lambda > 1/\\sqrt{2}\\), and Bazza has a winning strategy for \\(\\lambda < 1/\\sqrt{2}\\). (Neither player can guarantee winning for \\(\\lambda = 1/\\sqrt{2}\\).)\n\nWe divide the proof into two parts.\n\n¶ Alice’s strategy when \\(\\lambda \\geq 1/\\sqrt{... |
IMO-2025-6 | https://artofproblemsolving.com/wiki/index.php/2025_IMO_Problems/Problem_6 | Consider a 2025 x 2025 grid of unit squares. Matlida wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.
Determine the minimum number of tiles Matlida needs to place so that each row... | [
"The answer is \\(2112 = 2025 + 2 \\cdot 45 - 3\\). In general, the answer turns out to be \\(\\lceil n + 2\\sqrt{n} - 3 \\rceil\\), but when \\(n\\) is not a perfect square the solution is more complicated.\n\n**Remark.** The 2017 Romanian Masters in Math asked the same problem where the tiles are replaced by *sti... |
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