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+
+# IMO 2023 Solution Notes  
+
+Evan Chen 《陳誼廷》  
+
+18 October 2025  
+
+This is a compilation of solutions for the 2023 IMO. The ideas of the solution are a mix of my own work, the solutions provided by the competition organizers, and solutions found by the community. However, all the writing is maintained by me.  
+
+These notes will tend to be a bit more advanced and terse than the "official" solutions from the organizers. In particular, if a theorem or technique is not known to beginners but is still considered "standard", then I often prefer to use this theory anyways, rather than try to work around or conceal it. For example, in geometry problems I typically use directed angles without further comment, rather than awkwardly work around configuration issues. Similarly, sentences like "let \(\mathbb{R}\) denote the set of real numbers" are typically omitted entirely.  
+
+Corrections and comments are welcome!  
+
+## Contents  
+
+0 Problems 2  
+
+1 Solutions to Day 1 4  
+
+1.1 IMO 2023/1, proposed by Santiago Rodriguez (COL) 4  
+
+1.2 IMO 2023/2, proposed by Tiago Mourão and Nuno Arala (POR) 5  
+
+1.3 IMO 2023/3, proposed by Ivan Chan (MAS) 7  
+
+2 Solutions to Day 2 9  
+
+2.1 IMO 2023/4, proposed by Merlijn Staps (NLD) 9  
+
+2.2 IMO 2023/5, proposed by Merlijn Staps and Daniël Kroes (NLD) 11  
+
+2.3 IMO 2023/6, proposed by Ankan Bhattacharya, Luke Robitaille (USA) 14
+
+
+
+## Problems  
+
+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 then \(d_{i}\) divides \(d_{i + 1} + d_{i + 2}\) for every \(1\leq i\leq k - 2\)  
+
+2. Let \(A B C\) be an acute-angled triangle with \(A B< A C\) . Let \(\Omega\) be the circumcircle of \(A B C\) . Let \(S\) be the midpoint of the arc \(C B\) of \(\Omega\) containing \(A\) . The perpendicular from \(A\) to \(B C\) meets \(B S\) at \(D\) and meets \(\Omega\) again at \(E\neq A\) . The line through \(D\) parallel to \(B C\) meets line \(B E\) at \(L\) . Denote the circumcircle of triangle \(B D L\) by \(\omega\) . Let \(\omega\) meet \(\Omega\) again at \(P\neq B\) . Prove that the line tangent to \(\omega\) at \(P\) meets line \(B S\) on the internal angle bisector of \(\angle B A C\) .  
+
+3. For each integer \(k\geq 2\) , determine all infinite sequences of positive integers \(a_{1}\) , \(a_{2}\) , ... for which there exists a polynomial \(P\) of the form  
+
+\[P(x) = x^{k} + c_{k - 1}x^{k - 1} + \dots +c_{1}x + c_{0},\]  
+
+where \(c_{0}\) , \(c_{1}\) , ..., \(c_{k - 1}\) are non-negative integers, such that  
+
+\[P(a_{n}) = a_{n + 1}a_{n + 2}\cdot \cdot \cdot a_{n + k}\]  
+
+for every integer \(n\geq 1\)  
+
+4. Let \(x_{1}\) , \(x_{2}\) , ..., \(x_{2023}\) be pairwise different positive real numbers such that  
+
+\[a_{n} = \sqrt{(x_{1} + x_{2} + \cdot \cdot \cdot + x_{n})\left(\frac{1}{x_{1}} +\frac{1}{x_{2}} +\cdot \cdot \cdot +\frac{1}{x_{n}}\right)}\]  
+
+is an integer for every \(n = 1,2,\ldots ,2023\) . Prove that \(a_{2023}\geq 3034\)  
+
+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 \(1\leq i\leq n\) , the \(i^{\mathrm{th}}\) row contains exactly \(i\) circles, exactly one of which is colored red. A ninja path in a Japanese triangle is a sequence of \(n\) circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with \(n = 6\) , along with a ninja path in that triangle containing two red circles.  
+
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)
+  
+
+In terms of \(n\) , find the greatest \(k\) such that in each Japanese triangle there is a ninja path containing at least \(k\) red circles.  
+
+6. Let \(A B C\) be an equilateral triangle. Let \(A_{1}\) , \(B_{1}\) , \(C_{1}\) be interior points of \(A B C\) such that \(B A_{1} = A_{1}C\) , \(C B_{1} = B_{1}A\) , \(A C_{1} = C_{1}B\) , and  
+
+\[\angle B A_{1}C + \angle C B_{1}A + \angle A C_{1}B = 480^{\circ}.\]
+
+
+
+Let \(A_{2} = \overline{BC_{1}} \cap \overline{CB_{1}}\) , \(B_{2} = \overline{CA_{1}} \cap \overline{AC_{1}}\) , \(C_{2} = \overline{AB_{1}} \cap \overline{BA_{1}}\) . Prove that if triangle \(A_{1}B_{1}C_{1}\) is scalene, then the circumcircles of triangles \(AA_{1}A_{2}\) , \(BB_{1}B_{2}\) , and \(CC_{1}C_{2}\) all pass through two common points.
+
+
+
+## \(\S 1\) Solutions to Day 1  
+
+## \(\S 1.1\) IMO 2023/1, proposed by Santiago Rodriguez (COL)  
+
+Available online at https://aops.com/community/p28097575.  
+
+## Problem statement  
+
+Problem statementDetermine 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 then \(d_{i}\) divides \(d_{i + 1} + d_{i + 2}\) for every \(1\leq i\leq k - 2\) .  
+
+The answer is prime powers.  
+
+Verification that these work. When \(n = p^{e}\) , we get \(d_{i} = p^{i - 1}\) . The \(i^{\mathrm{th}}\) relationship reads  
+
+\[p^{i - 1}\mid p^{i} + p^{i + 1}\]  
+
+which is obviously true.  
+
+Proof that these are the only answers. Conversely, suppose \(n\) has at least two distinct prime divisors. Let \(p< q\) denote the two smallest ones, and let \(p^{e}\) be the largest power of \(p\) which both divides \(n\) and is less than \(q\) , hence \(e\geq 1\) . Then the smallest factors of \(n\) are 1, \(p\) , ..., \(p^{e}\) , \(q\) . So we are supposed to have  
+
+\[\frac{n}{q}\mid \frac{n}{p^{e}} +\frac{n}{p^{e - 1}} = \frac{(p + 1)n}{p^{e}}\]  
+
+which means that the ratio  
+
+\[\frac{q(p + 1)}{p^{e}}\]  
+
+needs to be an integer, which is obviously not possible.
+
+
+
+## \(\S 1.2\) IMO 2023/2, proposed by Tiago Mourão and Nuno Arala (POR)  
+
+Available online at https://aops.com/community/p28097552.  
+
+## Problem statement  
+
+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 \(BC\) meets line \(BE\) at \(L\) . Denote the circumcircle of triangle \(BDL\) by \(\omega\) . Let \(\omega\) meet \(\Omega\) again at \(P \neq B\) . Prove that the line tangent to \(\omega\) at \(P\) meets line \(BS\) on the internal angle bisector of \(\angle BAC\) .  
+
+Claim — We have \(LPS\) collinear.  
+
+Proof. Because \(\angle LPB = \angle LDB = \angle CBD = \angle CBS = \angle SCB = \angle SPB\) . \(\square\)  
+
+Let \(F\) be the antipode of \(A\) , so \(AMFS\) is a rectangle.  
+
+Claim — We have \(PDF\) collinear. (This lets us erase \(L\) .)  
+
+Proof. Because \(\angle SPD = \angle LPD = \angle LBD = \angle SBE = \angle FCS = \angle FPS\) . \(\square\)  
+
+Let us define \(X = \overline{AM} \cap \overline{BS}\) and complete chord \(\overline{PXQ}\) . We aim to show that \(\overline{PXQ}\) is tangent to \((PDLB)\) .  
+
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+  
+
+Claim (Main projective claim) — We have \(XP = XA\) .  
+
+Proof. Introduce \(Y = \overline{PDF} \cap \overline{AM}\) . Note that  
+
+\[-1 = (SM;EF) \stackrel{A}{=} (S,X;D,\overline{AF} \cap \overline{ES}) \stackrel{E}{=} (\infty X;YA)\]  
+
+where \(\infty = \overline{AM} \cap \overline{SF}\) is at infinity (because \(AMSF\) is a rectangle). Thus, \(XY = XA\) .
+
+
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+  
+
+Since \(\triangle APY\) is also right, we get \(XP = XA\) .  
+
+Alternative proof of claim without harmonic bundles, from Solution 9 of the marking scheme. With \(Y = \overline{PDF} \cap \overline{AM}\) defined as before, note that \(\overline{AE} \parallel \overline{SM}\) and \(\overline{AM} \parallel \overline{SF}\) (as AMFS is a rectangle) gives respectively the similar triangles  
+
+\[\triangle AXD\sim \triangle M X S,\qquad \triangle XDY\sim \triangle SDF.\]  
+
+From this we conclude  
+
+\[\frac{AX}{XD} = \frac{AX + XM}{XD + SX} = \frac{AM}{SD} = \frac{SF}{SD} = \frac{XY}{XD}.\]  
+
+So \(AX = XY\) and as before we conclude \(XP = XA\) .  
+
+From \(XP = XA\) , we conclude that \(\widehat{PM}\) and \(\widehat{AQ}\) have the same measure. Since \(\widehat{AS}\) and \(\widehat{EM}\) have the same measure, it follows \(\widehat{PE}\) and \(\widehat{SQ}\) have the same measure. The desired tangency then follows from  
+
+\[\angle QPL = \angle QPS = \angle PQE = \angle PFE = \angle PDL.\]  
+
+Remark (Logical ordering). This solution is split into two phases: the "synthetic phase" where we do a bunch of angle chasing, and the "projective phase" where we use cross- ratios because I like projective. For logical readability (so we write in only one logical direction), the projective phase is squeezed in two halves of the synthetic phase, but during an actual solve it's expected to complete the whole synthetic phase first (i.e. to reduce the problem to show \(XP = XA\) ).  
+
+Remark. There are quite a multitude of approaches for this problem; the marking scheme for this problem at the actual IMO had 13 different solutions.
+
+
+
+## \(\S 1.3\) IMO 2023/3, proposed by Ivan Chan (MAS)  
+
+Available online at https://aops.com/community/p28097600.  
+
+## Problem statement  
+
+For each integer \(k \geq 2\) , determine all infinite sequences of positive integers \(a_{1}\) , \(a_{2}\) , ... for which there exists a polynomial \(P\) of the form  
+
+\[P(x) = x^{k} + c_{k - 1}x^{k - 1} + \dots +c_{1}x + c_{0},\]  
+
+where \(c_{0}\) , \(c_{1}\) , ..., \(c_{k - 1}\) are non- negative integers, such that  
+
+\[P(a_{n}) = a_{n + 1}a_{n + 2}\dots a_{n + k}\]  
+
+for every integer \(n \geq 1\) .  
+
+The answer is \(a_{n}\) being an arithmetic progression. Indeed, if \(a_{n} = d(n - 1) + a_{1}\) for \(d \geq 0\) and \(n \geq 1\) , then  
+
+\[a_{n + 1}a_{n + 2}\dots a_{n + k} = (a_{n} + d)(a_{n} + 2d)\dots (a_{n} + kd)\]  
+
+so we can just take \(P(x) = (x + d)(x + 2d)\dots (x + kd)\) .  
+
+The converse direction takes a few parts.  
+
+Claim — Either \(a_{1} < a_{2} < \dots\) or the sequence is constant.  
+
+Proof. Note that  
+
+\[P(a_{n - 1}) = a_{n}a_{n + 1}\dots a_{n + k - 1\] \[P(a_{n}) = a_{n + 1}a_{n + 2}\dots a_{n + k\] \[\implies a_{n + k} = \frac{P(a_{n})}{P(a_{n - 1})}\cdot a_{n}.\]  
+
+Now the polynomial \(P\) is strictly increasing over \(\mathbb{N}\) .  
+
+So assume for contradiction there's an index \(n\) such that \(a_{n} < a_{n - 1}\) . Then in fact the above equation shows \(a_{n + k} < a_{n} < a_{n - 1}\) . Then there's an index \(\ell \in [n + 1, n + k]\) such that \(a_{\ell} < a_{\ell - 1}\) , and also \(a_{\ell} < a_{n}\) . Continuing in this way, we can an infinite descending subsequence of \((a_{n})\) , but that's impossible because we assumed integers.  
+
+Hence we have \(a_{1} \leq a_{2} \leq \dots\) . Now similarly, if \(a_{n} = a_{n - 1}\) for any index \(n\) , then \(a_{n + k} = a_{n}\) , ergo \(a_{n - 1} = a_{n} = a_{n + 1} = \dots = a_{n + k}\) . So the sequence is eventually constant, and then by downwards induction, it is fully constant. \(\square\)  
+
+Claim — There exists a constant \(C\) (depending only \(P\) , \(k\) ) such that we have \(a_{n + 1} \leq a_{n} + C\) .  
+
+Proof. Let \(C\) be a constant such that \(P(x) < x^{k} + Cx^{k - 1}\) for all \(x \in \mathbb{N}\) (for example \(C = c_{0} + c_{1} + \dots + c_{k - 1} + 1\) works). We have  
+
+\[a_{n + k} = \frac{P(a_{n})}{a_{n + 1}a_{n + 2}\dots a_{n + k - 1}}\]
+
+
+
+\[< \frac{P(a_{n})}{(a_{n} + 1)(a_{n} + 2)\ldots(a_{n} + k - 1)}\] \[< \frac{a_{n}^{k} + C\cdot a_{n}^{k - 1}}{(a_{n} + 1)(a_{n} + 2)\ldots(a_{n} + k - 1)}\] \[< a_{n} + C + 1.\]  
+
+Assume henceforth \(a_{n}\) is nonconstant, and hence unbounded. For each index \(n\) and term \(a_{n}\) in the sequence, consider the associated differences \(d_{1} = a_{n + 1} - a_{n}\) , \(d_{2} = a_{n + 2} - a_{n + 1}\) , ..., \(d_{k} = a_{n + k} - a_{n + k - 1}\) , which we denote by  
+
+\[\Delta (n):= (d_{1},\ldots ,d_{k}).\]  
+
+This \(\Delta\) can only take up to \(C^{k}\) different values. So in particular, some tuple \((d_{1},\ldots ,d_{n})\) must appear infinitely often as \(\Delta (n)\) ; for that tuple, we obtain  
+
+\[P(a_{N}) = (a_{N} + d_{1})(a_{N} + d_{1} + d_{2})\ldots (a_{N} + d_{1} + \dots +d_{k})\]  
+
+for infinitely many \(N\) . But because of that, we actually must have  
+
+\[P(X) = (X + d_{1})(X + d_{1} + d_{2})\ldots (X + d_{1} + \dots +d_{k}).\]  
+
+However, this also means that exactly one output to \(\Delta\) occurs infinitely often (because that output is determined by \(P\) ). Consequently, it follows that \(\Delta\) is eventually constant. For this to happen, \(a_{n}\) must eventually coincide with an arithmetic progression of some common difference \(d\) , and \(P(X) = (X + d)(X + 2d)\ldots (X + kd)\) . Finally, this implies by downwards induction that \(a_{n}\) is an arithmetic progression on all inputs.
+
+
+
+## \(\S 2\) Solutions to Day 2  
+
+## \(\S 2.1\) IMO 2023/4, proposed by Merlijn Staps (NLD)  
+
+Available online at https://aops.com/community/p28104298.  
+
+## Problem statement  
+
+Let \(x_{1}, x_{2}, \ldots , x_{2023}\) be pairwise different positive real numbers such that  
+
+\[a_{n} = \sqrt{(x_{1} + x_{2} + \cdots + x_{n})\left(\frac{1}{x_{1}} +\frac{1}{x_{2}} +\cdots +\frac{1}{x_{n}}\right)}\]  
+
+is an integer for every \(n = 1,2,\ldots ,2023\) . Prove that \(a_{2023}\geq 3034\)  
+
+Note that \(a_{n + 1} > \sqrt{\sum_{1}^{n}x_{i}\sum_{1}^{n}\frac{1}{x_{i}}} = a_{n}\) for all \(n\) , so that \(a_{n + 1}\geq a_{n} + 1\) . Observe \(a_{1} = 1\) . We are going to prove that  
+
+\[a_{2m + 1}\geq 3m + 1\qquad \mathrm{for~all~}m\geq 0\]  
+
+by induction on \(m\) , with the base case being clear.  
+
+We now present two variations of the induction. The first shorter solution compares \(a_{n + 2}\) directly to \(a_{n}\) , showing it increases by at least 3. Then we give a longer approach that compares \(a_{n + 1}\) to \(a_{n}\) , and shows it cannot increase by 1 twice in a row.  
+
+\(\P\) Induct- by- two solution. Let \(u = \sqrt{\frac{x_{n + 1}}{x_{n + 2}}}\neq 1\) . Note that by using Cauchy- Schwarz with three terms:  
+
+\[a_{n + 2}^{2} = \left[(x_{1} + \dots +x_{n}) + x_{n + 1} + x_{n + 2}\right]\left[\left(\frac{1}{x_{1}} +\dots +\frac{1}{x_{n}}\right) + \frac{1}{x_{n + 2}} +\frac{1}{x_{n + 1}}\right]\] \[\qquad \geq \left(\sqrt{(x_{1} + \dots +x_{n})\left(\frac{1}{x_{1}} +\dots +\frac{1}{x_{n}}\right)} +\sqrt{\frac{x_{n + 1}}{x_{n + 2}}} +\sqrt{\frac{x_{n + 2}}{x_{n + 1}}}\right)^{2}\] \[\qquad = \left(a_{n} + u + \frac{1}{u}\right)^{2}.\] \[\Rightarrow a_{n + 2}\geq a_{n} + u + \frac{1}{u} >a_{n} + 2\]  
+
+where the last equality \(u + \frac{1}{u} >2\) is by AM- GM, strict as \(u\neq 1\) . It follows that \(a_{n + 2}\geq a_{n} + 3\) , completing the proof.  
+
+\(\P\) Induct- by- one solution. The main claim is:  
+
+Claim — It's impossible to have \(a_{n} = c\) , \(a_{n + 1} = c + 1\) , \(a_{n + 2} = c + 2\) for any \(c\) and \(n\) .  
+
+Proof. Let \(p = x_{n + 1}\) and \(q = x_{n + 2}\) for brevity. Let \(s = \sum_{1}^{n}x_{i}\) and \(t = \sum_{1}^{n}\frac{1}{x_{n}}\) , so \(c^{2} = a_{n}^{2} = st\) .  
+
+From \(a_{n} = c\) and \(a_{n + 1} = c + 1\) we have  
+
+\[(c + 1)^{2} = a_{n + 1}^{2} = (p + s)\left(\frac{1}{p} +t\right)\]
+
+
+
+\[= s t + p t + \frac{1}{p} s + 1 = c^{2} + p t + \frac{1}{p} s + 1\] \[\overset {\mathrm{AM - GM}}{\geq} c^{2} + 2\sqrt{s t} + 1 = c^{2} + 2\sqrt{c^{2}} + 1 = (c + 1)^{2}.\]  
+
+Hence, equality must hold in the AM- GM we must have exactly  
+
+\[p t = \frac{1}{p} s = c.\]  
+
+If we repeat the argument again on \(a_{n + 1} = c + 1\) and \(a_{n + 2} = c + 2\) , then  
+
+\[q\left(\frac{1}{p} +t\right) = \frac{1}{q} (p + s) = c + 1.\]  
+
+However this forces \(\frac{p}{q} = \frac{q}{p} = 1\) which is impossible.
+
+
+
+## \(\S 2.2\) IMO 2023/5, proposed by Merlijn Staps and Daniël Kroes (NLD)  
+
+Available online at https://aops.com/community/p28104367.  
+
+## Problem statement  
+
+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 \(1\leq i\leq n\) , the \(i^{\mathrm{th}}\) row contains exactly \(i\) circles, exactly one of which is colored red. A ninja path in a Japanese triangle is a sequence of \(n\) circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with \(n = 6\) , along with a ninja path in that triangle containing two red circles.  
+
+![](data:image/jpeg;base64,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)
+  
+
+In terms of \(n\) , find the greatest \(k\) such that in each Japanese triangle there is a ninja path containing at least \(k\) red circles.  
+
+The answer is  
+
+\[k = \lfloor \log_2(n)\rfloor +1.\]  
+
+Construction. It suffices to find a Japanese triangle for \(n = 2^{e} - 1\) with the property that at most \(e\) red circles in any ninja path. The construction shown below for \(e = 4\) obviously generalizes, and works because in each of the sets \(\{1\}\) , \(\{2,3\}\) , \(\{4,5,6,7\}\) , ..., \(\{2^{e - 1}, \ldots , 2^{e} - 1\}\) , at most one red circle can be taken. (These sets are colored in different shades of red for visual clarity).  
+
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)
+
+
+
+
+Bound. Conversely, we show that in any Japanese triangle, one can find a ninja path containing at least  
+
+\[k = \lfloor \log_2(n)\rfloor +1.\]  
+
+The following short solution was posted at https://aops.com/community/p28134004, apparently first found by the team leader for Iran.  
+
+We construct a rooted binary tree \(T_{1}\) on the set of all circles as follows. For each row, other than the bottom row:  
+
+Connect the red circle to both circles under it; White circles to the left of the red circle in its row are connected to the left; White circles to the right of the red circle in its row are connected to the right.  
+
+The circles in the bottom row are all leaves of this tree. For example, the \(n = 6\) construction in the beginning gives the tree shown on the left half of the figure below:  
+
+![](data:image/jpeg;base64,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+  
+
+Now focus on only the red circles, as shown in the right half of the figure. We build a new rooted tree \(T_{2}\) where each red circle is joined to the red circle below it if there was a path of (zero or more) white circles in \(T_{1}\) between them. Then each red circle has at most 2 direct descendants in \(T_{2}\) . Hence the depth of the new tree \(T_{2}\) exceeds \(\log_2(n)\) , which produces the desired path.  
+
+Another recursive proof of bound, communicated by Helio Ng. We give another proof that \(\lfloor \log_2n\rfloor +1\) is always achievable. Define \(f(i,j)\) to be the maximum number of red circles contained in the portion of a ninja path from \((1,1)\) to \((i,j)\) , including the endpoints \((1,1)\) and \((i,j)\) . (If \((i,j)\) is not a valid circle in the triangle, define \(f(i,j) = 0\) for convenience.) An example is shown below with the values of \(f(i,j)\) drawn in the circles.  
+
+![](data:image/jpeg;base64,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)
+
+
+
+
+We have that  
+
+\[f(i,j) = \max \left\{f(i - 1,j - 1),f(i,j - 1)\right\} +\left\{ \begin{array}{l l}{1} & {\mathrm{if~}(i,j)\mathrm{~is~red~}}\\ {0} & {\mathrm{otherwise}} \end{array} \right.\]  
+
+since every ninja path passing through \((i,j)\) also passes through either \((i - 1,j - 1)\) or \((i,j - 1)\) . Now consider the quantity \(S_{j} = f(0,j) + \dots +f(j,j)\) . We obtain the following recurrence:  
+
+\[\mathrm{Claim} - S_{j + 1}\geq S_{j} + \left\lceil \frac{S_{j}}{j}\right\rceil +1.\]  
+
+Proof. Consider a maximal element \(f(m,j)\) of \(\{f(0,j),\ldots ,f(j,j)\}\) . We perform the following manipulations:  
+
+\[S_{j + 1} = \sum_{i = 0}^{j + 1}\max \left\{f(i - 1,j),f(i,j)\right\} +\sum_{i = 0}^{j + 1}\left\{ \begin{array}{l l}{1} & {\mathrm{if~}(i,j + 1)\mathrm{~is~red~}}\\ {0} & {\mathrm{otherwise}} \end{array} \right.\] \[\quad = \sum_{i = 0}^{m}\max \left\{f(i - 1,j),f(i,j)\right\} +\sum_{i = m + 1}^{j}\max \left\{f(i - 1,j),f(i,j)\right\} +1\] \[\quad \geq \sum_{i = 0}^{m}f(i,j) + \sum_{i = m + 1}^{j}f(i - 1,j) + 1\] \[\quad = S_{j} + f(m,j) + 1\] \[\quad \geq S_{j} + \left\lceil \frac{S_{j}}{j}\right\rceil +1\]  
+
+where the last inequality is due to Pigeonhole.  
+
+This is actually enough to solve the problem. Write \(n = 2^{c} + r\) , where \(0 \leq r \leq 2^{c} - 1\) .  
+
+Claim — \(S_{n} \geq cn + 2r + 1\) . In particular, \(\left\lceil \frac{S_{n}}{n} \right\rceil \geq c + 1\) .  
+
+Proof. First note that \(S_{n} \geq cn + 2r + 1\) implies \(\left\lceil \frac{S_{n}}{n} \right\rceil \geq c + 1\) because  
+
+\[\left\lceil \frac{S_{n}}{n}\right\rceil \geq \left\lceil \frac{cn + 2r + 1}{n}\right\rceil = c + \left\lceil \frac{2r + 1}{n}\right\rceil = c + 1.\]  
+
+We proceed by induction on \(n\) . The base case \(n = 1\) is clearly true as \(S_{1} = 1\) . Assuming that the claim holds for some \(n = j\) , we have  
+
+\[S_{j + 1}\geq S_{j} + \left\lceil \frac{S_{j}}{j}\right\rceil +1\] \[\qquad \geq cj + 2r + 1 + c + 1 + 1\] \[\qquad = c(j + 1) + 2(r + 1) + 1\]  
+
+so the claim is proved for \(n = j + 1\) if \(j + 1\) is not a power of 2. If \(j + 1 = 2^{c + 1}\) , then by writing \(c(j + 1) + 2(r + 1) + 1 = c(j + 1) + (j + 1) + 1 = (c + 2)(j + 1) + 1\) , the claim is also proved. \(\square\)  
+
+Now \(\left\lceil \frac{S_{n}}{n}\right\rceil \geq c + 1\) implies the existence of some ninja path containing at least \(c + 1\) red circles, and we are done.
+
+
+
+## \(\S 2.3\) IMO 2023/6, proposed by Ankan Bhattacharya, Luke Robitaille (USA)  
+
+Available online at https://aops.com/community/p28104331.  
+
+## Problem statement  
+
+Let \(A B C\) be an equilateral triangle. Let \(A_{1}\) \(B_{1}\) \(C_{1}\) be interior points of \(A B C\) such that \(B A_{1} = A_{1}C\) \(C B_{1} = B_{1}A\) \(A C_{1} = C_{1}B\) , and  
+
+\[\angle B A_{1}C + \angle C B_{1}A + \angle A C_{1}B = 480^{\circ}.\]  
+
+Let \(A_{2} = \overline{{B C_{1}}}\cap \overline{{C B_{1}}}\) \(B_{2} = \overline{{C A_{1}}}\cap \overline{{A C_{1}}}\) \(C_{2} = \overline{{A B_{1}}}\cap \overline{{B A_{1}}}\) . Prove that if triangle \(A_{1}B_{1}C_{1}\) is scalene, then the circumcircles of triangles \(A A_{1}A_{2}\) \(B B_{1}B_{2}\) , and \(C C_{1}C_{2}\) all pass through two common points.  
+
+This is the second official solution from the marking scheme, also communicated to me by Michael Ren. Define \(O\) as the center of \(A B C\) and set the angles  
+
+\[\alpha := \angle A_{1}C B = \angle C B A_{1}\] \[\beta := \angle A C B_{1} = \angle B_{1}A C\] \[\gamma := \angle C_{1}A B = \angle C_{1}B A\]  
+
+so that  
+
+\[\alpha +\beta +\gamma = 30^{\circ}.\]  
+
+In particular, \(\max (\alpha ,\beta ,\gamma)< 30^{\circ}\) , so it follows that \(A_{1}\) lies inside \(\triangle O B C\) , and similarly for the others. This means for example that \(C_{1}\) lies between \(B\) and \(A_{2}\) , and so on. Therefore the polygon \(A_{2}C_{1}B_{2}A_{1}C_{2}B_{1}\) is convex.
+
+
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+  
+
+We start by providing the "interpretation" for the \(480^{\circ}\) angle in the statement:  
+
+Claim — Point \(A_{1}\) is the circumcenter of \(\triangle A_{2}BC\) , and similarly for the others.  
+
+Proof. We have \(\angle B A_{1}C = 180^{\circ} - 2\alpha\) , and  
+
+\[\angle B A_{2}C = 180^{\circ} - \angle C B C_{1} - \angle B_{1}C B\] \[\qquad = 180^{\circ} - (60^{\circ} - \gamma) - (60^{\circ} - \beta)\] \[\qquad = 60^{\circ} + \beta +\gamma = 90^{\circ} - \alpha = \frac{1}{2}\angle B A_{1}C.\]  
+
+Since \(A_{1}\) lies inside \(\triangle B A_{2}C\) , it follows \(A_{1}\) is exactly the circumcenter.  
+
+Claim — Quadrilateral \(B_{2}C_{1}B_{1}C\) can be inscribed in a circle, say \(\gamma_{a}\) . Circles \(\gamma_{b}\) and \(\gamma_{c}\) can be defined similarly. Finally, these three circles are pairwise distinct.  
+
+Proof. Using directed angles now, we have  
+
+\[\angle B_{2}B_{1}C_{2} = 180^{\circ} - \angle A B_{1}B_{2} = 180^{\circ} - 2\angle A C B = 180^{\circ} - 2(60^{\circ} - \alpha) = 60^{\circ} + 2\alpha .\]  
+
+By the same token, \(\angle B_{2}C_{1}C_{2} = 60^{\circ} + 2\alpha\) . This establishes the existence of \(\gamma_{a}\) .  
+
+The proof for \(\gamma_{b}\) and \(\gamma_{c}\) is the same. Finally, to show the three circles are distinct, it would be enough to verify that the convex hexagon \(A_{2}C_{1}B_{2}A_{1}C_{2}B_{1}\) is not cyclic.
+
+
+
+Assume for contradiction it was cyclic. Then  
+
+\[360^{\circ} = \angle C_{2}A_{1}B_{2} + \angle B_{2}C_{1}A_{2} + \angle A_{2}B_{1}C_{2} = \angle B A_{1}C + \angle C B_{1}A + \angle A C_{1}B = 480^{\circ}\]  
+
+which is absurd. This contradiction eliminates the degenerate case, so the three circles are distinct. \(\square\)  
+
+For the remainder of the solution, let \(\mathrm{Pow}(P,\omega)\) denote the power of a point \(P\) with respect to a circle \(\omega\) .  
+
+Let line \(A A_{1}\) meet \(\gamma_{b}\) and \(\gamma_{c}\) again at \(X\) and \(Y\) , and set \(k_{a}:= \frac{A X}{A Y}\) . Consider the locus of all points \(P\) such that  
+
+\[\mathcal{C}_{a}:= \Big\{\mathrm{points} P\mathrm{in}\mathrm{the}\mathrm{plane}\mathrm{satisfying}\mathrm{Pow}(P,\gamma_{b}) = k_{a}\mathrm{Pow}(P,\gamma_{c})\Big\} .\]  
+
+We recall the coaxiality lemma \(^{1}\) , which states that (given \(\gamma_{b}\) and \(\gamma_{c}\) are not concentric) the locus \(\mathcal{C}_{a}\) must be either a circle (if \(k_{a}\neq 1\) ) or a line (if \(k_{a} = 1\) ).  
+
+On the other hand, \(A_{1}\) , \(A_{2}\) , and \(A\) all obviously lie on \(\mathcal{C}_{a}\) . (For \(A_{1}\) and \(A_{2}\) , the powers are both zero, and for the point \(A\) , we have \(\mathrm{Pow}(P,\gamma_{b}) = AX\cdot AA_{1}\) and \(\mathrm{Pow}(P,\gamma_{c}) = AY\cdot AA_{1}\) .) So \(\mathcal{C}_{a}\) must be exactly the circumcircle of \(\triangle AA_{1}A_{2}\) from the problem statement.  
+
+We turn to evaluating \(k_{a}\) more carefully. First, note that  
+
+\[\angle A_{1}X B_{1} = \angle A_{1}B_{2}B_{1} = \angle C B_{2}B_{1} = 90^{\circ} - \angle B_{2}A C = 90^{\circ} - (60^{\circ} - \gamma) = 30^{\circ} + \gamma .\]  
+
+Now using the law of sines, we derive  
+
+\[\frac{A X}{A B_{1}} = \frac{\sin\angle A B_{1}X}{\sin\angle A X B_{1}} = \frac{\sin(\angle A_{1}X B_{1} - \angle X A B_{1})}{\sin\angle A_{1}X B_{1}}\] \[\qquad = \frac{\sin((30^{\circ} + \gamma) - (30^{\circ} - \beta))}{\sin(30^{\circ} + \gamma)} = \frac{\sin(\beta + \gamma)}{\sin(30^{\circ} + \gamma)}.\]  
+
+Similarly, \(A Y = A C_{1}\cdot \frac{\sin(\beta + \gamma)}{\sin(30^{\circ} + \beta)}\) , so  
+
+\[k_{a} = \frac{A X}{A Y} = \frac{A B_{1}}{A C_{1}}\cdot \frac{\sin(30^{\circ} + \beta)}{\sin(30^{\circ} + \gamma)}.\]  
+
+Now define analogous constants \(k_{b}\) and \(k_{c}\) and circles \(\mathcal{C}_{b}\) and \(\mathcal{C}_{c}\) . Owing to the symmetry of the expressions, we have the key relation  
+
+\[k_{a}k_{b}k_{c} = 1.\]  
+
+In summary, the three circles in the problem statement may be described as  
+
+\[\mathcal{C}_{a} = (A A_{1}A_{2}) = \{\mathrm{points} P\mathrm{such}\mathrm{that}\mathrm{Pow}(P,\gamma_{b}) = k_{a}\mathrm{Pow}(P,\gamma_{c})\}\] \[\mathcal{C}_{b} = (B B_{1}B_{2}) = \{\mathrm{points} P\mathrm{such}\mathrm{that}\mathrm{Pow}(P,\gamma_{c}) = k_{b}\mathrm{Pow}(P,\gamma_{a})\}\] \[\mathcal{C}_{c} = (C C_{1}C_{2}) = \{\mathrm{points} P\mathrm{such}\mathrm{that}\mathrm{Pow}(P,\gamma_{a}) = k_{c}\mathrm{Pow}(P,\gamma_{b})\} .\]  
+
+Since \(k_{a}\) , \(k_{b}\) , \(k_{c}\) have product 1, it follows that any point on at least two of the circles must lie on the third circle as well. The convexity of hexagon \(A_{2}C_{1}B_{2}A_{1}C_{2}B_{1}\) mentioned earlier ensures these any two of these circles do intersect at two different points, completing the solution.
+
+