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+# IMO 2014 Solution Notes  
+
+Evan Chen 《陳誼廷》  
+
+18 October 2025  
+
+This is a compilation of solutions for the 2014 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 3  
+
+1.1 IMO 2014/1, proposed by Gerhard Woeginger (AUT) 3  
+
+1.2 IMO 2014/2, proposed by Tonči Kokan (HRV) 4  
+
+1.3 IMO 2014/3, proposed by ALi Zamani (IRN) 6  
+
+2 Solutions to Day 2 9  
+
+2.1 IMO 2014/4, proposed by Giorgi Arabidze (GEO) 9  
+
+2.2 IMO 2014/5, proposed by Gerhard Woeginger (LUX) 11  
+
+2.3 IMO 2014/6, proposed by Gerhard Woeginger (AUT) 13
+
+
+
+## Problems  
+
+1. Let \(a_{0}< a_{1}< a_{2}< \dots\) be an infinite sequence of positive integers. Prove that there exists a unique integer \(n\geq 1\) such that  
+
+\[a_{n}< \frac{a_{0} + a_{1} + a_{2} + \cdots + a_{n}}{n}\leq a_{n + 1}.\]  
+
+2. Let \(n\geq 2\) be an integer. Consider an \(n\times n\) chessboard consisting of \(n^{2}\) unit squares. A configuration of \(n\) rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer \(k\) such that, for each peaceful configuration of \(n\) rooks, there is a \(k\times k\) square which does not contain a rook on any of its \(k^{2}\) unit squares.  
+
+3. Convex quadrilateral \(A B C D\) has \(\angle A B C = \angle C D A = 90^{\circ}\) . Point \(H\) is the foot of the perpendicular from \(A\) to \(\overline{{B D}}\) . Points \(S\) and \(T\) lie on sides \(A B\) and \(A D\) , respectively, such that \(H\) lies inside triangle \(S C T\) and  
+
+\[\angle C H S - \angle C S B = 90^{\circ},\quad \angle T H C - \angle D T C = 90^{\circ}.\]  
+
+Prove that line \(B D\) is tangent to the circumcircle of triangle \(T S H\)  
+
+4. Let \(P\) and \(Q\) be on segment \(B C\) of an acute triangle \(A B C\) such that \(\angle P A B =\) \(\angle B C A\) and \(\angle C A Q = \angle A B C\) . Let \(M\) and \(N\) be points on \(\overline{{A P}}\) and \(\overline{{A Q}}\) , respectively, such that \(P\) is the midpoint of \(\overline{{A M}}\) and \(Q\) is the midpoint of \(\overline{{A N}}\) . Prove that \(\overline{{B M}}\) and \(\overline{{C N}}\) meet on the circumcircle of \(\triangle A B C\)  
+
+5. For every positive integer \(n\) , the Bank of Cape Town issues coins of denomination \(\frac{1}{n}\) . Given a finite collection of such coins (of not necessarily different denominations) with total value at most \(99 + \frac{1}{2}\) , prove that it is possible to split this collection into 100 or fewer groups, such that each group has total value at most 1.  
+
+6. A set of lines in the plane is in 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 finite regions. Prove that for all sufficiently large \(n\) , in any set of \(n\) lines in general position it is possible to colour at least \(\sqrt{n}\) lines blue in such a way that none of its finite regions has a completely blue boundary.
+
+
+
+## \(\S 1\) Solutions to Day 1  
+
+## \(\S 1.1\) IMO 2014/1, proposed by Gerhard Woeginger (AUT)  
+
+Available online at https://aops.com/community/p3542095.  
+
+## Problem statement  
+
+Let \(a_{0}< a_{1}< a_{2}< \dots\) be an infinite sequence of positive integers. Prove that there exists a unique integer \(n\geq 1\) such that  
+
+\[a_{n}< \frac{a_{0} + a_{1} + a_{2} + \cdots + a_{n}}{n}\leq a_{n + 1}.\]  
+
+Fedor Petrov presents the following nice solution. Let us define the sequence  
+
+\[b_{n} = (a_{n} - a_{n - 1}) + \cdot \cdot \cdot +(a_{n} - a_{1}).\]  
+
+Since \((a_{n})_{n}\) is increasing, we get \((b_{n})_{n}\) is strictly increasing, and moreover \(b_{1} = 0\) . The problem requires an \(n\) such that  
+
+\[b_{n}< a_{0}\leq b_{n + 1}\]  
+
+which obviously exists and is unique.
+
+
+
+## \(\S 1.2\) IMO 2014/2, proposed by Tonči Kokan (HRV)  
+
+Available online at https://aops.com/community/p3542094.  
+
+## Problem statement  
+
+Let \(n\geq 2\) be an integer. Consider an \(n\times n\) chessboard consisting of \(n^{2}\) unit squares. A configuration of \(n\) rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer \(k\) such that, for each peaceful configuration of \(n\) rooks, there is a \(k\times k\) square which does not contain a rook on any of its \(k^{2}\) unit squares.  
+
+The answer is \(k = \lfloor \sqrt{n - 1} \rfloor\) .  
+
+\(\P\) Proof that the property holds when \(n\geq k^{2} + 1\) . First, assume \(n > k^{2}\) for some \(k\) . We will prove we can find an empty \(k\times k\) square. Indeed, let \(R\) be a rook in the uppermost column, and draw \(k\) squares of size \(k\times k\) directly below it, aligned. There are at most \(k - 1\) rooks among these squares, as desired.  
+
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)
+  
+
+\(\P\) Construction for all \(n\leq k^{2}\) . We first give an example where for \(n = k^{2}\) showing there may be no empty \(k\times k\) square. We draw the example for \(k = 3\) (with the generalization being obvious);  
+
+![](data:image/jpeg;base64,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)
+  
+
+To show that this works, consider for each rook drawing an \(k\times k\) square of \(X\) 's whose bottom- right hand corner is the rook (these may go off the board). These indicate
+
+
+
+positions where one cannot place the upper- left hand corner of any square. It’s easy to see that these cover the entire board, except parts of the last \(k - 1\) columns, which don’t matter anyways.  
+
+It remains to check that \(n \leq k^{2}\) also all work (omitting this step is a common mistake). For this, we can delete rows and column to get an \(n \times n\) board, and then fill in any gaps where we accidentally deleted a rook.
+
+
+
+## \(\S 1.3\) IMO 2014/3, proposed by ALi Zamani (IRN)  
+
+Available online at https://aops.com/community/p3542092.  
+
+## Problem statement  
+
+Convex quadrilateral \(A B C D\) has \(\angle A B C = \angle C D A = 90^{\circ}\) . Point \(H\) is the foot of the perpendicular from \(A\) to \(\overline{{B D}}\) . Points \(S\) and \(T\) lie on sides \(A B\) and \(A D\) , respectively, such that \(H\) lies inside triangle \(S C T\) and  
+
+\[\angle C H S - \angle C S B = 90^{\circ}, \quad \angle T H C - \angle D T C = 90^{\circ}.\]  
+
+Prove that line \(B D\) is tangent to the circumcircle of triangle \(T S H\) .  
+
+First solution (mine). First we rewrite the angle condition in a suitable way.  
+
+Claim — We have \(\angle A T H = \angle T C H + 90^{\circ}\) . Thus the circumcenter of \(\triangle C T H\) lies on \(\overline{{A D}}\) . Similarly the circumcenter of \(\triangle C S H\) lies on \(\overline{{A B}}\) .  
+
+Proof.  
+
+\[\angle A T H = \angle D T H\] \[\qquad = \angle D T C + \angle C T H\] \[\qquad = \angle D T C - \angle T H C - \angle H C T\] \[\qquad = 90^{\circ} - \angle H C T = 90^{\circ} + \angle T C H.\]  
+
+which implies conclusion.  
+
+![](data:image/jpeg;base64,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+  
+
+Let the perpendicular bisector of \(T H\) meet \(A H\) at \(P\) now. It suffices to show that \(\frac{A P}{P H}\) is symmetric in \(b = A D\) and \(d = A B\) , because then \(P\) will be the circumcenter of \(\triangle T S H\) . To do this, set \(A H = \frac{b d}{2 R}\) and \(A C = 2 R\) .  
+
+Let \(O\) denote the circumcenter of \(\triangle C H T\) . Use the Law of Cosines on \(\triangle A C O\) and \(\triangle A H O\) , using variables \(x = A O\) and \(r = H O\) . We get that  
+
+\[r^{2} = x^{2} + A H^{2} - 2x\cdot A H\cdot \frac{d}{2R} = x^{2} + (2R)^{2} - 2b x.\]
+
+
+
+By the angle bisector theorem, \(\frac{AP}{PH} = \frac{AO}{HO}\) .  
+
+The rest is computation: notice that  
+
+\[r^{2} - x^{2} = h^{2} - 2xh\cdot \frac{d}{2R} = (2R)^{2} - 2bx\]  
+
+where \(h = AH = \frac{bd}{2R}\) , whence  
+
+\[x = \frac{(2R)^{2} - h^{2}}{2b - 2h\cdot\frac{d}{2R}}.\]  
+
+Moreover,  
+
+\[\frac{1}{2}\left(\frac{r^{2}}{x^{2}} -1\right) = \frac{1}{x}\left(\frac{2}{x} R^{2} - b\right).\]  
+
+Now, if we plug in the \(x\) in the right- hand side of the above, we obtain  
+
+\[\frac{2b - 2h\cdot\frac{d}{2R}}{4R^{2} - h^{2}}\left(\frac{2b - 2h\cdot\frac{d}{2R}}{4R^{2}\cdot h^{2}}\cdot 2R^{2} - b\right) = \frac{2h}{(4R^{2} - h^{2})^{2}}\left(b - h\cdot \frac{d}{2R}\right)\left(-2hdR + bh^{2}\right).\]  
+
+Pulling out a factor of \(- 2Rh\) from the rightmost term, we get something that is symmetric in \(b\) and \(d\) , as required.  
+
+\(\P\) Second solution (Victor Reis). Here is the fabled solution using inversion at \(H\) . First, we rephrase the angle conditions in the following ways:  
+
+\(\overline{AD} \perp (THC)\) , which is equivalent to the claim from the first solution.  
+
+\(\overline{AB} \perp (SHC)\) , by symmetry.  
+
+\(\overline{AC} \perp (ABCD)\) , by definition.  
+
+Now for concreteness we will use a negative inversion at \(H\) which swaps \(B\) and \(D\) and overlay it on the original diagram. As usual we denote inverses with stars.  
+
+Let us describe the inverted problem. We let \(M\) and \(N\) denote the midpoints of \(\overline{A^{*}B^{*}}\) and \(\overline{A^{*}D^{*}}\) , which are the centers of \((HA^{*}B^{*})\) and \((HA^{*}D^{*})\) . From \(\overline{T^{*}C^{*}} \perp (HA^{*}D^{*})\) , we know have \(C^{*}\) , \(M\) , \(T^{*}\) collinear. Similarly, \(C^{*}\) , \(N\) , \(S^{*}\) are collinear. We have that \((A^{*}HC^{*})\) is orthogonal to \((ABCD)\) which remains fixed. We wish to show \(\overline{T^{*}S^{*}}\) and \(\overline{MN}\) are parallel.  
+
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+
+
+
+
+Lot \(\omega\) denote the circumcircle of \(\triangle A^{*}H C^{*}\) , which is orthogonal to the original circle \((A B C D)\) . It would suffices to show \((A^{*}H C^{*})\) is an \(H\) - Apollonius circle with respect to \(\overline{{M N}}\) , from which we would get \(C^{*}M / H M = C^{*}N / H N\) .  
+
+However, \(\omega\) through \(H\) and \(A\) , hence it center lies on line \(M N\) . Moreover \(\omega\) is orthogonal to \((A^{*}M N)\) (since \((A^{*}M N)\) and \((A^{*}B D)\) are homothetic). This is enough (for example, if we let \(O\) denote the center of \(\omega\) , we now have \(\mathrm{r}(\omega)^{2} = O H^{2} = O M \cdot O N\) ). (Note in this proof that the fact that \(C^{*}\) lies on \((A B C D)\) is not relevant.)
+
+
+
+## \(\S 2\) Solutions to Day 2  
+
+## \(\S 2.1\) IMO 2014/4, proposed by Giorgi Arabidze (GEO)  
+
+Available online at https://aops.com/community/p3543136.  
+
+## Problem statement  
+
+Let \(P\) and \(Q\) be on segment \(BC\) of an acute triangle \(ABC\) such that \(\angle PAB = \angle BCA\) and \(\angle CAQ = \angle ABC\) . Let \(M\) and \(N\) be points on \(\overline{AP}\) and \(\overline{AQ}\) , respectively, such that \(P\) is the midpoint of \(\overline{AM}\) and \(Q\) is the midpoint of \(\overline{AN}\) . Prove that \(\overline{BM}\) and \(\overline{CN}\) meet on the circumcircle of \(\triangle ABC\) .  
+
+We give three solutions.  
+
+\(\P\) First solution by harmonic bundles. Let \(\overline{BM}\) intersect the circumcircle again at \(X\) .  
+
+![](data:image/jpeg;base64,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+  
+
+The angle conditions imply that the tangent to \((ABC)\) at \(B\) is parallel to \(\overline{AP}\) . Let \(\infty\) be the point at infinity along line \(AP\) . Then  
+
+\[-1 = (AM;P\infty)\stackrel {B}{=}(AX;BC).\]  
+
+Similarly, if \(\overline{CN}\) meets the circumcircle at \(Y\) then \((AY;BC) = - 1\) as well. Hence \(X = Y\) , which implies the problem.  
+
+\(\P\) Second solution by similar triangles. Once one observes \(\triangle CAQ \sim \triangle CBA\) , one can construct \(D\) the reflection of \(B\) across \(A\) , so that \(\triangle CAN \sim \triangle CBD\) . Similarly, letting \(E\) be the reflection of \(C\) across \(A\) , we get \(\triangle BAP \sim \triangle BCA \Rightarrow \triangle BAM \sim \triangle BCE\) . Now to show \(\angle ABM + \angle ACN = 180^\circ\) it suffices to show \(\angle EBC + \angle BCD = 180^\circ\) , which follows since \(BCDE\) is a parallelogram.
+
+
+
+Third solution by barycentric coordinates. Since \(PB = c^{2} / a\) we have  
+
+\[P = (0:a^{2} - c^{2}:c^{2})\]  
+
+so the reflection \(\vec{M} = 2\vec{P} - \vec{A}\) has coordinates  
+
+\[M = (-a^{2}:2(a^{2} - c^{2}):2c^{2}).\]  
+
+Similarly \(N = (-a^{2}:2b^{2}:2(b^{2} - a^{2}))\) . Thus  
+
+\[\overline{BM}\cap \overline{CN} = (-a^{2}:2b^{2}:2c^{2})\]  
+
+which clearly lies on the circumcircle, and is in fact the point identified in the first solution.
+
+
+
+## \(\S 2.2\) IMO 2014/5, proposed by Gerhard Woeginger (LUX)  
+
+Available online at https://aops.com/community/p3543144.  
+
+## Problem statement  
+
+For every positive integer \(n\) , the Bank of Cape Town issues coins of denomination \(\frac{1}{n}\) . Given a finite collection of such coins (of not necessarily different denominations) with total value at most \(99 + \frac{1}{2}\) , prove that it is possible to split this collection into 100 or fewer groups, such that each group has total value at most 1.  
+
+We'll prove the result for at most \(k - \frac{k}{2k + 1}\) with \(k\) groups. First, 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 \(\frac{1}{2m + 1}\) appears \(2m + 1\) times, group it into a single group and induct downwards.  
+
+Apply this operation repeatedly until it cannot be done anymore.  
+
+Now construct boxes \(B_{0}\) , \(B_{1}\) , ..., \(B_{k - 1}\) . In box \(B_{0}\) put any coins of size \(\frac{1}{2}\) (clearly there is at most one). More generally for \(m \geq 0\) , \(B_{m}\) , put coins of size \(\frac{1}{2m + 1}\) and \(\frac{1}{2m + 2}\) (there at most \(2m\) of the former and at most one of the latter). Note that  
+
+\[\mathrm{total~weight~in~}B_{m}\leq 2m\cdot \frac{1}{2m + 1} +\frac{1}{2m + 2} < 1.\]  
+
+Finally, place the remaining "light" coins of size at most \(\frac{1}{2k + 1}\) in a pile.  
+
+Remark. One way to explain where this boxing comes from is to imagine what happens after applying the operation repeatedly until it can't be done any more. We expect the most troublesome situation would be if the leftover coins are as large as possible, which looks like  
+
+\[\frac{1}{2},\quad \frac{1}{3}\times 2,\quad \frac{1}{4},\quad \frac{1}{5}\times 4,\quad \frac{1}{6},\quad \frac{1}{7}\times 6,\quad \frac{1}{8},\quad \dots\]  
+
+Seeing this the boxing \(B_{i}\) is quite reasonable because it groups these into groups of almost 1 without too much effort:  
+
+\[B_{0}\longrightarrow \frac{1}{2\] \[B_{1}\longrightarrow \frac{1}{3}\cdot 2 + \frac{1}{4} = \frac{11}{12\] \[B_{2}\longrightarrow \frac{1}{5}\cdot 4 + \frac{1}{6} = \frac{29}{30\] \[B_{3}\longrightarrow \frac{1}{7}\cdot 6 + \frac{1}{8} = \frac{55}{56\] \[\vdots\]  
+
+Then just toss coins from the pile into the boxes arbitrarily, other than the proviso that no box should have its weight exceed 1. We claim this uses up all coins in the pile. Assume not, and that some coin remains in the pile when all the boxes are saturated. Then all the boxes must have at least \(1 - \frac{1}{2k + 1}\) , meaning the total amount in the boxes is strictly greater than  
+
+\[k\left(1 - \frac{1}{2k + 1}\right) > k - \frac{1}{2}\]
+
+
+
+which is a contradiction.  
+
+Remark. This gets a stronger bound \(k - \frac{k}{2k + 1}\) than the requested \(k - \frac{1}{2}\) .
+
+
+
+## \(\S 2.3\) IMO 2014/6, proposed by Gerhard Woeginger (AUT)  
+
+Available online at https://aops.com/community/p3543151.  
+
+## Problem statement  
+
+A set of lines in the plane is in 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 finite regions. Prove that for all sufficiently large \(n\) , in any set of \(n\) lines in general position it is possible to colour at least \(\sqrt{n}\) lines blue in such a way that none of its finite regions has a completely blue boundary.  
+
+Suppose we have colored \(k\) of the lines blue, and that it is not possible to color any additional lines. That means any of the \(n - k\) non- blue lines is the side of some finite region with an otherwise entirely blue perimeter. For each such line \(\ell\) , select one such region, and take the next counterclockwise vertex; this is the intersection of two blue lines \(v\) . We'll say \(\ell\) is the eyelid of \(v\) .  
+
+![](data:image/jpeg;base64,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)
+  
+
+You can prove without too much difficulty that every intersection of two blue lines has at most two eyelids. Since there are \(\binom{k}{2}\) such intersections, we see that  
+
+\[n - k\leq 2\binom{k}{2} = k^{2} - k\]  
+
+so \(n\leq k^{2}\) , as required.  
+
+Remark. In fact, \(k = \sqrt{n}\) is "sharp for greedy algorithms", as illustrated below for \(k = 3\) :  
+
+![](data:image/jpeg;base64,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)
+
+
+