{
  "index": "2005-A-6",
  "type": "COMB",
  "tag": [
    "COMB",
    "GEO"
  ],
  "difficulty": "",
  "question": "Let $n$ be given, $n \\geq 4$, and suppose that $P_1, P_2, \\dots, P_n$\nare $n$ randomly, independently and uniformly, chosen points on a circle.\nConsider the convex $n$-gon whose vertices are the $P_i$. What is the\nprobability that at least one of the vertex angles of this polygon is\nacute?",
  "solution": "\\textbf{First solution:}\nThe angle at a vertex $P$ is acute if and only if all of the other points\nlie on an open semicircle. We first deduce from this that if there are any\ntwo acute angles at all, they must occur consecutively. Suppose the contrary;\nlabel the vertices $Q_1, \\dots, Q_n$ in counterclockwise order (starting anywhere),\nand suppose that the angles at $Q_1$ and $Q_i$ are acute for some $i$\nwith $3 \\leq i \\leq n-1$.\nThen the open semicircle starting at $Q_2$ and proceeding counterclockwise\nmust contain all of $Q_3, \\dots, Q_n$, while the open semicircle starting at\n$Q_i$ and proceeding counterclockwise must contain $Q_{i+1}, \\dots,\nQ_n, Q_1, \\dots, Q_{i-1}$. Thus two open semicircles cover the entire circle,\ncontradiction.\n\nIt follows that if the polygon has at least one acute angle, then\nit has either one acute angle or two acute angles\noccurring consecutively. In particular, there\nis a unique pair of consecutive vertices $Q_1, Q_2$ in counterclockwise order\nfor which $\\angle Q_2$ is acute and $\\angle Q_1$ is not acute.\nThen the remaining points all lie in the arc from the antipode of $Q_1$\nto $Q_1$, but $Q_2$ cannot lie in the arc, and the remaining points\ncannot all lie in the arc from the antipode of $Q_1$ to the antipode\nof $Q_2$. Given the choice of $Q_1, Q_2$, let $x$ be the measure of\nthe counterclockwise arc from $Q_1$ to $Q_2$; then the probability that\nthe other points fall into position is\n$2^{-n+2} - x^{n-2}$ if $x \\leq 1/2$ and 0 otherwise.\n\nHence the probability that the polygon has at least one acute angle with\na \\emph{given} choice of which two points will act as $Q_1$ and $Q_2$ is\n\\[\n\\int_0^{1/2} (2^{-n+2} - x^{n-2})\\,dx\n= \\frac{n-2}{n-1} 2^{-n+1}.\n\\]\nSince there are $n(n-1)$ choices for which two points act as $Q_1$ and $Q_2$,\nthe probability of at least one acute angle is $n(n-2) 2^{-n+1}$.\n\n\\textbf{Second solution:}\n(by Calvin Lin)\nAs in the first solution, we may compute the probability that for a\nparticular one of the points $Q_1$, the angle at $Q_1$ is not acute but\nthe following angle is, and then multiply by $n$.\nImagine picking the points by first choosing $Q_1$, then\npicking $n-1$ \\emph{pairs}\nof antipodal points and then picking one\nmember of each pair. Let $R_2, \\dots, R_n$ be the points of the pairs\nwhich lie in the semicircle, taken in order away from $Q_1$,\nand let $S_2, \\dots, S_n$ be the antipodes of these. Then to\nget the desired situation,\nwe must choose from the pairs to end up with all but one of the\n$S_i$, and we cannot take $R_n$ and the other $S_i$ or else $\\angle Q_1$ will be\nacute. That gives us $(n-2)$ good choices out of $2^{n-1}$; since we could\nhave chosen $Q_1$ to be any of the $n$ points, the probability is again\n$n(n-2) 2^{-n+1}$.",
  "vars": [
    "x",
    "i",
    "P_1",
    "P_2",
    "P_n",
    "P_i",
    "Q_1",
    "Q_2",
    "Q_3",
    "Q_i",
    "Q_i-1",
    "Q_i+1",
    "Q_n",
    "R_2",
    "R_n",
    "S_2",
    "S_i",
    "S_n"
  ],
  "params": [
    "n"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "n": "totalpoints",
        "x": "arcmeasure",
        "i": "indexvar",
        "P_1": "firstpoint",
        "P_2": "secondpoint",
        "P_n": "lastpoint",
        "P_i": "ithpoint",
        "Q_1": "firstvertex",
        "Q_2": "secondvertex",
        "Q_3": "thirdvertex",
        "Q_i": "indexvertex",
        "Q_i-1": "prevvertex",
        "Q_{i-1}": "prevvertex",
        "Q_i+1": "nextvertex",
        "Q_{i+1}": "nextvertex",
        "Q_n": "lastvertex",
        "R_2": "secondrpoint",
        "R_n": "lastrpoint",
        "S_2": "secondsopoint",
        "S_i": "indexspoint",
        "S_n": "lastspoint"
      },
      "question": "Let $totalpoints$ be given, $totalpoints \\geq 4$, and suppose that $firstpoint, secondpoint, \\dots, lastpoint$\nare $totalpoints$ randomly, independently and uniformly, chosen points on a circle.\nConsider the convex $totalpoints$-gon whose vertices are the $ithpoint$. What is the\nprobability that at least one of the vertex angles of this polygon is\nacute?",
      "solution": "\\textbf{First solution:}\nThe angle at a vertex $P$ is acute if and only if all of the other points\nlie on an open semicircle. We first deduce from this that if there are any\ntwo acute angles at all, they must occur consecutively. Suppose the contrary;\nlabel the vertices $firstvertex, \\dots, lastvertex$ in counterclockwise order (starting anywhere),\nand suppose that the angles at $firstvertex$ and $indexvertex$ are acute for some $indexvar$\nwith $3 \\leq indexvar \\leq totalpoints-1$.\nThen the open semicircle starting at $secondvertex$ and proceeding counterclockwise\nmust contain all of $thirdvertex, \\dots, lastvertex$, while the open semicircle starting at\n$indexvertex$ and proceeding counterclockwise must contain $nextvertex, \\dots,\nlastvertex, firstvertex, \\dots, prevvertex$. Thus two open semicircles cover the entire circle,\ncontradiction.\n\nIt follows that if the polygon has at least one acute angle, then\nit has either one acute angle or two acute angles\noccurring consecutively. In particular, there\nis a unique pair of consecutive vertices $firstvertex, secondvertex$ in counterclockwise order\nfor which $\\angle secondvertex$ is acute and $\\angle firstvertex$ is not acute.\nThen the remaining points all lie in the arc from the antipode of $firstvertex$\nto $firstvertex$, but $secondvertex$ cannot lie in the arc, and the remaining points\ncannot all lie in the arc from the antipode of $firstvertex$ to the antipode\nof $secondvertex$. Given the choice of $firstvertex, secondvertex$, let $arcmeasure$ be the measure of\nthe counterclockwise arc from $firstvertex$ to $secondvertex$; then the probability that\nthe other points fall into position is\n$2^{-totalpoints+2} - arcmeasure^{totalpoints-2}$ if $arcmeasure \\leq 1/2$ and 0 otherwise.\n\nHence the probability that the polygon has at least one acute angle with\na \\emph{given} choice of which two points will act as $firstvertex$ and $secondvertex$ is\n\\[\n\\int_0^{1/2} \\left(2^{-totalpoints+2} - arcmeasure^{totalpoints-2}\\right)\\,d arcmeasure\n= \\frac{totalpoints-2}{totalpoints-1} 2^{-totalpoints+1}.\n\\]\nSince there are $totalpoints(totalpoints-1)$ choices for which two points act as $firstvertex$ and $secondvertex$,\nthe probability of at least one acute angle is $totalpoints(totalpoints-2) 2^{-totalpoints+1}$.\n\n\\textbf{Second solution:}\n(by Calvin Lin)\nAs in the first solution, we may compute the probability that for a\nparticular one of the points $firstvertex$, the angle at $firstvertex$ is not acute but\nthe following angle is, and then multiply by $totalpoints$.\nImagine picking the points by first choosing $firstvertex$, then\npicking $totalpoints-1$ \\emph{pairs}\nof antipodal points and then picking one\nmember of each pair. Let $secondrpoint, \\dots, lastrpoint$ be the points of the pairs\nwhich lie in the semicircle, taken in order away from $firstvertex$,\nand let $secondsopoint, \\dots, lastspoint$ be the antipodes of these. Then to\nget the desired situation,\nwe must choose from the pairs to end up with all but one of the\n$indexspoint$, and we cannot take $lastrpoint$ and the other $indexspoint$ or else $\\angle firstvertex$ will be\nacute. That gives us $(totalpoints-2)$ good choices out of $2^{totalpoints-1}$; since we could\nhave chosen $firstvertex$ to be any of the $totalpoints$ points, the probability is again\n$totalpoints(totalpoints-2) 2^{-totalpoints+1}$.}",
      "confidence": 0.11
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "thunderbolt",
        "i": "lighthouse",
        "P_1": "sunflower",
        "P_2": "blackboard",
        "P_n": "rainshower",
        "P_i": "moonstone",
        "Q_1": "honeycomb",
        "Q_2": "horseshoe",
        "Q_3": "cinnamon",
        "Q_i": "willowtree",
        "Q_i-1": "driftwood",
        "Q_i+1": "dandelion",
        "Q_n": "starflower",
        "R_2": "marshmallow",
        "R_n": "stonebridge",
        "S_2": "featherbed",
        "S_i": "gingerroot",
        "S_n": "silverfish",
        "n": "parchment"
      },
      "question": "Let $parchment$ be given, $parchment \\geq 4$, and suppose that $sunflower, blackboard, \\dots, rainshower$\nare $parchment$ randomly, independently and uniformly, chosen points on a circle.\nConsider the convex $parchment$-gon whose vertices are the $moonstone$. What is the\nprobability that at least one of the vertex angles of this polygon is\nacute?",
      "solution": "\\textbf{First solution:}\nThe angle at a vertex $P$ is acute if and only if all of the other points\nlie on an open semicircle. We first deduce from this that if there are any\ntwo acute angles at all, they must occur consecutively. Suppose the contrary;\nlabel the vertices $honeycomb, \\dots, starflower$ in counterclockwise order (starting anywhere),\nand suppose that the angles at $honeycomb$ and $willowtree$ are acute for some $lighthouse$\nwith $3 \\leq lighthouse \\leq parchment-1$.\nThen the open semicircle starting at $horseshoe$ and proceeding counterclockwise\nmust contain all of $cinnamon, \\dots, starflower$, while the open semicircle starting at\n$willowtree$ and proceeding counterclockwise must contain $dandelion, \\dots,\nstarflower, honeycomb, \\dots, driftwood$. Thus two open semicircles cover the entire circle,\ncontradiction.\n\nIt follows that if the polygon has at least one acute angle, then\nit has either one acute angle or two acute angles\noccurring consecutively. In particular, there\nis a unique pair of consecutive vertices $honeycomb, horseshoe$ in counterclockwise order\nfor which $\\angle horseshoe$ is acute and $\\angle honeycomb$ is not acute.\nThen the remaining points all lie in the arc from the antipode of $honeycomb$\nto $honeycomb$, but $horseshoe$ cannot lie in the arc, and the remaining points\ncannot all lie in the arc from the antipode of $honeycomb$ to the antipode\nof $horseshoe$. Given the choice of $honeycomb, horseshoe$, let $thunderbolt$ be the measure of\nthe counterclockwise arc from $honeycomb$ to $horseshoe$; then the probability that\nthe other points fall into position is\n$2^{-parchment+2} - thunderbolt^{parchment-2}$ if $thunderbolt \\leq 1/2$ and 0 otherwise.\n\nHence the probability that the polygon has at least one acute angle with\na \\emph{given} choice of which two points will act as $honeycomb$ and $horseshoe$ is\n\\[\n\\int_0^{1/2} (2^{-parchment+2} - thunderbolt^{parchment-2})\\,dthunderbolt\n= \\frac{parchment-2}{parchment-1} 2^{-parchment+1}.\n\\]\nSince there are $parchment(parchment-1)$ choices for which two points act as $honeycomb$ and $horseshoe$,\nthe probability of at least one acute angle is $parchment(parchment-2) 2^{-parchment+1}$.\n\n\\textbf{Second solution:}\n(by Calvin Lin)\nAs in the first solution, we may compute the probability that for a\nparticular one of the points $honeycomb$, the angle at $honeycomb$ is not acute but\nthe following angle is, and then multiply by $parchment$.\nImagine picking the points by first choosing $honeycomb$, then\npicking $parchment-1$ \\emph{pairs}\nof antipodal points and then picking one\nmember of each pair. Let $marshmallow, \\dots, stonebridge$ be the points of the pairs\nwhich lie in the semicircle, taken in order away from $honeycomb$,\nand let $featherbed, \\dots, silverfish$ be the antipodes of these. Then to\nget the desired situation,\nwe must choose from the pairs to end up with all but one of the\n$gingerroot$, and we cannot take $stonebridge$ and the other $gingerroot$ or else $\\angle honeycomb$ will be\nacute. That gives us $(parchment-2)$ good choices out of $2^{parchment-1}$; since we could\nhave chosen $honeycomb$ to be any of the $parchment$ points, the probability is again\n$parchment(parchment-2) 2^{-parchment+1}$.}"
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "nullangle",
        "i": "unindexed",
        "P_1": "fixedpoint",
        "P_2": "staticpoint",
        "P_n": "terminalpoint",
        "P_i": "determinedpoint",
        "Q_1": "concavepoint",
        "Q_2": "flatpoint",
        "Q_3": "emptyvertex",
        "Q_i": "lockedpoint",
        "Q_i-1": "forwardpoint",
        "Q_i+1": "backwardpoint",
        "Q_n": "startingpoint",
        "R_2": "outerbegin",
        "R_n": "outerfinal",
        "S_2": "proximaltwo",
        "S_i": "proximalmid",
        "S_n": "proximalmax",
        "n": "voidnumber"
      },
      "question": "Let $voidnumber$ be given, $voidnumber \\geq 4$, and suppose that $fixedpoint, staticpoint, \\dots, terminalpoint$\nare $voidnumber$ randomly, independently and uniformly, chosen points on a circle.\nConsider the convex $voidnumber$-gon whose vertices are the $determinedpoint$. What is the\nprobability that at least one of the vertex angles of this polygon is\nacute?",
      "solution": "\\textbf{First solution:}\nThe angle at a vertex $P$ is acute if and only if all of the other points\nlie on an open semicircle. We first deduce from this that if there are any\ntwo acute angles at all, they must occur consecutively. Suppose the contrary;\nlabel the vertices $concavepoint, \\dots, startingpoint$ in counterclockwise order (starting anywhere),\nand suppose that the angles at $concavepoint$ and $lockedpoint$ are acute for some $unindexed$\nwith $3 \\leq unindexed \\leq voidnumber-1$.\nThen the open semicircle starting at $flatpoint$ and proceeding counterclockwise\nmust contain all of $emptyvertex, \\dots, startingpoint$, while the open semicircle starting at\n$lockedpoint$ and proceeding counterclockwise must contain $backwardpoint, \\dots,\nstartingpoint, concavepoint, \\dots, forwardpoint$. Thus two open semicircles cover the entire circle,\ncontradiction.\n\nIt follows that if the polygon has at least one acute angle, then\nit has either one acute angle or two acute angles\noccurring consecutively. In particular, there\nis a unique pair of consecutive vertices $concavepoint, flatpoint$ in counterclockwise order\nfor which $\\angle flatpoint$ is acute and $\\angle concavepoint$ is not acute.\nThen the remaining points all lie in the arc from the antipode of $concavepoint$\nto $concavepoint$, but $flatpoint$ cannot lie in the arc, and the remaining points\ncannot all lie in the arc from the antipode of $concavepoint$ to the antipode\nof $flatpoint$. Given the choice of $concavepoint, flatpoint$, let $nullangle$ be the measure of\nthe counterclockwise arc from $concavepoint$ to $flatpoint$; then the probability that\nthe other points fall into position is\n$2^{-\\voidnumber+2} - nullangle^{\\voidnumber-2}$ if $nullangle \\leq 1/2$ and 0 otherwise.\n\nHence the probability that the polygon has at least one acute angle with\na \\emph{given} choice of which two points will act as $concavepoint$ and $flatpoint$ is\n\\[\n\\int_0^{1/2} \\bigl(2^{-\\voidnumber+2} - nullangle^{\\voidnumber-2}\\bigr)\\,d nullangle\n= \\frac{\\voidnumber-2}{\\voidnumber-1} 2^{-\\voidnumber+1}.\n\\]\nSince there are $\\voidnumber(\\voidnumber-1)$ choices for which two points act as $concavepoint$ and $flatpoint$,\nthe probability of at least one acute angle is $\\voidnumber(\\voidnumber-2) 2^{-\\voidnumber+1}$.\n\n\\textbf{Second solution:}\n(by Calvin Lin)\nAs in the first solution, we may compute the probability that for a\nparticular one of the points $concavepoint$, the angle at $concavepoint$ is not acute but\nthe following angle is, and then multiply by $voidnumber$.\nImagine picking the points by first choosing $concavepoint$, then\npicking $voidnumber-1$ \\emph{pairs}\nof antipodal points and then picking one\nmember of each pair. Let $outerbegin, \\dots, outerfinal$ be the points of the pairs\nwhich lie in the semicircle, taken in order away from $concavepoint$,\nand let $proximaltwo, \\dots, proximalmax$ be the antipodes of these. Then to\nget the desired situation,\nwe must choose from the pairs to end up with all but one of the\n$proximalmid$, and we cannot take $outerfinal$ and the other $proximalmid$ or else $\\angle concavepoint$ will be\nacute. That gives us $(\\voidnumber-2)$ good choices out of $2^{\\voidnumber-1}$; since we could\nhave chosen $concavepoint$ to be any of the $\\voidnumber$ points, the probability is again\n$\\voidnumber(\\voidnumber-2) 2^{-\\voidnumber+1}$.}"
    },
    "garbled_string": {
      "map": {
        "x": "plxqrewv",
        "i": "mznvqoro",
        "P_1": "floszjkd",
        "P_2": "gnivpqal",
        "P_n": "hjtycwem",
        "P_i": "kbxrdoan",
        "Q_1": "opdfzcam",
        "Q_2": "rqisvalb",
        "Q_3": "tehvymku",
        "Q_i": "ugosnqre",
        "Q_i-1": "vhqkplds",
        "Q_i+1": "wynrszbe",
        "Q_n": "xjdfqoeu",
        "R_2": "ysalipzg",
        "R_n": "zpwemvct",
        "S_2": "aibgzkrl",
        "S_i": "bqncxvto",
        "S_n": "cuvhykam",
        "n": "kajdpeoq"
      },
      "question": "Let $kajdpeoq$ be given, $kajdpeoq \\geq 4$, and suppose that $floszjkd, gnivpqal, \\dots, hjtycwem$\nare $kajdpeoq$ randomly, independently and uniformly, chosen points on a circle.\nConsider the convex $kajdpeoq$-gon whose vertices are the $kbxrdoan$. What is the\nprobability that at least one of the vertex angles of this polygon is\nacute?",
      "solution": "\\textbf{First solution:}\nThe angle at a vertex $P$ is acute if and only if all of the other points\nlie on an open semicircle. We first deduce from this that if there are any\ntwo acute angles at all, they must occur consecutively. Suppose the contrary;\nlabel the vertices $opdfzcam, \\dots, xjdfqoeu$ in counterclockwise order (starting anywhere),\nand suppose that the angles at $opdfzcam$ and $ugosnqre$ are acute for some $mznvqoro$\nwith $3 \\leq mznvqoro \\leq kajdpeoq-1$.\nThen the open semicircle starting at $rqisvalb$ and proceeding counterclockwise\nmust contain all of $tehvymku, \\dots, xjdfqoeu$, while the open semicircle starting at\n$ugosnqre$ and proceeding counterclockwise must contain $wynrszbe, \\dots,\nxjdfqoeu, opdfzcam, \\dots, vhqkplds$. Thus two open semicircles cover the entire circle,\ncontradiction.\n\nIt follows that if the polygon has at least one acute angle, then\nit has either one acute angle or two acute angles\noccurring consecutively. In particular, there\nis a unique pair of consecutive vertices $opdfzcam, rqisvalb$ in counterclockwise order\nfor which $\\angle rqisvalb$ is acute and $\\angle opdfzcam$ is not acute.\nThen the remaining points all lie in the arc from the antipode of $opdfzcam$\nto $opdfzcam$, but $rqisvalb$ cannot lie in the arc, and the remaining points\ncannot all lie in the arc from the antipode of $opdfzcam$ to the antipode\nof $rqisvalb$. Given the choice of $opdfzcam, rqisvalb$, let $plxqrewv$ be the measure of\nthe counterclockwise arc from $opdfzcam$ to $rqisvalb$; then the probability that\nthe other points fall into position is\n$2^{-kajdpeoq+2} - plxqrewv^{kajdpeoq-2}$ if $plxqrewv \\leq 1/2$ and 0 otherwise.\n\nHence the probability that the polygon has at least one acute angle with\na \\emph{given} choice of which two points will act as $opdfzcam$ and $rqisvalb$ is\n\\[\n\\int_0^{1/2} (2^{-kajdpeoq+2} - plxqrewv^{kajdpeoq-2})\\,dplxqrewv\n= \\frac{kajdpeoq-2}{kajdpeoq-1} 2^{-kajdpeoq+1}.\n\\]\nSince there are $kajdpeoq(kajdpeoq-1)$ choices for which two points act as $opdfzcam$ and $rqisvalb$,\nthe probability of at least one acute angle is $kajdpeoq(kajdpeoq-2) 2^{-kajdpeoq+1}$.\n\n\\textbf{Second solution:}\n(by Calvin Lin)\nAs in the first solution, we may compute the probability that for a\nparticular one of the points $opdfzcam$, the angle at $opdfzcam$ is not acute but\nthe following angle is, and then multiply by $kajdpeoq$.\nImagine picking the points by first choosing $opdfzcam$, then\npicking $kajdpeoq-1$ \\emph{pairs}\nof antipodal points and then picking one\nmember of each pair. Let $ysalipzg, \\dots, zpwemvct$ be the points of the pairs\nwhich lie in the semicircle, taken in order away from $opdfzcam$,\nand let $aibgzkrl, \\dots, cuvhykam$ be the antipodes of these. Then to\nget the desired situation,\nwe must choose from the pairs to end up with all but one of the\n$bqncxvto$, and we cannot take $ysalipzg$ and the other $bqncxvto$ or else $\\angle opdfzcam$ will be\nacute. That gives us $(kajdpeoq-2)$ good choices out of $2^{kajdpeoq-1}$; since we could\nhave chosen $opdfzcam$ to be any of the $kajdpeoq$ points, the probability is again\n$kajdpeoq(kajdpeoq-2) 2^{-kajdpeoq+1}$.}",
      "confidence": "0.15"
    },
    "kernel_variant": {
      "question": "Let $n$ be an integer with $n\\ge 4$ and mark a distinguished point $A$ on a circle whose total length is normalised to $1$.  \nChoose the remaining $n-1$ points independently and uniformly at random on the circle and, starting with $A$, label the vertices in clockwise order  \n\\[\nV_1=A,\\;V_2,\\ldots ,V_n .\n\\]\nJoin successive vertices to obtain the cyclic, convex $n$-gon $V_1V_2\\ldots V_n$.\n\n(a)\\;Compute the probability that the two consecutive interior angles at $V_1$ and $V_2$ are both acute (that is, each is strictly smaller than $90^{\\circ}$).\n\n(b)\\;Show that a cyclic $n$-gon can never possess three acute interior angles and that, if two acute angles exist, they must be consecutive.  \nDeduce the probability that the polygon has \\emph{exactly} two acute interior angles, that these two angles are consecutive and that no other interior angle is acute.",
      "solution": "\\textbf{Notation.}  \nIndices are understood modulo $n$.  Let  \n\\[\nX_1:=\\operatorname{arc}(V_1V_2),\\;X_2:=\\operatorname{arc}(V_2V_3),\\;\\ldots ,\\;\nX_n:=\\operatorname{arc}(V_nV_1),\n\\]\nall measured clockwise.  Because the $n-1$ additional points are chosen independently and uniformly, the random vector  \n\\[\n(X_1,\\ldots ,X_n)\n\\]\nis uniformly distributed over the standard $(n-1)$-simplex\n\\[\nx_i>0,\\qquad\\sum_{i=1}^{n}x_i=1,\n\\]\nthat is, it has Dirichlet law $\\mathcal D(1,1,\\ldots ,1)$ with constant density $(n-1)!$.\n\n\\textbf{Inscribed–angle criterion.}  \nFor a cyclic polygon the interior angle at $V_i$ equals  \n\\[\n\\angle V_i=\\pi\\bigl(1-(X_{i-1}+X_i)\\bigr). \\tag{1}\n\\]\nHence $\\angle V_i$ is acute $(<\\tfrac{\\pi}{2})$ iff  \n\\[\nX_{i-1}+X_i>\\tfrac12. \\tag{2}\n\\]\n\n%---------------------------------------------------------------\n\\textbf{Part (a) – two \\emph{prescribed}, consecutive angles are acute}\n%---------------------------------------------------------------\n\nWe require  \n\\[\n\\mathbb P:=\\mathbb P\\bigl(X_n+X_1>\\tfrac12,\\;X_1+X_2>\\tfrac12\\bigr). \\tag{3}\n\\]\n\n\\emph{Step 1: condition on $X_1$.}  \nThe marginal density of $X_1$ is $\\mathrm{Beta}(1,n-1)$, viz.  \n\\[\nf_1(x)=(n-1)(1-x)^{n-2},\\qquad 0<x<1. \\tag{4}\n\\]\nWrite $\\mathbb P=\\int_{0}^{1}P(x)f_1(x)\\,dx$, where  \n\\[\nP(x)=\\mathbb P\\!\\Bigl(X_n>\\tfrac12-x,\\;X_2>\\tfrac12-x\\;\\Bigm|\\;X_1=x\\Bigr). \\tag{5}\n\\]\n\n\\emph{Step 2: conditional Dirichlet distribution.}  \nGiven $X_1=x$, the remaining mass $1-x$ is split among  \n\\[\nX_n,\\;X_2,\\;R:=\\sum_{i=3}^{\\,n-1}X_i,\n\\]\nand  \n\\[\n\\Bigl(\\tfrac{X_n}{1-x},\\tfrac{X_2}{1-x},\\tfrac{R}{1-x}\\Bigr)\n\\]\nis Dirichlet$(1,1,m)$ with $m:=n-3$, density  \n\\[\ng(y_1,y_2,r)=(m+1)m\\,r^{\\,m-1},\\qquad y_1,y_2,r>0,\\;y_1+y_2+r=1. \\tag{6}\n\\]\n\n\\emph{Step 3: rewrite the restrictions.}  \nIntroduce  \n\\[\n\\theta(x):=\\frac{\\max\\{0,\\;\\tfrac12-x\\}}{1-x}. \\tag{7}\n\\]\nThen (5) becomes $y_1\\ge\\theta(x)$ and $y_2\\ge\\theta(x)$.  One checks $0\\le\\theta(x)\\le\\frac12$, so the constraints are feasible.\n\n\\emph{Lemma.}  \nFor $\\bigl(y_1,y_2,r\\bigr)\\sim\\text{Dirichlet}(1,1,m)$ with $m\\ge1$ and $0\\le\\theta\\le\\frac12$,\n\\[\n\\mathbb P\\bigl(y_1\\ge\\theta,\\;y_2\\ge\\theta\\bigr)=(1-2\\theta)^{\\,m+1}. \\tag{8}\n\\]\n\n\\emph{Proof of the lemma.}  \nSet $u=y_1+y_2,\\;v=y_1$; the Jacobian equals $1$.  The domain is $2\\theta\\le u\\le1$, $\\theta\\le v\\le u-\\theta$.  Thus  \n\\[\n\\begin{aligned}\n\\mathbb P\n&=(m+1)m\\int_{u=2\\theta}^{1}\\int_{v=\\theta}^{u-\\theta}(1-u)^{m-1}\\,dv\\,du\\\\\n&=(m+1)m\\int_{u=2\\theta}^{1}(u-2\\theta)(1-u)^{m-1}\\,du\\\\\n&=(1-2\\theta)^{\\,m+1},\n\\end{aligned}\n\\]\nthe last step being an elementary Beta–integral computation. \\qed\n\nApplying the lemma with $\\theta=\\theta(x)$ gives  \n\\[\nP(x)=\n\\begin{cases}\n1, & x\\ge\\tfrac12,\\\\[6pt]\n\\bigl(\\dfrac{x}{1-x}\\bigr)^{n-2}, & 0<x<\\tfrac12.\n\\end{cases}\\tag{9}\n\\]\n\n\\emph{Step 4: integrate over $x$.}  \nUsing (4) and (9),\n\\[\n\\begin{aligned}\n\\mathbb P\n&=(n-1)\\int_{0}^{1/2}x^{\\,n-2}\\,dx+(n-1)\\int_{1/2}^{1}(1-x)^{n-2}\\,dx\\\\\n&=2\\!\\int_{0}^{1/2}(n-1)x^{\\,n-2}\\,dx\n=2^{-(n-2)}.\n\\end{aligned}\\tag{10}\n\\]\nHence  \n\\[\n\\boxed{\\,\n\\mathbb P\\bigl(\\angle V_1\\text{ and }\\angle V_2\\text{ are acute}\\bigr)=2^{-(n-2)}\\,}. \\tag{11}\n\\]\n\n%---------------------------------------------------------------\n\\textbf{Geometry interlude – no three acute angles, and two must be consecutive}\n%---------------------------------------------------------------\n\n\\textbf{Proposition.}  \nIn any cyclic, convex $n$-gon ($n\\ge4$):\n\n(i)\\;No three interior angles are acute.\n\n(ii)\\;If two interior angles are acute, the corresponding vertices must be consecutive.\n\n\\emph{Proof.}  \nRewrite (2) as  \n\\[\nX_{i-1}+X_i>\\tfrac12\\quad\\Longleftrightarrow\\quad\n\\text{“angle at }V_i\\text{ is acute.”} \\tag{12}\n\\]\n\n(ii)\\;Suppose $V_i$ and $V_j$ are acute with $i<j$ and $j\\notin\\{i\\pm1\\}$.  \nThen (12) yields  \n\\[\nX_{i-1}+X_i>\\tfrac12,\\qquad X_{j-1}+X_j>\\tfrac12. \\tag{13}\n\\]\nThe four arcs in (13) are disjoint, hence their sum exceeds $1$, contradicting $\\sum_{k=1}^{n}X_k=1$.  Thus two acute angles must be consecutive.\n\n(i)\\;Assume to the contrary that the \\emph{three} consecutive vertices $V_{i-1},V_i,V_{i+1}$ are all acute.  By (12),\n\\[\nX_{i-2}+X_{i-1}>\\tfrac12,\\qquad X_i+X_{i+1}>\\tfrac12. \\tag{14}\n\\]\nThe pairs of arcs in (14) are disjoint, so\n\\[\n(X_{i-2}+X_{i-1})+(X_i+X_{i+1})>1,\n\\]\nagain contradicting $\\sum_{k=1}^{n}X_k=1$.  Hence three acute angles are impossible, and assertion (i) is proved. \\qed\n\n%---------------------------------------------------------------\n\\textbf{Part (b) – exactly two acute angles, which are consecutive}\n%---------------------------------------------------------------\n\nLet  \n\\[\nE:=\\bigl\\{\\text{the polygon has exactly two acute interior angles}\\bigr\\}.\n\\]\nBy the proposition, $E$ is the disjoint union of the $n$ events  \n\\[\nE_k:=\\bigl\\{\\angle V_k\\text{ and }\\angle V_{k+1}\\text{ are acute}\\bigr\\},\n\\qquad k=1,2,\\ldots ,n.\n\\]\nRotational symmetry gives  \n\\[\n\\mathbb P(E_k)=\\mathbb P(E_1)=2^{-(n-2)},\n\\]\nwhence  \n\\[\n\\boxed{\\;\n\\mathbb P(E)=n\\cdot2^{-(n-2)}\\,}. \\tag{15}\n\\]\nFor $n=4$ the $n$ events $E_k$ are mutually exclusive, and (15) yields $4\\cdot2^{-2}=1$ in agreement with the geometry.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.793738",
        "was_fixed": false,
        "difficulty_analysis": "1.  The original kernel variant asked only for the probability that a *single* specified angle is acute.  \n    Here we require a *joint* event involving two overlapping angle conditions plus the exclusion of all other possibilities, which is strictly stronger.\n\n2.  Solving part (a) demands\n    •  translating angle conditions into **overlapping linear inequalities** in the Dirichlet-distributed arc variables,  \n    •  marginalising a Dirichlet distribution with *unequal parameters* (1,1,1,n−3),  \n    •  executing a **three–dimensional integral with a non-constant weight** r^{n−4},  \n    •  using properties of the **beta–function** for evaluation.\n\n   None of these steps appears in the original problem or its kernel variant.\n\n3.  Part (b) combines the result of part (a) with a **geometric extremal lemma about cyclic polygons**, showing how probabilistic calculations interact with a non-trivial structural restriction.\n\n4.  The appearance of higher-order Dirichlet and Beta integrals, together with geometric reasoning, means pattern-matching shortcuts fail; a solver must know—or derive—advanced facts about Dirichlet marginals and carry out genuine multivariate integration.\n\n5.  The final answers (2^{−(n−1)} and n·2^{−(n−1)}) are compact, but getting there requires substantially more sophisticated analysis, making this version *significantly harder* than the original."
      }
    },
    "original_kernel_variant": {
      "question": "Let $n$ be an integer with $n\\ge 4$ and mark a distinguished point $A$ on a circle whose total length is normalised to $1$.  \nChoose the remaining $n-1$ points independently and uniformly at random on the circle and, starting with $A$, label the vertices in clockwise order  \n\\[\nV_1=A,\\;V_2,\\ldots ,V_n .\n\\]\nJoin successive vertices to obtain the cyclic, convex $n$-gon $V_1V_2\\ldots V_n$.\n\n(a)\\;Compute the probability that the two consecutive interior angles at $V_1$ and $V_2$ are both acute (that is, each is strictly smaller than $90^{\\circ}$).\n\n(b)\\;Show that a cyclic $n$-gon can never possess three acute interior angles and that, if two acute angles exist, they must be consecutive.  \nDeduce the probability that the polygon has \\emph{exactly} two acute interior angles, that these two angles are consecutive and that no other interior angle is acute.",
      "solution": "\\textbf{Notation.}  \nIndices are understood modulo $n$.  Let  \n\\[\nX_1:=\\operatorname{arc}(V_1V_2),\\;X_2:=\\operatorname{arc}(V_2V_3),\\;\\ldots ,\\;\nX_n:=\\operatorname{arc}(V_nV_1),\n\\]\nall measured clockwise.  Because the $n-1$ additional points are chosen independently and uniformly, the random vector  \n\\[\n(X_1,\\ldots ,X_n)\n\\]\nis uniformly distributed over the standard $(n-1)$-simplex\n\\[\nx_i>0,\\qquad\\sum_{i=1}^{n}x_i=1,\n\\]\nthat is, it has Dirichlet law $\\mathcal D(1,1,\\ldots ,1)$ with constant density $(n-1)!$.\n\n\\textbf{Inscribed–angle criterion.}  \nFor a cyclic polygon the interior angle at $V_i$ equals  \n\\[\n\\angle V_i=\\pi\\bigl(1-(X_{i-1}+X_i)\\bigr). \\tag{1}\n\\]\nHence $\\angle V_i$ is acute $(<\\tfrac{\\pi}{2})$ iff  \n\\[\nX_{i-1}+X_i>\\tfrac12. \\tag{2}\n\\]\n\n%---------------------------------------------------------------\n\\textbf{Part (a) – two \\emph{prescribed}, consecutive angles are acute}\n%---------------------------------------------------------------\n\nWe require  \n\\[\n\\mathbb P:=\\mathbb P\\bigl(X_n+X_1>\\tfrac12,\\;X_1+X_2>\\tfrac12\\bigr). \\tag{3}\n\\]\n\n\\emph{Step 1: condition on $X_1$.}  \nThe marginal density of $X_1$ is $\\mathrm{Beta}(1,n-1)$, viz.  \n\\[\nf_1(x)=(n-1)(1-x)^{n-2},\\qquad 0<x<1. \\tag{4}\n\\]\nWrite $\\mathbb P=\\int_{0}^{1}P(x)f_1(x)\\,dx$, where  \n\\[\nP(x)=\\mathbb P\\!\\Bigl(X_n>\\tfrac12-x,\\;X_2>\\tfrac12-x\\;\\Bigm|\\;X_1=x\\Bigr). \\tag{5}\n\\]\n\n\\emph{Step 2: conditional Dirichlet distribution.}  \nGiven $X_1=x$, the remaining mass $1-x$ is split among  \n\\[\nX_n,\\;X_2,\\;R:=\\sum_{i=3}^{\\,n-1}X_i,\n\\]\nand  \n\\[\n\\Bigl(\\tfrac{X_n}{1-x},\\tfrac{X_2}{1-x},\\tfrac{R}{1-x}\\Bigr)\n\\]\nis Dirichlet$(1,1,m)$ with $m:=n-3$, density  \n\\[\ng(y_1,y_2,r)=(m+1)m\\,r^{\\,m-1},\\qquad y_1,y_2,r>0,\\;y_1+y_2+r=1. \\tag{6}\n\\]\n\n\\emph{Step 3: rewrite the restrictions.}  \nIntroduce  \n\\[\n\\theta(x):=\\frac{\\max\\{0,\\;\\tfrac12-x\\}}{1-x}. \\tag{7}\n\\]\nThen (5) becomes $y_1\\ge\\theta(x)$ and $y_2\\ge\\theta(x)$.  One checks $0\\le\\theta(x)\\le\\frac12$, so the constraints are feasible.\n\n\\emph{Lemma.}  \nFor $\\bigl(y_1,y_2,r\\bigr)\\sim\\text{Dirichlet}(1,1,m)$ with $m\\ge1$ and $0\\le\\theta\\le\\frac12$,\n\\[\n\\mathbb P\\bigl(y_1\\ge\\theta,\\;y_2\\ge\\theta\\bigr)=(1-2\\theta)^{\\,m+1}. \\tag{8}\n\\]\n\n\\emph{Proof of the lemma.}  \nSet $u=y_1+y_2,\\;v=y_1$; the Jacobian equals $1$.  The domain is $2\\theta\\le u\\le1$, $\\theta\\le v\\le u-\\theta$.  Thus  \n\\[\n\\begin{aligned}\n\\mathbb P\n&=(m+1)m\\int_{u=2\\theta}^{1}\\int_{v=\\theta}^{u-\\theta}(1-u)^{m-1}\\,dv\\,du\\\\\n&=(m+1)m\\int_{u=2\\theta}^{1}(u-2\\theta)(1-u)^{m-1}\\,du\\\\\n&=(1-2\\theta)^{\\,m+1},\n\\end{aligned}\n\\]\nthe last step being an elementary Beta–integral computation. \\qed\n\nApplying the lemma with $\\theta=\\theta(x)$ gives  \n\\[\nP(x)=\n\\begin{cases}\n1, & x\\ge\\tfrac12,\\\\[6pt]\n\\bigl(\\dfrac{x}{1-x}\\bigr)^{n-2}, & 0<x<\\tfrac12.\n\\end{cases}\\tag{9}\n\\]\n\n\\emph{Step 4: integrate over $x$.}  \nUsing (4) and (9),\n\\[\n\\begin{aligned}\n\\mathbb P\n&=(n-1)\\int_{0}^{1/2}x^{\\,n-2}\\,dx+(n-1)\\int_{1/2}^{1}(1-x)^{n-2}\\,dx\\\\\n&=2\\!\\int_{0}^{1/2}(n-1)x^{\\,n-2}\\,dx\n=2^{-(n-2)}.\n\\end{aligned}\\tag{10}\n\\]\nHence  \n\\[\n\\boxed{\\,\n\\mathbb P\\bigl(\\angle V_1\\text{ and }\\angle V_2\\text{ are acute}\\bigr)=2^{-(n-2)}\\,}. \\tag{11}\n\\]\n\n%---------------------------------------------------------------\n\\textbf{Geometry interlude – no three acute angles, and two must be consecutive}\n%---------------------------------------------------------------\n\n\\textbf{Proposition.}  \nIn any cyclic, convex $n$-gon ($n\\ge4$):\n\n(i)\\;No three interior angles are acute.\n\n(ii)\\;If two interior angles are acute, the corresponding vertices must be consecutive.\n\n\\emph{Proof.}  \nRewrite (2) as  \n\\[\nX_{i-1}+X_i>\\tfrac12\\quad\\Longleftrightarrow\\quad\n\\text{“angle at }V_i\\text{ is acute.”} \\tag{12}\n\\]\n\n(ii)\\;Suppose $V_i$ and $V_j$ are acute with $i<j$ and $j\\notin\\{i\\pm1\\}$.  \nThen (12) yields  \n\\[\nX_{i-1}+X_i>\\tfrac12,\\qquad X_{j-1}+X_j>\\tfrac12. \\tag{13}\n\\]\nThe four arcs in (13) are disjoint, hence their sum exceeds $1$, contradicting $\\sum_{k=1}^{n}X_k=1$.  Thus two acute angles must be consecutive.\n\n(i)\\;Assume to the contrary that the \\emph{three} consecutive vertices $V_{i-1},V_i,V_{i+1}$ are all acute.  By (12),\n\\[\nX_{i-2}+X_{i-1}>\\tfrac12,\\qquad X_i+X_{i+1}>\\tfrac12. \\tag{14}\n\\]\nThe pairs of arcs in (14) are disjoint, so\n\\[\n(X_{i-2}+X_{i-1})+(X_i+X_{i+1})>1,\n\\]\nagain contradicting $\\sum_{k=1}^{n}X_k=1$.  Hence three acute angles are impossible, and assertion (i) is proved. \\qed\n\n%---------------------------------------------------------------\n\\textbf{Part (b) – exactly two acute angles, which are consecutive}\n%---------------------------------------------------------------\n\nLet  \n\\[\nE:=\\bigl\\{\\text{the polygon has exactly two acute interior angles}\\bigr\\}.\n\\]\nBy the proposition, $E$ is the disjoint union of the $n$ events  \n\\[\nE_k:=\\bigl\\{\\angle V_k\\text{ and }\\angle V_{k+1}\\text{ are acute}\\bigr\\},\n\\qquad k=1,2,\\ldots ,n.\n\\]\nRotational symmetry gives  \n\\[\n\\mathbb P(E_k)=\\mathbb P(E_1)=2^{-(n-2)},\n\\]\nwhence  \n\\[\n\\boxed{\\;\n\\mathbb P(E)=n\\cdot2^{-(n-2)}\\,}. \\tag{15}\n\\]\nFor $n=4$ the $n$ events $E_k$ are mutually exclusive, and (15) yields $4\\cdot2^{-2}=1$ in agreement with the geometry.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.606174",
        "was_fixed": false,
        "difficulty_analysis": "1.  The original kernel variant asked only for the probability that a *single* specified angle is acute.  \n    Here we require a *joint* event involving two overlapping angle conditions plus the exclusion of all other possibilities, which is strictly stronger.\n\n2.  Solving part (a) demands\n    •  translating angle conditions into **overlapping linear inequalities** in the Dirichlet-distributed arc variables,  \n    •  marginalising a Dirichlet distribution with *unequal parameters* (1,1,1,n−3),  \n    •  executing a **three–dimensional integral with a non-constant weight** r^{n−4},  \n    •  using properties of the **beta–function** for evaluation.\n\n   None of these steps appears in the original problem or its kernel variant.\n\n3.  Part (b) combines the result of part (a) with a **geometric extremal lemma about cyclic polygons**, showing how probabilistic calculations interact with a non-trivial structural restriction.\n\n4.  The appearance of higher-order Dirichlet and Beta integrals, together with geometric reasoning, means pattern-matching shortcuts fail; a solver must know—or derive—advanced facts about Dirichlet marginals and carry out genuine multivariate integration.\n\n5.  The final answers (2^{−(n−1)} and n·2^{−(n−1)}) are compact, but getting there requires substantially more sophisticated analysis, making this version *significantly harder* than the original."
      }
    }
  },
  "checked": true,
  "problem_type": "calculation",
  "iteratively_fixed": true
}