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2023 Canadian Intermediate
Mathematics Contest
Solutions

Wednesday, November 15, 2023
(in North America and South America)

Thursday, November 16, 2023
(outside of North American and South America)

©2023 University of Waterloo


Part A

  1. The bus leaves at exactly 7:43 a.m., which means that after the bus has travelled for exactly 17 minutes, it will be 8:00 a.m.
    The bus arrives at its destination at exactly 8:22 a.m., which is exactly 22 minutes after 8:00 a.m.
    Therefore, the number of minutes the trip took was 17+22=39.

    Answer: 39

  2. Given that a4=18, we have 2a4=18 or a2=18, and so a=36.

    Answer: 36

  3. We label the vertices of the figure by the letters A through H as shown.

    The vertex in the top-left corner of the 8-sided polygon is labelled A, and moving around the vertices of the polygon in the clockwise direction, the remaining vertices are labelled B through H, in order.

    From the information given in the questions, AB=5x+1, BC=2x, CD=2, FG=4, GH=3x, and AH=4x. We will set DE=y and EF=z.
    Because of the fact that two line segments meet at a right angle if they meet at all, the sum of the lengths of the horizontal edges CD, EF, and GH must equal the length AB.
    Therefore, 2+z+3x=5x+1, from which we obtain z=5x+1(3x+2)=2x1.
    By similar reasoning, AH+FG=BC+DE, so 4x+4=2x+y, from which we obtain y=4x+42x=2x+4.
    Using that y=2x+4 and z=2x1, we can add the side lengths to get the following expression for the perimeter: (5x+1)+2x+2+(2x+4)+(2x1)+4+3x+4x which can be simplified to get 18x+10.
    The perimeter is given to be 82, so 18x+10=82, which means 18x=72 or x=4.

    Answer: 4

  4. Since 3×3×3=27, the larger cube is made by arranging the 27 smaller cubes in three layers so that each layer consists of 9 small cubes arranged in a 3×3 square.
    This means each face of the larger cube is made up of 3×3=9 faces of small cubes.
    A cube has six faces, so the surface of the cube is made up of 6×9=54 faces of small cubes.
    Of the 27 small cubes, 1 is completely concealed inside the larger cube, so none of its faces contribute to the surface of the larger cube.
    The remaining 26 small cubes each contribute 1, 2, or 3 of their faces to the surface of the larger cube.
    There are 6 small cubes that contribute 1 face to the surface of the larger cube, 12 that contribute 2 faces, and 8 that contribute 3 faces.
    Observe that 6+12+8=26, so this accounts for the remaining small cubes.
    The 6 small cubes that contribute 1 face are in the centres of the faces of the larger cube.
    The 12 small cubes that contribute 2 faces are on the edges of the larger cube.
    The 8 small cubes that contribute 3 faces are on the corners of the larger cube.
    Observe that 6×1+12×2+8×3=54, which is the correct total number of faces of small cubes that we expect on the surface of the larger cube.
    The goal is to minimize the sum of the integers showing on the surface of the larger cube. This means we want to position each small cube so that the sum of the integers on its visible faces is as small as possible.
    Each small cube has a face with the integer 1 on it. This means the 6 small cubes that each contribute 1 face to the surface of the larger cube can be positioned so that the number on this face is 1.
    The 12 small cubes that contribute 2 faces to the surface of the cube contribute 2 faces that share an edge.
    The smallest possible total of the integers on 2 faces of a small cube is 1+2=3.
    The faces showing 1 and 2 also share an edge (as indicated in the net given in the problem), so this means these 12 small cubes can be oriented so that they each contribute a total of 1+2=3 to the total on the surface of the larger cube.
    Each of the 8 small cubes that contribute three faces to the surface of the larger cube contribute 3 faces that meet at a vertex.
    The smallest possible sum of the integers on 3 faces of a small cube is 1+2+3=6.
    The faces that have the integers 1, 2, and 3 on them meet at a vertex.
    This means these 8 cubes can be oriented so that they each contribute 1+2+3=6 to the total on the surface of the larger cube.
    By minimizing the total for each of the 26 cubes that contributes to the surface of the larger cube, we minimize the overall total. Using the arguments above, the minimum possible total of the integers on the surface of the larger cube is 6×1+12×3+8×6=90 Below is an image of what the larger cube could look like when the small cubes are arranged to minimize the total of the visible integers.

    A description of the image follows.

    Answer: 90

  5. Let the distance between P and Q be x km and the distance between Q and R be y km as shown in the diagram below.

    A horizontal line segment
of length x connects point P to point Q on the right. A slanted line segment of length y connects point Q to point R above and to the right.

    The total distance travelled by the hiker was (2x+2y) km.
    The total distance walked on flat ground was 2x km and this was done at a speed of 4 km/h.
    Therefore, the total time the hiker spent walking on flat ground was 2x km4 km/h=x2 h.
    The total distance walked uphill was y km and this was done at a speed of 3 km/h.
    Therefore, the total time spent walking uphill was y3 h.
    The total distance walked downhill was y km and this was done at a speed of 6 km/h.
    Therefore, the total time spent walking downhill was y6 h.
    Since the hiker left point P at 1:00 p.m. and returned to point P at 8:00 p.m., the total time for the trip, including the hour spent at the viewing platform, was 7 hours.
    Of these 7 hours, 71=6 of them were spent walking.
    The total times spent walking on flat ground, uphill, and downhill were x2 h, y3 h, and y6 h.
    Therefore, we have x2+y3+y6=6 Combining the left side into a single fraction, we get 3x+2y+y6=6 or 3x+3y6=6.
    This simplifies to x+y2=6 or x+y=12, and after multiplying both sides of this equation by 2, we get 2(x+y)=24 or 2x+2y=24.
    From earlier, the total distance travelled by the hiker was (2x+2y) km, so the answer is 24 km.

    Answer: 24 km

  6. In the diagram below, a dashed vertical line and a dashed horizontal line has been drawn through each of A and B.
    These four lines, together with the y-axis, divide the set of points with positive integer coordinates into six regions. These regions are labelled 1 through 6 as shown in the diagram.

    A dashed horizontal line passes through the dots in row 1 (the bottom row) and another passes through the dots in row 5. A dashed vertical line passes through the dots in column 2 and another passes through the dots in column 4. This forms 6 rectangular regions in the space above row 1. From left to right, the top three regions are labelled 1, 2, and 3, and the bottom three regions are labelled 4, 5, and 6.

    We will use the convention that a point on the horizontal boundary between two regions is considered to be in the top region, and a point on the vertical boundary between two regions is considered to be in the rightmost region.
    This means, for example, that point B is in Region 2 and the point (4,3) is in Region 6.
    There are three other regions between the x-axis and the line with equation y=1. However, following the convention stated above, there are no points with positive integer coordinates in any of these three regions.

    For each of these six regions, we will examine the values of dC for points C in that region. When we refer to the "positive difference" between two values, we mean the larger value minus the smaller value. We will allow the “positive difference” to be 0 if values are equal.

    First, we make some general observations about how dC can be computed.
    The length of the shortest path from A to C is equal to the sum of the positive difference between the x-coordinates of A and C and the positive difference between the y-coordinates of A and C.
    Suppose the positive difference between the x-coordinates is h and the positive difference between the y-coordinates is v.
    Then there must be at least h horizontal moves of length 1 and v vertical moves of length 1 in any path from A to C, so the length of the shortest path is at least h+v.
    However, a path of length h+v is always possible because we can always move horizontally from A by h units (in the appropriate direction) and then move vertically by v units.
    For the example in the problem statement, the coordinates of C are (3,4) and the coordinates of A are (4,1), and the length of the shortest path between the two is (43)+(41)=4.
    Similarly, the length of the shortest path from C to B is equal to the sum of the positive difference between the x-coordinates of C and B and the positive difference between the y-coordinates of C and B.

    Region 1:

    If C(x,y) is in Region 1, then x=1 and y5. By the discussion above, the length of the shortest path from A to C is (41)+(y1)=2+y.
    The length of the shortest path from C to B is (21)+(y5)=4+y.
    Adding these lengths, we get that dC=(2+y)+(4+y)=2y2 when C is in Region 1.
    For the point C(1,5), we thus have dC=2(5)2=8.
    For the point C(1,6), we have dC=2(6)2=10.
    Continuing in this way, if C and E are points in Region 1 with C one unit above E, then dC is exactly two more than dE.
    Therefore, if C is in Region 1, then dC is an even integer that is at least 8. Moreover, for every even integer N8, there is exactly one point C in Region 1 such that dC=N.

    Region 2:

    If C(x,y) is in Region 2, then x=2 or x=3 and y5.
    The length of the shortest path from A to C is (4x)+(y1)=3x+y.
    The length of the shortest path from C to B is (x2)+(y5)=7+x+y.
    In general, when C(x,y) is in Region 2, dC=(3x+y)+(7+x+y)=4+2y, which does not depend on the value of x.
    For both of the points C(2,5) and C(3,5), we have dC=4+2(5)=6.
    For the points C(2,6) and C(3,6), dC=8.
    By reasoning similar to that which was used when examining Region 1, we see that for every even integer N6, there are exactly two points C in Region 2 such that dC=N.

    Region 6:

    If C(x,y) is in Region 6, then x4 and y is equal to one of 1, 2, 3, or 4.
    Using similar calculations to the previous cases, one can check that dC=2x2, which does not depend on y.
    Similar to the observations for Regions 1 and 2, the smallest possible value of dC is when x=4, which gives dC=2(4)2=6.
    As well, if x increases by 1, then dC increases by 2.
    Therefore, for every even positive integer N6, there are exactly 4 (1 for each possible value of y) points C in Region 6 such that dC=N.

    Finally, we will examine Region 3. We will not need to carefully examine Region 4 or Region 5 to answer the question. This will be explained after Region 3 is examined.

    Region 3:

    If C(x,y) is in Region 3, then x4 and y5.
    In this case, it can be checked that dC=2x+2y12.
    The smallest possible value of dC is when x and y are both as small as possible. This occurs when (x,y)=(4,5), which gives dC=2(4)+2(5)12=6.
    If exactly one of x or y increases by 1 (and the other does not change), then dC increases by 2.
    Thus, for N=8, there are 2 points C for which dC=N. These are the points one unit above (4,5) and one unit to the right of (4,5), which are (4,6) and (5,5), respectively.
    To find points C with dC=10, we can move two units up from (4,5), or move one unit up and one unit to the right of (4,5), or move two units to the right of (4,5).
    This corresponds to the 3 points (6,5), (5,6), and (4,7).
    Thus, there are 3 points C in Region 3 with dC=10.
    Continuing in this way, if k is a positive integer, the points C in Region 3 that satisfy dC=6+2k are

    of which there are exactly k+1.

    We now suppose N has the property that there are 2023 points C for which dC=N.
    There are only 12 points in total in Regions 4 and 5.
    Regions 1, 2, and 6 contain at most 1, 2, and 4, points C, respectively, that have dC=N.
    Therefore, Region 3 must contain at least 202312124=2004 of the points.
    From above, we have that for each positive integer k, there are k+1 points C in Region 3 with the property that dC=6+2k.
    We need k+1 to be at least 2004, which means k must be at least 2003, so N=dC is at least 6+2(2003)=4012.
    It is easily checked that no points C in Region 4 or Region 5 have dC as large as 4012, so this means every point C with dC=N is in Region 1, Region 2, Region 3, or Region 6.
    Moreover, since N4012, we can be sure that there is exactly one point C in Region 1 with dC=N, there are exactly two points C in Region 2 with dC=N, and exactly four points C in Region 6 with dC=N.
    Therefore, there are exactly 2023124=2016 points C in Region 3 with dC=N.
    This means k+1=2016 or k=2015, so dC=6+2(2015)=4036.

    Answer: 4036

Part B

    1. Since ABCD is as square with side length AB=10 cm, its area is (10 cm)2=100 cm2.
      Since EFGH is a square with side length EF=8 cm, is area is (8 cm)2=64 cm2.
      The shaded region is the region inside ABCD that lies outside square EFGH, so its area is the difference between the two areas, or 100 cm264 cm2=36 cm2.

    2. The area of ABCD is (13 cm)2=169 cm2 and the area of the shaded region is 120 cm2.
      The area of EFGH is 169 cm2120 cm2=49 cm2.
      Since EFGH is a square, its side length is 49 cm2=7 cm.

    3. The area of the shaded region is given to be 34 of the total area of ABCD, so the area of EFGH is 134=14 of the area of ABCD.
      The area of ABCD is (18 cm)2=324 cm2.
      The area of EFGH is 14 of the area of ABCD, so the area of EFGH is 3244 cm2=81 cm2.
      Since EFGH is a square, its side length is 81 cm2=9 cm.

    4. The area of ABCD is a2 cm2 and the area of EFGH is b2 cm2.
      The given condition means that a2b2=64100a2, which can be rearranged to 36100a2=b2.
      Taking square roots of both sides and noting that a>0 and b>0, we have 36100a2=b2 or 610a=b, and so 35a=b.
      Multiplying both sides of this equation by 53 gives a=53b.
      Dividing both sides of this equation by b gives ab=53.

    1. The RD sum of 4281 is 4281+1824=6105.

    2. The integer with digits ABCD is equal to 1000A+100B+10C+D.
      The integer with digits DCBA is equal to 1000D+100C+10B+A.
      The RD sum of ABCD is ABCD+DCBA=(1000A+100B+10C+D)+(1000D+100C+10B+A)(group like terms)=1001A+110B+110C+1001D(factor)=1001(A+D)+110(B+C) and so m=1001 and n=110.

    3. Suppose the RD sum of ABCD is 3883.
      From part (b), we have that 1001(A+D)+110(B+C)=3883.
      If A+D were 4 or greater, then 1001(A+D) would be at least 4004, which is larger than 3883.
      Therefore, A+D is no larger than 3.
      Since A and D are both nonzero digits, they are each at least 1, so A1 and D1, which implies A+D1+1=2.
      This means A+D=2 or A+D=3.
      If A+D=2, then 1001(A+D)=2002.
      Substituting this into 1001(A+D)+110(B+C)=3883, we get 2002+110(B+C)=3883 or 110(B+C)=1881.
      We also know that B+C is an integer, so 110(B+C) is a multiple of 10, so it cannot be equal to 1881 which is not a multiple of 10.
      Therefore, it is not possible that A+D=2, so we must have A+D=3.
      Using that A+D=3, we get 1001(A+D)=3003, so 110(B+C)=38831001(A+D)=38833003=880 Dividing 110(B+C)=880 on both sides by 110, we conclude that B+C=8.
      Since A and D are positive digits that have a sum of 3, we must have A=1 and D=2 or A=2 and D=1.
      Since B and C are digits that have a sum of 8 but there is no restriction that they must be positive, we have that (B,C) can be any of the pairs (0,8), (1,7), (2,6), (3,5), (4,4), (5,3), (6,2), (7,1), or (8,0).
      There are 2 possibilities for the values of A and D, and 9 possibilities for the values of B and C, Therefore, there are 2×9=18 integers with an RD sum of 3883. They are listed below. 1082,1172,1262,1352,1442,1532,1622,1712,1802 2081,2171,2261,2351,2441,2531,2621,2711,2801

    4. Suppose that n is a four-digit integer and that ABCD is an integer with RD sum equal to n.
      From part (b), this means 1001(A+D)+110(B+C)=n. Note that A and D are both non-zero.
      Suppose there is some other integer, EFGH, with an RD sum equal to n.
      Again by part (b), this means 1001(E+H)+110(F+G)=n.
      Therefore, we have 1001(A+D)+110(B+C)=1001(E+H)+110(F+G).
      For convenience, set w=A+D, x=B+C, y=E+H, and z=F+G.
      The equation above simplifies to 1001w+110x=1001y+110z.
      Rearranging this equation, we get 1001w1001y=110z110x.
      After factoring, we have 1001(wy)=110(zx), and after dividing both sides of this equation, we get 91(wy)=10(zx).
      Observe that 91=7×13.
      The quantities wy and zx are both integers, which means 91(wy) and 10(zx) are both integers.
      Moreover, since 91(wy) is a multiple of 13, we must also have that 10(zx) is a multiple of 13.
      The integer 13 is prime and 10 does not have a factor of 13, so zx must have a factor of 13.
      By similar reasoning, zx must have a factor of 7, which means zx is a multiple of 7×13=91.
      Now recall that each of x and z is the sum of two digits.
      This means that each of x and z is an integer from 0 to 18 inclusive.
      The difference between two integers that are at least 0 and at most 18 must lie between 018=18 and 180=18.
      We have shown that zx is a multiple of 91 that lies between 18 and 18.
      The only such integer is 0, which shows that zx=0 or z=x.
      Since zx=0, 10(zx)=0, which means 91(wy)=0.
      Therefore, wy=0 as well, so w=y.
      We have shown that if ABCD and EFGH have the same RD sum, then A+D=E+H and B+C=F+G.
      In other words, if n is the RD sum of a four digit integer, then the values of A+D and B+C are uniquely determined.
      This means, to count the number of four-digit RD sums, we can count the number of pairs (w,x) with the property that 1001w+110x9999, subject to the condition that w and x are the sum of digits, with w being the sum of positive digits.

      Thus, the answer to the question is the number of pairs (w,x) with 2w18 and 0x18 that satisfy 1001w+110x9999.

      If w=2, then 1001w+110x=2002+110x, so we want 2002+110x9999 or 110x99992002 or 110x7997.
      The largest that x can be is 18, so the largest that 110x can be is 18×110=1980, which is less than 7997.
      Therefore, if w=2, then x can be any integer from 0 through 18.
      Thus, we get 19 pairs (w,x) when w=2.

      If w=3, then 1001w+110x=3003+110x, which means we need 110x6996.
      As with the case when w=2, every possible value of x from 0 through 18 satisfies this inequality, so we get 19 pairs in this case as well.

      If w=4, similar reasoning implies that 110x5995, so we again get 19 pairs.

      Continuing this reasoning, we find that we get 19 pairs for each of w=5, w=6, w=7, and w=8.

      If w=9, then we must have that 9009+110x9999, so 110x99999009 or 110x990.
      It is not difficult to see that x can be any integer from 0 through 9 inclusive, but if x10, then 110x990 is not true. Therefore, we get 10 pairs in this case.

      If w10, then 1001w is at least 10010, so there is no way to choose x so that 1001w+110x has four digits.

      Therefore, there are no more cases to consider, and the number of possible four-digit RD sums is 19+19+19+19+19+19+19+10=143

    1. The positive multiples of 7 that are no larger than 72=49 are 7,14,21,28,35,42,49 None of these integers is in an earlier row, so they are exactly the integers in Row 7.
      The positive multiples of 8 that are no larger than 82=64 are 8,16,24,32,40,48,56,64 Of these, 8, 16, and 24 are in earlier rows, so Row 8 is 32,40,48,56,64.
      The positive multiples of 9 that are no larger than 92=81 are 9,18,27,36,45,54,63,72,81 Of these, 9, 18, and 36 are in earlier rows, so Row 9 is 27,45,54,63,72,81.
      The positive multiples of 10 that are no larger than 102=100 are 10,20,30,40,50,60,70,80,90,100 Of these, 10, 20, 30, and 40 are in earlier rows, so the smallest integer in Row 10 is 50.

    2. Consider an integer n3. Notice that n, n1, and n2 are all positive integers.

      Factoring, the quantity n2n, we get n2n=n(n1).
      Since n>n1, n(n1)>(n1)(n1)=(n1)2, which means that n(n1) is too large to be in Row n1.
      This also implies that n(n1) is larger than each of 12, 22, 32, and so on up to (n2)2, so we conclude that n(n1) is not in any of the first n1 rows.
      We also have that n(n1) is a multiple of n, and since n is positive, n2n<n2.
      Therefore, n2n is a multiple of n so it satisfies (i), it does not exceed n2 so it satisfies (ii), and it is not in an earlier row, so it satisfies (iii).
      This shows that n2n is in Row n.

      By factoring n22n, we have n22n=n(n2).
      Similar to the reasoning above, n>n2 implies that n(n2)>(n2)(n2)=(n2)2.
      This implies that n22n is larger than k2 for all positive integers k with kn2.
      As well, since n is positive, n22n<n2, so n22n is a multiple of n that does not exceed n2.
      By the given conditions on Row n, n22n must be in Row n provided it is not in an earlier row. We have already argued that it cannot be in any of Rows 1 through n2.
      Therefore, we can show that n22n is in Row n by showing that it is not in Row n1.

      We will now justify that for all n3, the integer n22n is not a multiple of n1, which implies n22n is not in Row n1.
      Consider the quantity (n1)2. We can expand it as follows: (n1)2=(n1)(n1)=(n1)n+(n1)(1)=n2n+(n)+(1)(1)=n22n+1 We can now rearrange (n1)2=n22n+1 to get n22n=(n1)21, which shows that n22n and (n1)2 are consecutive integers.
      Note that any two multiples of n1 must differ by at least n1.
      We are assuming that n3, which implies that n12, and so any two multiples of n1 must differ by at least 2.
      Since (n1)2 and n22n differ by 1, they cannot both be multiples of n1.
      The integer (n1)2 is a multiple of n1, and so we conclude that n22n is not a multiple of n1.

    3. We will show that n=30 is the largest integer with the property that n210n is not in Row n.
      When n=30, n210n=30210×30=600.
      Since 600=24×25, we see that 600 is a multiple of 25 that is less than 252.
      Therefore, 600 is in Row 25 or an earlier row, so n=30 has the property that n210n is not in Row n.

      In order to show that n=30 is the largest such integer, we need to show that all integers n with the property that n210n is not in Row n satisfy n30.

      For the remainder of the solution, we will assume that n is a positive integer with the property that n210n is not in Row n. We will deduce from this assumption that n30.
      Note that if n10, then n210n=n(n10)0, which means n210n is not in any row, so we will also assume that n11 in order to ensure that n210n>0.
      Since n(n10)=n210n and n is positive, we must have n210n<n2. This means n210n must occur somewhere in the first n rows.
      Since n>n10 and n10 is positive, we have that n(n10)>(n10)(n10)=(n10)2.
      This shows that n(n10) is too large to be in Row m for any mn10.
      The previous two sentences combine to show that n210n must be in one of Row n9, Row n8, and so on up to Row n. We are assuming that n210n is not in Row n, so it must be in one of Rows n9 through n1.
      This means we have 9 cases to consider: n210n is in Row n9, in Row n8, and so on to Row n1.
      For each integer k from 1 through 9, we will consider the case when n210n is in Row nk, and in each of these cases, we will show that n30. This will show that no matter which of the 9 rows n210n is in, it cannot be larger than 30.

      Suppose n210n is in Row n9. Among other things, this means n210n must be a multiple of n9.
      The expression n210n does not have an obvious factor of n9. It will be useful to find an algebraic expression that is a multiple of n9 and is “close” to n210n.
      After some trial and error, you might notice that (n9)(n1)=n210n+9, which can be rearranged to get 9=(n9)(n1)(n210n).
      We are assuming n210n is a multiple of n9, so the expression (n9)(n1)(n210n) is the difference of two multiples of n9.
      The difference between two multiples of n9 is itself a multiple of 9, which means 9=(n9)(n1)(n210n) must be a multiple of n9.
      If 9 is a multiple of n9, then we must have that 9n9, which can be rearranged to get that n18.
      We have now shown that if n210n is in Row n9, then n18, and so n30.

      This kind of reasoning can be applied to the remaining eight cases.
      For example, suppose n210n is in Row n8.
      Observe that (n8)(n2)=n210n+16, and so 16=(n8)(n2)(n210n).
      If n210n in Row n8, then n210n is a multiple of n8, which means the expression 16=(n8)(n2)(n210n) is a multiple of n8.
      If 16 is a multiple of n8, then n816, so n24 which implies n30.

      We will now quickly include the results of examining the remaining seven cases. The calculations are similar to those for Rows n9 and n8 and are not included.

      • n210n is in Row n7: 21=(n7)(n3)(n210n), so 21 is a multiple of n7. Therefore, n721 or n28, and so n30.

      • n210n is in Row n6: 24=(n6)(n4)(n210n), so 24 is a multiple of n6. Therefore, n624 or n30.

      • n210n is in Row n5: 25=(n5)(n5)(n210n), so 25 is a multiple of n5. Therefore, n525 or n30.

      • n210n is in Row n4: 24=(n4)(n6)(n210n), so 24 is a multiple of n4. Therefore, n424 or n28, and so n30.

      • n210n is in Row n3: 21=(n3)(n7)(n210n), so 21 is a multiple of n3. Therefore, n321, or n24, and so n30.

      • n210n is in Row n2: 16=(n2)(n8)(n210n), so 16 is a multiple of n2. Therefore, n216 or n18, and so n30.

      • n210n is in Row n1: 9=(n1)(n9)(n210n), so 9 is a multiple of n1. Therefore, n19 or n10, and so n30.

      To recap, we assumed that n210n is not in Row n, then deduced that it must be in Row k for some k from n9 through n1. We then considered each each of these 9 possibilities individually and showed that n30 in each case.

      Therefore, if n210n is not in Row n, then n30. Since n=30 has the property that n210n is not in Row n, we can conclude that 30 is the largest integer with this property.