Solutions

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85.

Find all pairs (a, b) of positive integers with a ¹ b for which the system

cosax + cosbx = 0

asinax + bsinbx = 0 .

Solution 1. Suppose that the system is solvable; note that x = 0 is not a solution. Then cosax = -cosbx so that sinax = esinbx where e = ±1. Hence (a + eb)sinbx = 0. Since a and b are positive and unequal, a + eb ¹ 0, so that sinbx = 0. Hence bx = np for some integer n. Also sinax = 0, so that ax = mp for some integer m. Hence, we must have an = bm for some integers m and n. Since cosmp = -cosnp, m and n must have opposite parity.

Suppose that a = 2^u p and b = 2^v q with u and v unequal integers and p and q odd. Then x = 2^-wp. where w is the minimum of u and v satisfies the system of equations.

Solution 2. First, observe that x ¹ 0 for any solution. If the system is satisfied, then

0 = cosax + cosbx = 2cos 1 2 (a+b)x cos 1 2 (a-b)x Þ cos 1 2 (a + b)x = 0    or      cos 1 2 (a-b)x = 0 Þ 1 2 (a + b)x = (2k+1) p 2     or      1 2 (a - b)x = (2k+1) p 2 Þ ax ±bx = (2k + 1)p    for some integer  k Þ sinax = -sin(±bx) = -±sinbx Þ 0 = a sinax + b sinbx = (a -±b)sinbx .

Since a ¹ b and a + b > 0, 0 = sinbx = sinax, so that ax = mp and bx = np for some integers m and n. Since 0 = cosax + cosbx = (-1)^m + (-1)ⁿ, the integers m and n must have different parity. Hence

x = mp a = np b

where m and n are integers not both even or both odd. Since x ¹ 0, a/b = m/n, so a/b in lowest terms must have numerator and denominator of different parities.

We now show that, for any pair a, b satisfying this condition, there is a solution. Wolog, let a = 2uw and v = (2v+1)w, where the greatest common divisor of 2u and 2v+1 is 1, and w is an arbitrary positive integer. Suppose that x = p/w. Then

cosax + cosbx = cos2up+ cos(2v+1)p = 1 - 1 = 0

and

a sinax + b sinbx = a sin2up+ b sin(2v+1)p = 0 + 0 = 0

as desired.

Solution 3. Since cos² ax = cos² bx and a² sin² ax + b² sin² bx, then

a2 cos2 bx + b2 sin2 bx = a2 Þ(b2 - a2)sin2 bx = 0

so that bx = np for some integer. Similarly ax = mp. The solution can be completed as before.

Comment. Note that there are two parts to the solution of this problem, and your write-up should make sure that these are carefully delineated. First, assuming that there is a solution, you derive necessary conditions on a and b that the two equations are consistent. Then, you assume these conditions on a and b, and then display a solution to the two equations. A complete solution requires noting that suitable numbers a and b actually do lead to a solution.

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86.

Let ABCD be a convex quadrilateral with AB = AD and CB = CD. Prove that

(a) it is possible to inscribe a circle in it;

(b) it is possible to circumscribe a circle about it if and only if AB ^BC;

(c) if AB ^BC and R and r are the respective radii of the circumscribed and inscribed circles, then the distance between the centres of the two circles is equal to the square root of R² + r² - rÖ[(r² + 4R²)].

Comment. Most students picked up the typo in part (c) in which AC was given instead of the intended BC. I am sorry for the mistake. However, this does happen from time to time even on competitions, and you should be alert. From the context of this problem, the intention was probably pretty clear (in fact, some of you might not have realized that there was an error). The rule in such a situation is that, if you feel that there is an error, make a reasonable nontrivial interpretation of the problem, state it clearly and solve it.

Solution 1. (a) Triangles ABC and ADC are congruent (SSS) with the congruence implemented by a reflection in AC. Hence AC bisects angles DAB and DCB. The angle bisectors of ÐADB and ÐABC are reflected images and intersect in I, a point on AC. Since I is equidistant from the four sides of the kite ABCD, it is the centre of its incircle.

(b) If AB ^BC, then the circle with diameter AC passes through B. By symmetry about AC, it must pass through D as well. Conversely, let \frak C be the circumcircle of ABCD. The circle goes to itself under reflection in AC, so AC must be a diameter of \frak C. Hence ÐABC = ÐADC = 90^°.

(c) Let I be the incentre and O the circumcentre of ABCD; both lie on AC. Suppose that J and K are the respective feet of the perpendiculars to AB and BC from I, and P and Q the respective feet of the perpendiculars to AB and BC from O. Let x = |AB | and y = |BC |. Then

x y = x - r r Þ xy = r(x + y) .

Since |AO | = |OC | = R, 4R² = x² + y². Noting that x and y both exceed r, we have that

(x + y)2 = x2 + y2 + 2xy = 4R2 + 2r(x + y) Þ (x + y - r)2 = r2 + 4R2 Þ x + y = r +   _______ Ör2 + 4R2    .

Now

|IO |2 = |JP |2 + |KQ |2 = (|JB |- 1 2 |AB |)2+ (|KB |- 1 2 |BC |)2 = 2r2 - r(x + y) + 1 4 (x2 + y2) = r2 - r   _______ Ör2 + 4R2   + R2  ,

yielding the desired result.

Solution 2. (a) Since triangles ADB and CDB are isosceles, the angle bisectors of A and C right bisect the base BD and so they coincide. The line AC is an axis of reflective symmetry that interchanges B and D, and also interchanges the angle bisectors of B and D. The point P where one of the bisectors intersects the axis AC is fixed by the reflection and so lies on the other bisector. Hence, P is common to all four angle bisectors, and so is equidistant from the four sides of the quadrilateral. Thus, we can inscribe a circle inside ABCD with centre P.

(b) Since AC is a line of symmetry, ÐB = ÐD. Note that, ABCD has a circumcircle Û pairs of opposite angles sum to 180^° ÛÐB + ÐD = 180^° Û ÐB = ÐD = 90^°. This establishes the result.

(c) [R. Barrington Leigh] Let a, b and c be the respective lengths of the segments BC, AC and AB. Let O and I be, respectively, the circumcentre and the incentre for the quadrilateral. Note that both points lie on the diagonal AC. Wolog, we may take a ³ c.

We observe that ÐABI = 45^° and that BI is the hypotenuse of an isosceles right triangle with arms of length r. We have, by the Law of Cosines,

d2 = R2 + 2r2 - 2Ö2Rr cosÐIBO = R2 + 2r2 - 2Ö2Rr é ê ë cos æ ç è ÐABO- p 4 ö ÷ ø ù ú û = R2 + 2r2 - 2Ö2Rr [cosÐABO (1/Ö2)+ sinÐABO (1/Ö2)] = R2 + 2r2 - 2Rr [cosÐABO + cosÐCBO] = R2 + 2r2 - 2Rr [cosÐBAC + cosÐBCA] = R2 + 2r2 - 2Rr æ ç è a+c b ö ÷ ø = R2 + 2r2 - r(a + c) ,

since b = 2R and both triangle AOB and COB are isosceles with arms of length R.

By looking at the area of DABC in two ways, we have that ac = r(a+c). Now

(a + c - r)2 = a2 + c2 + r2 + 2[ac - r(a+c)] = a2 + c2 + r2 = 4R2 + r2 ,

so that a + c = r + Ö[(4R² + r²)]. (The positive root is selected as a and c both exceed r.) Hence

d2 = R2 + 2r2 - r[r +   _______ Ö4R2 + r2   ] = R2 + r2 - r   _______ Ör2 + 4R2    .

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87.

Prove that, if the real numbers a, b, c, satisfy the equation

ëna û+ ënb û = ënc û

for each positive integer n, then at least one of a and b is an integer.

Solution. We first show that a + b = c. Suppose, if possible, that c > a + b. Let n ³ (c - a - b)^-1. Then

ënc û > nc - 1 > n(a + b) > ëna û+ ënb û

yielding a contradiction. Similarly, if c < a + b and n ³ 2(a + b - c)^-1, then

ënc û £ nc £ na - 1 + nb - 1 < ëna û+ ënb û

again yielding a contradiction. Hence a + b = c.

Let a = ëa û+ a, b = ëb û+ b and c = ëc û+g. From the condition for n = 1, we have ëa û+ ëb û = ëc û. Then, ëna û = ënëa û+ naû = nëa û+ ënaû with similar equations for b and c. Putting this together gives that ënaû+ ënbû = ëngû, for all n ³ 1. As in the first part of the solution, we have that a+ b = g, from which { na} +{ nb} = { ng} for all n ³ 1, where { x } denotes the fractional part x - ëx û of x.

We first show that a, b and g cannot all be positive and rational. For, if they were rational, then for some positive integers i, j, k with k ³ 2, we would have a = i/k, b = j/k and g = (i+j)/k. Then ëkaû = i = ë(k-1)aû+ 1, with similar relations for b and g. Thus,

ëkaû+ ëkbû- ëkgû- [ë(k-1)aû+ ë(k-1)bû- ë(k-1)gû] = 1

so that ënaû+ ënbû = ëngû must fail for either n = k or n = k-1.

Wolog, suppose that all of a, b, g are positive with a irrational. Let p be a nonnegative integer for which a+ pb < 1 £ a+ (p+1)b and suppose that e = 1 - (a+ pb). Since a+ b = g < 1, it follows that p ³ 1.

We show that there is a positive integer m ³ 2 for which a+ pb < { ma}. Let t = é1/eù and consider the intervals [i/t, (i+1)/t) where 0 £ i £ t-1. By the Pigeonhole Principle, one of these intervals must contain two of the numbers { 2a},{ 4a}, ¼, { 2(t+1)a}, say { qa} and { ra} with q ³ r + 2. Thus, |{ qa} - { ra} | < 1/t £ e. Since

{ qa} - { ra} = (q - r)a-ëqaû+ ëraû = (q - r)a±I

for some integer I, either { (q - r)a} < e or { (q - r)a} > 1 - e.

In the first case, we can find a positive integer s for which 1 > s { (q - r) a} > 1 - e. Since

s(q - r)a = së(q - r)aû+ s{ (q - r)a} ,

it follows that

{ s(q-r)a} = s { (q - r) a} > 1 - e .

Thus, in either case, we can find m ³ 2 for which a+ pb = 1 - e < { ma}.

Now, { ma} > a,

{ mg} = { ma} + { mb} ³ { ma} > a+ pb ³ a+ b = g

and

{ mb} = { mg} - { ma} < 1 - (a+ pb) = 1 - [a+ (p+1)b- b] £ 1 - (1 - b) = b .

Hence,

ëmaû = ma- { ma} = (m-1)a- ({ ma} - a) < (m-1)a

so that ë(m-1) û = ëmaû;

ëmgû = mg- { mg} = (m-1)g- ({ mg} - g) < (m-1)g

so that ë(m-1)gû = ëmgû; and ëmbû = mb- { mb} > (m-1)b ³ ë(m-1) bû, so that ë(m-1)bû+ 1 = ëmbû. Hence ënaû+ ënbû = ëngû must fail for either n = m or n = m-1.

The only remaining possibility is that either a or b vanishes, i.e., that a or b is an integer. This possibility is feasible when the other two variables have the same fractional part.

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88.

Let I be a real interval of length 1/n. Prove that I contains no more than ¹/₂(n+1) irreducible fractions of the form p/q with p and q positive integers, 1 £ q £ n and the greatest common divisor of p and q equal to 1.

Comment. The statement of the problem needs a slight correction. The result does not apply for closed intervals of length 1/n whose endpoints are consecutive fractions with denominator n. The interval [1/3, 2/3] is a counterexample. So we need to strengthen the hypothesis to exclude this case, say by requiring that the interval be open (i.e., not include its endpoints), or by supposing that not both endpoints are rational. Alternatively, we could change the bound to ¹/₂(n+3) and ask under what circumstances this bound is achieved.

Solution 1. We first establish a lemma: Let 1 £ q £ n. Then there exists a positive integer m for which ¹/₂(n+1) £ mq £ n. For, let m = ën/q û. If q > n/2, then m = 1 and clearly ¹/₂(n+1) £ mq = q £ n. If q £ n/2, then (n/q) - 1 < m £ (n/q), so that

n 2 £ n - q < mq £ n

and ¹/₂(n+1) £ mq £ n.

Let p/q and p¢/q¢ be two irreducible fractions in I with m and m¢ corresponding integers as determined by the lemma. Suppose, if possible, that mq = m¢q¢. Then

ê ê ê p q - p¢ q¢ ê ê ê = ê ê ê mp mq - m¢p¢ mq ê ê ê ³ 1 mq ³ 1 n  ,

contradicting the fact that no two fractions in I can be distant at least ¹/_n.

It follows that the mapping p/q ® mq from the set of irreducible fractions in I into the set of integers in the interval [(n+1)/2, n] is one-one. But the latter set has at most n - ((n+1)/2) + 1 = (n+1)/2 elements, and the result follows.

Solution 2. [M. Zaharia] For 1 £ i £ ¹/₂(n+1), define

Si = { 2j(2i - 1) : j = 0, 1, 2, ¼} .

(Thus, S₁ = { 1, 2, 4, 8, ¼}, S₃ = { 3, 6, 12, 24, ¼} and S₅ = { 5, 10, 20, 40, ¼}, for example.) We show that each S_i contains at most one denominator not exceeding n among the irreducible fractions in I. For suppose

a 2u(2i - 1)     and      b 2v(2i - 1)

are distinct irreducible fractions in I, with u £ v. Then

ê ê ê a 2u(2i - 1) - b 2v(2i - 1) ê ê ê = ê ê ê 2v-ua - b 2v(2i - 1) ê ê ê ³ 1 2v(2i - 1) ³ 1 n  .

But I cannot contain two fractions separated by a distance of 1/n or larger. Thus, we get a contradiction, and it follows that there cannot be more than one fraction with a denominator in each of the at most (n+1)/2 sets S_i. The result follows.

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89.

Prove that there is only one triple of positive integers, each exceeding 1, for which the product of any two of the numbers plus one is divisible by the third.

Solution 1. Let a, b, c be three numbers with the desired property; wolog, suppose that 2 £ a £ b £ c. Since a |(bc + 1), a has greatest common divisor 1 with each of b and c. Similarly, the greatest common divisor of b and c is 1. Since ab + bc + ca + 1 is a multiple of each of a, b, c, it follows that ab + bc + ca + 1 is a multiple of abc. Therefore, abc £ ab + bc + ca + 1.

Since a, b, c are distinct and so 2 £ a < b < c, we must have a ³ 2 and b ³ 3. Suppose, if possible that b ³ 4, so that c ³ 5. Then abc ³ 40 and

ab + bc + ca + 1 £ abc 5 + abc 2 + abc 4 + 1 £ 19abc 20 + abc 40 < abc ,

a contradiction. Hence b must equal 3 and a must equal 2. Since c |(ab+1), c must equal 7. The triple (a, b, c) = (2, 3, 7) satisfies the desired condition and is the only triple that does so.

Solution 2. As in Solution 1, we show that abc £ ab + bc + ca + 1, so that

1 £ 1 a + 1 b + 1  c + 1 abc  .

However, if a ³ 3, the right ride of the inequality cannot exceed (1/3) + (1/4) + (1/5) + (1/60) = 4/5 < 1. Hence a = 2. If a = 2 and b ³ 4, then b ³ 5 (why?) and the right side cannot exceed (1/2) + (1/5)+ (1/7) + (1/70) = 6/7 < 1. Hence (a, b) = (2, 3). If now c exceeds 7, then c ³ 11 and the right side of the inequality cannot exceed (1/2) + (1/3) + (1/11) + (1/66) = 31/33 < 1. Hence c £ 7, and we are now led to the solution.

Solution 3. [P. Du] As in Solution 1, we show that a, b, c are pairwise coprime and that ab + bc + ca + 1 is a multiple of abc. Assume 2 £ a < b < c. Then abc > ac and bc(a - 1) ³ bc > ac ³ a(b + 1) > ab + 1, whence, adding these, we get 2abc - bc > ac + ab + 1, so that 2abc > ab + bc + ca + 1. Hence,

abc = ab + bc + ca + 1 = bc - (c-a)b + bc + bc - (b-a)c + 1 = 3bc - (c-a)b - (c-b)a + 1 < 3bc

so that a < 3. Hence a = 2. Plugging this into the equation yields

bc = 2b + 2c + 1 = 4c - 2(c - b) + 1 < 4c

so that b < 4. Hence b = 3, and we find that 6c = 6 + 5c + 1 or c = 7.

Solution 4. [O. Ivrii] As before, we show that a, b and c are pairwise coprime, and take 2 £ a < b < c. Then bc |ab + ac + 1. Since bc > ac + 1 and bc > ab + 1, we have that 2bc > ab + ac + 1. Hence bc = ab + ac + 1, so that (bc + 1) + ab = 2(ab + 1) + ac. Since a divides each of bc + 1, ab and ac, but is coprime with ab + 1, it follows that a divides 2. Hence a = 2 and

bc = 2(b + c) + 1 Þ (b - 2)(c - 2) = 5Þ (b, c) = (3, 7) .

Solution 5. As above, we can take 2 £ a < b < c. Since

(ab + 1) c · (ca + 1) b = a2 + a c + a b + 1 bc  ,

we see that (a/c) + (a/b) + (1/(bc)) is a positive integer less than 3.

Suppose, if possible, that (a/c) + (a/b) + (1/(bc)) = 2. Then ab + ac + 1 = 2bc, whence b(c - a) + c(b - a) = 1, an impossibility. Hence a(b + c) + 1 = bc, so that

2 = (bc + 1) - a(b + c) .

Since both terms on the right are divisible by a, 2 must be a multiple of a. Hence a = 2, and we obtain (b - 2)(c - 2) = 5, so that (b, c) = (3, 7).