03.12
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1. How many positive integers have square less than 107? 2. The squares of a chessboard have side 4. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board? 3. What is the remainder on dividing 1234567 + 891011 by 12?
1. Find all polynomials f(x) such that f(2x) = f '(x) f ''(x). 2. ABCD is a square side 1. P and Q lie on the side AB and R lies on the side CD. What are the possible values for the circumradius of PQR? 3. Find all pairs (m, n) of integers such that n2 - 3mn + m - n = 0.
1. Let S be the system of equations (1) y(x4 - y2 + x2) = x, (2) x(x4 - y2 + x2) = 1. Take S' to be the system of equations (1) and x·(1) - y·(2) (or y = x2). Show that S and S' do not have the same set of solutions and explain why. 2. Show that x1/xn + x2/xn-1 + x3/xn-2 + ... + xn/x1 ≥ n for any positive reals x1, x2, ... , ...
02.41
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Make Learning Math A Fun Time For Your Child Your child can have fun learning math. Learning math could be difficult to kids sometimes. Considering a different approach on getting your child to learn math and become smarter might be a good idea. why not have your child play and learn math at the same time? Math do not have to be a dull anymore.
Search Amazon.com for Math Games for Kids&nb...
20.26
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Introduction to 4 digit division: Meaning of the term division is dividing the group into equal parts. Use of the term division is dividing. During the division we can get the quotient and the reminder. We can do division in single digit, double digit, treble digit, 4 digits and 5 digits etc. Let us see 4 digit divisions in this article.
02.32
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1. Knowing that 23 October 1948 was a Saturday, which is more frequent for New Year's Day, Sunday or Monday? 2. A convex polyhedron has no diagonals (every pair of vertices are connected by an edge). Prove that it is a tetrahedron.
1. Prove that 462n+1 + 296·132n+1 is divisible by 1947. 2. Show that any graph with 6 points has a triangle or three points which are not joined to each other. 3. What is the smallest number of disks radius ½ that can cover a disk radius 1?
1. Show that a graph has an even number of points of odd degree. 2. P is any point inside an acute-angled triangle. D is the maximum and d is the minimum distance PX for X on the perimeter. Show that D ≥ 2d, and find when D = 2d.
1. Show that no triangle has two sides each shorter than its corresponding altitude (from the opposite vertex). 2. a, b, c, d are integers. For all integers m, n we can find integers h, k such that ah + bk = m and ch + dk = n. Show that |ad - bc| = 1.
1. Prove that (1+x)(1+x2)(1+x4) ... (1+x2n-1) = 1 + x + x2 + x3 + ... + x2n-1. 2. The a parallelogram has its vertices at lattice points and there is at least one other lattice point inside the parallelogram or on its sides. Show that its area is greater than 1.
1. Each button in a box is big or small, and white or black. There is at least one big button, at least one small button, at least one white button, and at least one black button. Show that there are two buttons with different size and color. 2. m and n are different positive integers. Show that 22m + 1 and 22n + 1 are coprime.
1. Show that (a + a')(c + c') ≥ (b + b')2 for all real numbers a, a', b, b', c, c' such that aa' > 0, ac ≥ b2, a'c' ≥ b'2. 2. Find the highest power of 2 dividing 2n! 3. ABC is...
01.14
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1. Show that 1/(1·2) + 1/(3·4) + 1/(5·6) + ... + 1/( (2n-1)·2n) = 1/(n+1) + 1/(n+2) + ... + 1/(2n). 2. ABC is a triangle. Show that, if the point P inside the triangle is such that the triangles PAB, PBC, PCA have equal area, then P must be the centroid.
1. x1, x2, ... , xn is any sequence of positive reals and yi is any permutation of xi. Show that ∑ xi/yi ≥ n. 2. S is a finite set of points in the plane. Show that there is at most one point P in the plane such that if A is any point of S, then there is a point A' in S with P the midpoint of AA'.
1. E is the product 2·4·6 ... 2n, and D is the product 1·3·5 ... (2n-1). Show that, for some m, D·2m is a multiple of E. 2. Given a circle, find the inscribed polygon with the largest sum of the squares of its sides.
1. If x2 + y2 = u2 + v2 = 1 and xu + yv = 0 for real x, y, u, v, find xy + uv. 2. S is a set of 16 squares on an 8 x 8 chessboard such that there are just 2 squares of S in each row and column. Show that 8 black pawns and 8 white pawns can be placed on these squares so that there is just one white pawn and one black pawn in each row and column.
1. Show that if m is a multiple of an, then (a + 1)m -1 is a multiple of an+1. 2. ABC is a triangle with AB and AC unequal. AM, AX, AD are the median, angle bisector and altitude. Show that X always lies between D and M, and that if the triangle is acute-angled, then angle MAX < angle DAX.
1. Prove that there is just one solution in integers m > n to 2/p = 1/m + 1/n for p an odd prime. 2. Show that an odd square cannot be expressed as the sum of five odd squares. 3. Find the...
00.56
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1. How many integers (1) have 5 decimal digits, (2) have last digit 6, and (3) are divisible by 3? 2. A straight line is drawn on an 8 x 8 chessboard. What is the largest possible number of the unit squares with interior points on the line?
1. Coins denomination 1, 2, 10, 20 and 50 are available. How many ways are there of paying 100? 2. Show that ∑i=0 to k nCi (-x)i is positive for all 0 ≤ x < 1/n and all k ≤ n, where nCi is the binomial coefficient.
By Mo WillemsCall number: ER WillemsBest for kids ages 4-8He has done it again! Mo Willems has produced another book that makes me laugh my pants off. You may remember Mo from his excellent Pigeon books.* His early reader series Elephant & Piggie is a great introduction for first time readers, with Gerald the Elephant and his dear friend Piggie speak in simple speech bubbles and solve complex problems such as having a bird on one’s head and being invited to a party for the first time. (What if it is a fancy-pool-costume party? You MUST be ready!)In this newest installment, Elephant and Piggie realize they are in a book. They discover they...
00.47
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1. a, b, c, d are each relatively prime to n = ad - bc, and r and s are integers. Show ar + bs is a multiple of n iff cr + ds is a multiple of n. 2. Find the sum of all four digit numbers (written in base 10) which contain only the digits 1 - 5 and contain no digit twice.
1. Show that for any integers m, n the equations w + x + 2y + 2z = m, 2w - 2x + y - z = n have a solution in integers. 2. Show that the product of four consecutive integers cannot be a square. 3. A circle...
00.34
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1. Show that given four non-coplanar points A, B, P, Q there is a plane with A, B on one side and P, Q on the other, and with all the points the same distance from the plane. 2. Show that we cannot factorise x4 + 2x2 + 2x + 2 as the product of two quadratics with integer coefficients.
1. The positive integers a, b, c are such that there are triangles with sides an, bn, cn for all positive integers n. Show that at least two of a, b, c must be equal. 2. What is the locus of the point (in the plane), the sum of whose distances to a given point and line is fixed? 3. Given three points in the plane, how does one construct three distinct...
00.25
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1. AC is the long diagonal of a parallelogram ABCD. The perpendiculars from C meet the lines AB and AD at P and Q respectively. Show that AC2 = AB·AP + AD·AQ. 2. Find three distinct positive integers a, b, c such that 1/a + 1/b + 1/c is an integer.
1. a, b are integers. The solutions of y - 2x = a, y2 - xy + x2 = b are rational. Show that they must be integers. 2. A square has 10s digit 7. What is its units digit? 3. A, B are two points inside a given...
00.22
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1. a, b are positive reals. Show that 1/x + 1/(x-a) + 1/(x+b) = 0 has two real roots one in [a/3, 2a/3] and the other in [-2b/3, -b/3]. 2. ABC is a triangle. The bisector of ∠C meets AB at D. Show that CD2 < CA·CB.
1. Given any reals A, B, C, show that An2 + Bn + C < n! for all sufficiently large integers n. 2. A triangle lies entirely inside a polygon. Show that its perimeter does not exceed the perimeter of the polygon. 3. Show that a triangle inscribed in a parallelogram has area at most half that of the parallelogram. SolutionsProblem 1Given any reals A, B, C, show...
00.18
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1. Circles C and C' meet at A and B. The arc AB of C' divides the area inside C into two equal parts. Show that its length is greater than the diameter of C. 2. a, b, c are reals such that |ax2 + bx + c| ≤ 1 for all x ≤ |1|. Show that |2ax + b| ≤ 4 for all |x| ≤ 1.
1. Prove that n! n! > nn for n > 2. 2. Let A and B be diagonally opposite vertices of a cube. Prove that the midpoints of the 6 edges not containing A or B form a regular (planar) hexagon. 3. If d is the...
00.08
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1. Real numbers a, b, c, A, B, C satisfy b2 < ac and aC - 2bB + cA = 0. Show that B2 ≥ AC. 2. L1, L2, L3, L4 are diameters of a circle C radius 1 and the angle between any two is either π/2 or π/4. If P is a point on the circle, show that the sum of the fourth powers of the distances from P to the four diameters is 3/2.
1. α, β, γ are real and satisfy α2 + β2 + γ2 = 1. Show that -1/2 ≤ αβ + βγ + γα ≤ 1. 2. If ac, bc + ad, bd = 0 (mod n) show that bc, ad = 0 (mod n). 3. ABC is a triangle with angle C = 120o. Find the...
23.59
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1. Show that the quadratic x2 + 2mx + 2n has no rational roots for odd integers m, n. 2. Let r be the radius of a circle through three points of a parallelogram. Show that any point inside the parallelogram is a distance ≤ r from at least one of its vertices.
1. Let α be a real number, not an odd multiple of π. Prove that tan α/2 is rational iff cos α and sin α are rational. 2. Show that the centers of the squares on the outside of the four sides of a rhombus form a square.
1. For what positive integers m, n can we find positive integers a, b, c such that a + mb = n and a + b = mc. Show that there is at most one such solution for each m, n. 2. Divide the unit square into 9 equal squares (forming a 3 x 3 array) and color the central square red. Now subdivide each of the 8 uncolored squares into 9 equal squares and color each...
02.58
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1. Show that 5 divides 1n + 2n + 3n + 4n iff 4 does not divide n. 2. Let α = cot π/8, β = cosec π/8. Show that α satisfies a quadratic and β a quartic, both with integral coefficients and leading coefficient 1.
1. d is not divisible by 5. For some integer n, a n3 + b n2 + c n + d is divisible by 5. Show that for some integer m, a + b m + c m2 + d m3 is divisible by 5. 2. Construct the triangle ABC given c, r and r', where c = |AB|, r is the radius of the inscribed circle, and r' is the radius of the other circle tangent to the segment AB and the lines BC and...
02.48
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1. ABCDE is a regular pentagon (with vertices in that order) inscribed in a circle of radius 1. Show that AB·AC = √5. 2. The roots of the quadratic x2 - (a + d) x + ad - bc = 0 are α and β. Show that α3 and β3 are the roots of x2 - (a3 + d3 + 3abc + 3bcd) x + (ad - bc)3 = 0.
1. For which positive integers n does 3 divide 2n + 1? 2. Triangles ABC, PQR satisfy (1) ∠A = ∠P, (2) |∠B - ∠C| < |∠Q - ∠R|. Show that sin A + sin B + sin C > sin P + sin Q + sin R. What angles A, B, C maximise sin A + sin B + sin C?
1. ABC is a right-angled triangle. Show that sin A sin B sin(A - B) + sin B sin C sin(B - C) + sin C sin A sin(C - A) + sin(A - B) sin(B - C) sin(C - A) = 0. 2. ABC is an arbitrary triangle. Show that sin(A/2) sin(B/2) sin(C/2) < 1/4. 3. The line L contains the distinct points A, B, C, D in that order. Construct a rectangle whose sides (or their...
02.36
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1. n cards are dealt to two players. Each player has at least one card. How many possible hands can the first player have? 2. ABC is a right-angled triangle. Construct a point P inside ABC so that the angles PAB, PBC, PCA are equal.
1. Show that { (m, n): 17 divides 2m + 3n} = { (m, n): 17 divides 9m + 5n}. 2. Given a circle C, and two points A, B inside it, construct a right-angled triangle PQR with vertices on C and hypoteneuse QR such that A lies on the side PQ and B lies on the side PR. For which A, B is this not possible?
Please plan to join us for our annual Family Resource Fair on Saturday, October 16, 2010 from 10:00 a.m. to 2:00 p.m. in the meeting rooms on the first floor of the library. There will be 30 family-friendly community organizations present, along with books and activities for kids and raffle prizes donated by local Beaverton businesses. This is a great opportunity to discover all the resources and services for families that are available in the Beaverton area. We look forward to seeing you there! -...
21.13
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15th Iberoamerican Mathematical Olympiad Problems 2000A1. Label the vertices of a regular n-gon from 1 to n > 3. Draw all the diagonals. Show that if n is odd then we can label each side and diagonal with a number from 1 to n different from the labels of its endpoints so that at each vertex the sides and diagonals all have different labels.
A1. Find all positive integers n < 1000 such that the cube of the sum of the digits of n equals n2. A2. Given two circles C and C' we say that C bisects C' if their common chord is a diameter of C'. Show that for any two circles which are not concentric, there are infinitely many circles which bisect them both. Find the locus of the centers of the ...
03.31
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A1. Find the smallest positive integer n so that a cube with side n can be divided into 1996 cubes each with side a positive integer. A2. M is the midpoint of the median AD of the triangle ABC. The ray BM meets AC at N. Show that AB is tangent to the circumcircle of NBC iff BM/MN = (BC/BN)2.
Arithmetic FourA game like Fraction Four but instead of fraction questions the player must answer arithmetic questions (addition, subtraction, multiplication, division) to earn a piece to place on the board. Arithmetic Quiz Arithmetic Quiz gives the user randomized questions to answer on arithmetic with whole numbers and integers. Bounded Fraction Poin...
03.00
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A1. Find all possible values for the sum of the digits of a square. A2. n > 1. Find all solutions in real numbers x1, x2, ... , xn+1 all at least 1 such that: (1) x11/2 + x21/3 + x31/4 + ... + xn1/(n+1) = n xn+11/2; and (2) (x1 + x2 + ... + xn)/n = xn+1.
A1. Show that there is a number 1 < b < 1993 such that if 1994 is written in base b then all its digits are the same. Show that there is no number 1 < b < 1992 such that if 1993 is written in base b then all its digits are the same. A2. ABCD is a cyclic quadrilateral. A circle whose center is on the side AB touches the other three sides. Show that AB = AD...
02.27
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A1. The sides of a triangle form an arithmetic progression. The altitudes also form an arithmetic progression. Show that the triangle must be equilateral. A2. The positive integers a, b, c, d, p, q satisfy ad - bc = 1 and a/b > p/q > c/d. Show that q ≥ b + d and that if q = b + d, then p = a + c.
A1. Find f(x) such that f(x)2f( (1-x)/(1+x) ) = 64x for x not 0, ±1. A2. In the triangle ABC, the midpoints of AC and AB are M and N respectively. BM and CN meet at P. Show that if it is possible to inscribe a circle in the quadrilateral AMPN (touching every side), then ABC is isosceles.
A1. Find all integer solutions to: a + b + c = 24, a2 + b2 + c2 = 210, abc = 440. A2. P is a point inside the equilateral triangle ABC such that PA = 5, PB = 7, PC = 8. Find AB. A3. Find the roots r1, r2, r3, r4 of the equation 4x4 - ax3 + bx2 - cx + 5 = 0, given that they are positive reals satisfying r1/2 + r2/4 + r3/5 + r4/8...
20.08
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a3 + b3 = c3 a3 + b3 = c3 + d3 This has no solutions when a, b and c are whole numbers 13 +123 = 93 + 103 93 +153 = 23 + 163 Can you find others?
How can you tell if a number divides exactly by 2? Answer: If it ends in an even number. How can you tell if a number divides exactly by 5? Answer: If it ends in a 0 or a 5. ...
02.01
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1. Let A = 44...4 (2n digits) and B = 88...8 (n digits). Show that A + 2B + 4 is a square. 2. A1, A2, ... , An are points in the plane, so that if we take the points in any order B1, B2, ... , Bn, then the broken line B1B2...Bn does not intersect itself. What is the largest possible value of n?
1. The triangle ABC has CA = CB. P is a point on the circumcircle between A and B (and on the opposite side of the line AB to C). D is the foot of the perpendicular from C to PB. Show that PA + PB = 2·PD. 2. The circles center O1 and O2 meet at A and B with the centers on opposite sides of AB. The lines BO1 and BO2 meet their respective circles...
01.56
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1. Find all positive integers a, b, c such that a3 + b3 + c3 = 2001. 2. ABC is a triangle with ∠C = 90o and CA ≠ CB. CH is an altitude and CL is an angle bisector. Show that for X ≠ C on the line CL, we have ∠XAC ≠ ∠XBC. Show that for Y ≠ C on the line CH we have ∠YAC ≠ ∠YBC.
1. x and y are positive reals such that x3 + y3 + (x + y)3 + 30xy = 2000. Show that x + y = 10. 2. Find all positive integers n such that n3 + 33 is a perfect square. 3. ABC is a triangle. E, F are...
01.51
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1. a, b, c are distinct reals and there are reals x, y such that a3 + ax + y = 0, b3 + bx + y = 0 and c3 + cx + y = 0. Show that a + b + c = 0. 2. Let an = 23n + 36n+2 + 56n+2. Find gcd(a0, a1, a2, ... , a1999).
1. The number 11...122...25 has 1997 1s and 1998 2s. Show that it is a perfect square. 2. The convex pentagon ABCDE has AB = AE = CD = 1 and angle ABC = angle DEA = 90o and BC + DE = 1. Find its area. ...
01.46
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1. Show that given any 9 points inside a square side 1 we can always find three which form a triangle with area < 1/8. 2. Given reals x, y with (x2 + y2)/(x2 - y2) + (x2 - y2)/(x2 + y2) = k, find (x8 + y8)/(x8 - y8) + (x8 - y8)/(x8 + y8) in terms of k.
A1. Find the smallest positive integer N which is a multiple of 83 and is such that N2 has exactly 63 positive divisors. A2. Given two circles (neither inside the other) with different radii, a line L, and k > 0, show how to construct a line L' parallel to L so that L intersects the two circles in chords with total length k.
A1. Let S be the set of all points in the plane with integer coordinates. Let T be the set of all segments AB, where A, B ∈ S and AB has integer length. Prove that we cannot find two elements of T making an angle 45o. Is the same true in three dimensions?
A1. Show that √x + √y + √(xy) = √x + √(y + xy + 2y√x). Hence show that √3 + √(10 + 2√3) = √(5 + √22) + √(8 - √22 + 2√(15 - 3√22)). A2. Every point of the plane is painted with one of three colors. Can we always find two points a distance 1 apart which are the same color?
The Tilting House by Tom LlewellynThe Peshik family has finally found a house they can afford. There are a few problems. All the floors in the house tilt. And there are funny diagrams drawn on almost every wall. But they decide to buy the house anyway and so begins their adventure in "The Tilting House." Slanting floors and pictures on the wall are just the start of their troubles. Mr. Peshik accidently makes an enemy out of a family of talking rats living in the attic. Josh and Aaron, the Peshik boys, find some magic grow powder which turns a hiking trip into a survival test. Next, the dimmer switch in the dining room goes on the...
02.28
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1. One face of a pyramid with square base and all edges 2 is glued to a face of a regular tetrahedron with edge length 2 to form a polyhedron. What is the total edge length of the polyhedron? 2. P is a point on the circumcircle of the square ABCD between C and D. Show that PA2 - PB2 = PB·PD - PA·PC.
1. A chess board is placed on top of an identical board and rotated through 45 degrees about its center. What is the area which is black in both boards? 2. AB is a diameter of the circle C. M and N are any two points on the circle. The chord MA' is perpendicular to the line NA and the chord MB' is perpendicular to the line NB. Show that AA' and BB' are ...
02.14
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1. P is a polygon. Its sides do not intersect except at its vertices, and no three vertices lie on a line. The pair of sides AB, PQ is called special if (1) AB and PQ do not share a vertex and (2) either the line AB intersects the segment PQ or the line PQ intersects the segment AB. Show that the number of special pairs is even.
1. 50 watches, all keeping perfect time, lie on a table. Show that there is a moment when the sum of the distances from the center of the table to the center of each dial equals the sum of the distances from the center of the table to the tip of each minute hand.
1. (1) O is the circumcenter of the triangle ABC. The triangle is rotated about O to give a new triangle A'B'C'. The lines AB and A'B' intersect at C'', BC and B'C' intersect at A'', and CA and C'A' intersect at B''. Show that A''B''C'' is similar to ABC. (2) O is the center of the circle through ABCD. ABCD is rotated about O to give the quadrilateral A'B'C'...
01.08
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6th All Russian Mathematical Olympiad Problems 19661. There are an odd number of soldiers on an exercise. The distance between every pair of soldiers is different. Each soldier watches his nearest neighbour. Prove that at least one soldier is not being watched.
5th All Russian Mathematical Olympiad Problems 19651. (a) Each of x1, ... , xn is -1, 0 or 1. What is the minimal possible value of the sum of all xixj with 1 ≤ i < j ≤ n? (b) Is the answer the same if the xi are real numbers satisfying 0 ≤ |xi| ≤ 1 for 1 ≤ i ≤ n?
4th All Russian Mathematical Olympiad Problems 19641. In the triangle ABC, the length of the altitude from A is not less than BC, and the length of the altitude from B is not less than AC. Find the angles. 2. If m, k, n are natural numbers and n>1, prove that we cannot have m(m+1) = kn. 3. Reduce each of the first billion natural numbers (billion = 109)...
00.59
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2nd All Russian Mathematical Olympiad Problems 19621. ABCD is any convex quadrilateral. Construct a new quadrilateral as follows. Take A' so that A is the midpoint of DA'; similarly, B' so that B is the midpoint of AB'; C' so that C is the midpoint of BC'; and D' so that D is the midpoint of CD'. Show that the area of A'B'C'D' is five times the area of ABCD.
1. Given 12 vertices and 16 edges arranged as follows:Draw any curve which does not pass through any vertex. Prove that the curve cannot intersect each edge just once. Intersection means that the curve crosses the edge from one side to the other. For example, a circle which had one of the edges as tangent would not intersect that edge.
In the last column, I discussed ellipses and how drawing them involves the fluid, fairly fast movement of the hand, letting your reflexes carry out the kind of rounded shape you intend to make. Now we’ll move on to shading the pot that we previously described in simple outline, using curving lines that are like segments of the ellipse.James McMullanThese are what I think of as “cat strok...
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