Note 1: Problems are not in order of difficulty.
Note 2: You will have until midnight on Friday, June 18, 2021 to submit solutions.
Problem 35A. Proposed by DC
In trapezoid ABCD, the bases are AB=7 cm and CD=3 cm. The circle with the origin at A and radius AD intersects diagonal AC at M and N. Calculate the value of the product CM×CN.
Problem 41A. Proposed by Alexander Monteith-Pistor
Find all functions f:N→N such that f(1)=1 and
f(pkm)=k−1∑i=0f(pim)
for all p,k,m∈N where p is a prime which does not divide m.
Problem 42A. Proposed by Vedaant Srivastava
Given positive reals a,b,c, prove that
a(a3+1)2b+6c+b(b3+1)2c+6a+c(c3+1)2a+6b≥18(a3+b3+c3+3).
Problem 43A. Proposed by Alexandru Benescu
Let ABCDA′B′C′D′ be a cube , R the midpoint of BB′, S the midpoint of AD and T on C′D′, such that C′TD′T=2. Find cos(∠SC,TR).
Problem 44A. Proposed by Gabriel Crisan
Let us consider the sum:
Sn=31!+2!+3!+42!+3!+4!+⋯+n+2n!+(n+1)!+(n+2)!. Solve the equation:
4x=[112+S10]−x2, where [y] is the floor of y∈R.
Problem 45A. Proposed by Alexandru Benescu
Let a and b be positive real numbers such that a(a+2b−2)=8b(b+1). Find the value of ab+1.
Problem 46A. Proposed by Alexandru Benescu
Let a, b and c be positive real numbers such that abc(a+b+c)≥a2+b2+c2. Find the minimum of the expression a2+b2+c2.
Problem 47A. Proposed by Max Jiang
Find all integer solutions (a,b,c)∈Z3 for
1+3b+2c+6bc=1a+3ab+2ac+25abc.
Problem 48A. Proposed by Vedaant Srivastava
Let G be the set of all lattice points (x,y) on the Cartesian plane where 0≤x,y≤2021. Suppose that there are 2022 roadblocks positioned at points in
G such that no two roadblocks have the same x or y coordinate. Anna starts at the point (0,0), attempting to reach the point (2021,2021) through a sequence
of moves. In each move, she moves one unit up, down, left, or right, such that she always remains in G.
Given that Anna cannot visit a lattice point which is occupied by a roadblock, determine all configurations
of the roadblocks in which Anna is unable to reach her destination.
Problem 49A. Proposed by Alexander Monteith-Pistor
Let T1 and T2 be triangles. We write T1↪T2 if the interior of T2 can be divided into triangles,
all of which are similar to T1. Prove that there exists an infinite set of triangles S such that, for any distinct T,T′∈S, T↪T′ is false.
Problem 50A. Proposed by Nicholas Sullivan
Let O be the intersection point of the two diagonals of non-degenerate quadrilateral ABCD. Next, let E, F, G and H be the
midpoints of AB, BC, CD and DA respectively. If EG and FH intersect at O, show that ABCD is a parallelogram.
Problem 51A. Proposed by Andy Kim
Let a,b,c be positive reals. Prove
a2b+b2c+c2a≥a+b+c.
Problem 52A. Proposed by Andy Kim
Let n be a positive integer, and let
an=1⋅(n1)+⋯+n⋅(nn)=n∑i=1i⋅(ni)
a) Prove that an is divisible by n.
b) Find a value for an in terms of n.
Problem 53A. Proposed by Nikola Milijevic
The positive integers a1,a2,…,an are not greater than 2021, with the property that lcm(ai,aj) > 2021 for all i,j,i≠j. Show that: n∑i=11ai<2
Note 1: Problems are not in order of difficulty.
Note 2: You will have until midnight on Friday, June 18, 2021 to submit solutions.
Problem 39B. Proposed by Alexander Monteith-Pistor
For n∈N, let S(n) and P(n) denote the sum and product of the digits of n (respectively). For how many k∈N do there exist positive integers n1,...,nk satisfying
k∑i=1ni=2021
k∑i=1S(ni)=k∑i=1P(ni)
Problem 40B. Proposed by Vedaant Srivastava
Two identical rows of numbers are written on a chalkboard, each comprised of the natural numbers from 1 to 10! inclusive. Determine the number of ways to pick one number from each row such that the product of the two numbers is divisible by 10!
Problem 42B. Proposed by Andy Kim
Define an L-region of size n as an L-shaped region with two
sides of length 2n and four sides of length n, and define an L-tile to be a tile with the
same shape as an L-region of size 1 (i.e. a 2×2 square with one 1×1 square missing). Prove that an L-region of size n can be tiled with L-tiles for all positive integers n.
Problem 46B. Proposed by Ana Maria Popa
For positive numbers x, y, and z show that:
S=(x+z−y)24+(x+y−z)24+(y+z−x)24+(x+z−y)24y4+
+(x+y−z)24z4+(y+z−x)24x4≥32.
Problem 47B. Proposed by Andrei Radu Vasile
Given a, b, c positive real numbers such that a+b+c=15, prove that:
aa2+50+bb2+50+cc2+50≤15.
Problem 48B. Proposed by Alexandru Benescu
Let x and p be positive integers with p prime, such that:
xxx⋯x=ppp⋯p, where x appears p times and p appears x times.
Prove that x2+p2+2(x−p−xp) is nonnegative.
Problem 49B. Proposed by Cosmina Ghitescu
Solve the equation 2021+2x=7y5z, where x,y,z∈N.
Problem 50B. Proposed by Cosmina Ghitescu
Let a, b, c be the lengths of the sides of a triangle and a, b, c∈Q+, such that they verify the system:
{ab+c=4(a+bc)(b+1)2a+cb+1+√ac=7+4√310
Find the area of the triangle.
Problem 51B. Proposed by Cosmina Ghitescu
Find the minimum of the expression
E=1−cos2A2cosA+cos2A+1+1−cos2B2cosB+cos2B+1+1−cos2C2cosC+cos2C+1, where A,B, and C are the angles of a triangle.
Problem 52B. Proposed by Daisy Sheng
Find the general form for the integer k such that the expression 2n+1+5n+2⋅32n+4⋅k+(2k+1)⋅(47⋅3)n+3
is divisible by 2021 for all positive integers n that are odd multiples of 3. For reference, 2021=43⋅47.
Problem 53B. Proposed by Daisy Sheng
Quadrilateral ABCD is constructed with M as the midpoint of AD and 2AB>AD. Let the circumcircle of triangle ACD intersect AB at K and BD at N, where arc
KN=30∘ and ∠ABD=45∘ (see figure below).
If CD2+2AC⋅AB=4AB2+AC⋅CD and cos(m(∠BAD))=AB−ACAD, prove that BM=CM.
Note: Inspired by a TST Problem for Girls' Math Team Canada.
Problem 54B. Proposed by Max Jiang
Find all functions f:R→R that satisfy the following property:
Given an ordered pair (x,y)∈R2, we have either
f(x)−f(y)=f(x2−y2)f(x)f(y)=f(x2y2)
or
f(y)≠0f(x)+f(y)=f(x2−y2)f(x)/f(y)=f(x2y2).
Note: it is possible that an ordered pair (x,y),x≠y satisfies the first set of conditions while the ordered pair (y,x) satisfies the other set.
Problem 55B. Proposed by Vedaant Srivastava
Determine all functions f:Q→Q such that
2f(x+y)+2f(x−y)=f(2x)+f(2y)
for all x,y∈Q.
Problem 56B. Proposed by Alexander Monteith-Pistor
A game is played with white and black pieces and a chessboard (8 by 8). There is an unlimited number of identical black pieces and identical white pieces. To obtain a starting position, any number of black pieces are placed on one half of the board and any number of white pieces are placed on the other half (at most one piece per square). A piece is called matched if its color is the same of the square it is on. If a piece is not matched then it is mismatched. How many starting positions satisfy the following condition
# of matched pieces -# of mismatched pieces = 16
(your answer should be a binomial coefficient).
Problem 57B. Proposed by Nicholas Sullivan
Consider two perpendicular vectors a,b in R3. If these vectors have components a=(sinα,sinβ,sinγ) and b=(cosα,cosβ,cosγ) respectively, and
sin2(α−β)+sin2(β−γ)+sin2(γ−α)=2,
then find |a|4+|b|4.