Issue #7 - July 15, 2021
(Last issue for Year 1)

This journal issue includes problems from our collaboration with the students at Upper School in Bucharest, Romania. The students’ submissions were coordinated by Prof. Mihaela Berindeanu.

Problems A (Not open to Grade 11 and 12 students)

Note 1: Problems are not in order of difficulty.
Note 2: You will have until midnight on Sunday, August 15, 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 $CM \times CN$.

Problem 42A. Proposed by Vedaant Srivastava
Given positive reals $a, b, c$, prove that $$\frac{a(a^3+1)}{2b+6c} + \frac{b(b^3+1)}{2c+6a} + \frac{c(c^3+1)}{2a+6b} \ge \frac{1}{8}(a^3+b^3+c^3+3).$$

Problem 43A. Proposed by Alexandru Benescu
Let $ABCDA'B'C'D'$ be a cube, $R$ be the midpoint of $BB'$, and $S$ be the midpoint of $AD$. Point $T$ is on $C'D'$ such that $\frac{C'T}{D'T}=2$. Find $\cos\angle (SC, TR)$.

Problem 48A. Proposed by Vedaant Srivastava
Let $G$ be the set of all lattice points $(x, y)$ on the Cartesian plane where $0 \le x, y \le 2021$. 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 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 53A. Proposed by Nikola Milijevic
The positive integers \(a_1, a_2, \dots, a_n\) are not greater than 2021, with the property that lcm\((a_i,a_j)\) $>$ 2021 for all \(i, j, i \neq j \). Show that: \[ \sum_{i=1}^{n} \frac{1}{a_i} < 2 \]
Problem 54A. Proposed by Cosmina Ghitescu
Consider a point $M$ in the interior of equilateral triangle $ABC$ such that $ m(\angle MAC)= 45^{\circ}$ and $ m(\angle MCB)= 30^{\circ}$. Take $N \in (AB)$ such that $ m(\angle NMC)= 120^{\circ}$. If $AM \cap CB=\{D\}$, find the value of the ratio $\frac{MN}{BD}$.

Problem 55A. Proposed by Eliza Andreea Radu
Consider the triangle $ABC$ with $AB=4$ cm, $BC=6$ cm and $AC=5$ cm. Take $M \in (AB)$ and $N \in (AC)$ such that $\cos(\angle AMN) =\frac{3}{4}$. The feet of the perpendiculars drawn from $B$ to $MN$, $NC$, and $MC$ are $P$, $Q$, and $R$, respectively. What does $P, Q, R$ form?

Problem 56A. Proposed by Cosmina Ghitescu
Let $ABCDA'B'C'D'$ be a cube with $M \in (BC)$ and $N \in (DD')$ such that $\frac{CM}{MB}=\frac{D'N}{ND}=k$. If $AC \cap DM= \{R\}$ and $CN \cap DC'= \{T\}$, find the value of $k$ such that $RT || (ABC')$.

Problem 57A. Proposed by Eliza Andreea Radu
Consider the tetrahedron $VABC$ with a volume equal to 4 such that $ m(\angle ACB)= 45^{\circ}$ and $\frac{AC+3\sqrt{2}(BC+VB)}{\sqrt{18}}=6$. Find the distance from $B$ to the plane $(VAC)$.

Problem 58A. Proposed by Aida Dragomirescu
Find $a, b, c\in\mathbb{R}$ with the property that $\left(3a-10 \right) ^2+\left(3b-10 \right) ^2+\left(3c-10 \right) ^2+150\leq3\left(ab+ac+bc \right)$.

Problem 59A. Proposed by Irina Daria Avram Popa
Find the area of a triangle with sides $a, b, c$ knowing that $$\left(3a-b+c \right)^2+ \left(3b-c+a \right)^2+\left(3c-a+b \right)^2+\dfrac{1}{3}\leq2\left(a+b+c \right). $$
Problem 60A. Proposed by Daniel Alexandru Guba
If $a,b,c\in\left[5,\infty \right)$, prove that $\dfrac{a+b}{ab+4c+15}+\dfrac{a+c}{ac+4b+15}+\dfrac{b+c}{bc+4a+15}\leq\dfrac{1}{2}$.

Problem 61A. Proposed by Gabriel Crisan
Let $M$ be a point in the interior of triangle $ABC$ such that angles $ABM$ and $ACM$ are congruent. From $M$, build $MP$ perpendicular to side $AB$, where $P \in AB$, and build $MQ$ perpendicular to side $AC$, where $Q \in AC$. If $S$ and $K$ are the midpoints of $BC$ and $PQ$, respectively, prove that $SK$ is perpendicular to $PQ$.

Problem 62A. Proposed by Andrei Croitoru
Let $a_1$,$a_2$ . . . $a_n$ be a string and $n \in$ $\mathbb{N}$. Knowing that $a_1$=14 and \begin{equation} a_{n+1}= \cfrac{(n+3)\cdot a_n +26}{n+1}\hspace{20pt}\forall n\geq 1, \nonumber \end{equation} find all numbers $n$ such that $a_n$ $\in$ $\mathbb{N}$.

Problem 63A. Proposed by Eliza Andreea Radu
Prove that $7^{7^{2021}}+1$ and $7^{7^{2024}}+50$ are coprime.

Problem 64A. Proposed by Aida Dragomirescu
Solve in $\mathbb{R}$ the following system $\left\lbrace\begin{array}{l} x^4+256=4y^3+64z \\ y^4+256=4z^3+64x \\ z^4+256=4x^3+64y \end{array}\right.$

Problem 65A. Proposed by Daisy Sheng
The regular hexagon $ABCDEF$, with vertices labeled in a counterclockwise order, has coordinates $A(4 + \sqrt 3, 2)$ and $B(8+\sqrt 3, 4)$. Diagonal $CF$ is extended until it intersects the $y$-axis at $M$. Find the area of triangle $BAM$.

 

Problems B (Open to all Grades)

Note 1: Problems are not in order of difficulty.
Note 2: You will have until midnight on Sunday, August 15, 2021 to submit solutions.

Problem 39B. Proposed by Alexander Monteith-Pistor
For $n \in \mathbb{N}$, let $S(n)$ and $P(n)$ denote the sum and product of the digits of $n$ (respectively). For how many $k \in \mathbb{N}$ do there exist positive integers $n_1, ..., n_k$ satisfying $$\sum_{i=1}^k n_i = 2021$$ $$\sum_{i=1}^k S(n_i) = \sum_{i=1}^k P(n_i)$$

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 $2 \times 2$ square with one $1 \times 1$ square missing). Prove that an $L$-region of size $n$ can be tiled with $L$-tiles for all positive integers $n$.

Problem 49B. Proposed by Cosmina Ghitescu
Solve the equation $2021+2^x=7^y5^z$, where $x$,$y$,$z \in \mathbb{N}$.

Problem 52B. Proposed by Daisy Sheng
Find the general form for the integer $k$ such that the expression $$2^{n+1}+5^{n+2}\cdot3^{2n+4}\cdot k + (2k+1)\cdot (47 \cdot 3)^{n+3}$$ is divisible by 2021 for all positive integers $n$ that are odd multiples of $3$. For reference, $2021 = 43 \cdot 47$.

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 $KN=30^{\circ}$ and $ \angle ABD = 45^{\circ}$ (see figure below). If $CD^2 + 2AC\cdot AB = 4AB^2 + AC \cdot CD$ and $\cos(m(\angle BAD))=\frac{AB-AC}{AD}$, prove that $BM=CM$.
Note: Inspired by a TST Problem for Girls' Math Team Canada.

Problem 54B. Proposed by Max Jiang
Find all functions $f:\mathbb{R}\to\mathbb{R}$ that satisfy the following property: Given an ordered pair $(x,y)\in\mathbb{R}^2$, we have either \begin{align*} f(x)-f(y)&=f(x^2-y^2) \\ f(x)f(y)&=f(x^2y^2) \end{align*} or \begin{align*} f(y)&\ne0 \\ f(x)+f(y)&=f(x^2-y^2) \\ f(x)/f(y)&=f(x^2y^2). \end{align*} Note: it is possible that an ordered pair $(x,y),x\ne y$ satisfies the first set of conditions while the ordered pair $(y,x)$ satisfies the other set.

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 58B. Proposed by Aida Dragomirescu
Solve in $\mathbb{R}$ the system consisting of the following equations \begin{align} \sum_{i=1}^{4} x_i^4=4\left(\sum _{i=1}^{4} x_i^3 \right)-108 \\ \sum_{i=1}^{4} x_i^2=6\left(\sum_{i=1}^{4} x_i \right)-36 \end{align}

Problem 59B. Proposed by Alexandru Benescu
Prove that $$i) \frac{a^5}{b^3}+ \frac{b^5}{c^3}+ \frac{c^5}{a^3} \geq ab+bc+ca $$ $$ii) \frac{a^{x+y+1}}{b^x}+ \frac{b^{x+y+1}}{c^x}+ \frac{c^{x+y+1}}{a^x} \geq a^yb+b^yc+c^ya $$ where $a, b, c \in \mathbb{R_{+}}$ and $x,y \in \mathbb{N}$.

Problem 60B. Proposed by Cosmina Ghitescu
Consider the triangle ABC with sides $a>b>c$ satisfying the condition $$\sin^2A+\sin^2B+\sin^2C=2.$$ Prove that $$\frac{3\sqrt{3}}{\sqrt{2}+1}(\sin A+\sin B+\sin C)<\frac{8R^2}{S}.$$
Problem 61B. Proposed by Alexandru Benescu
Let $a$, $b$ and $c$ be the sides of a triangle and $R$ be the radius of its circumscribed circle. Prove that $$ (a+b+c)\cdot\sqrt[3]{(abc)^2}<2R\cdot(ab+bc+ca).$$
Problem 62B. Proposed by Eliza Andreea Radu
If $a_1, a_2, \ldots , a_{2001} \in \mathbb{R_{+}}$ such that $ \sum_{i=1}^{2021}a_i>2021$, prove that $$a_{1}^{2^{2021}} \cdot 1 \cdot 2 + a_{2}^{2^{2021}} \cdot 2 \cdot 3 + \ldots + a_{2021}^{2^{2021}} \cdot 2021 \cdot 2022 > 4086462.$$
Problem 63B. Proposed by Alexandru Benescu
Prove that $$ \frac {\sqrt{1^6+1}}{1^2}+\frac {\sqrt{2^6+1}}{2^2}+\frac {\sqrt{3^6+1}}{3^2}+\ldots \frac {\sqrt{2020^6+1}}{2020^2} > \frac {\sqrt{(2021^2\cdot1010)^2+2020^2}}{2021}.$$

Problem 64B. Proposed by Pavel Ciurea
In each $1 \times 1$ square of an $n \times n$ board, the number $1$ or $-1$ is written. We define a move as the change of the signs of all the numbers on a diagonal. Determine for which configurations there is a finite series of moves that lead to the product of the numbers in each row and each column being 1.

Problem 65B. Proposed by Stefan-Ionel Dumitrescu
If $a, b, c\in\mathbb{R_{+}}$ such that $abc = a + b + c$ prove that $$\sqrt{ab + bc + ca} \bigg(\frac{1}{3} + \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}\bigg) \geq 4.$$

Problem 66B. Proposed by Stefan-Ionel Dumitrescu
Consider right-angled triangle $ABC$ with $m(\angle BAC) = 90^{\circ}$. Extend $AC$ to $D$, where $D$ and $C$ are on different sides of $AB$, such that $AB =AD$. Similarly, extend $AB$ to $E$, where $E$ and $B$ are on different sides of $AC$, such that $AC =AE$. Let $M$ be the midpoint of $BD$, $N$ be the midpoint of $CE$, and $BN \cap CM = \{U\}$. Take $I \in (AU)$, where $A$ and $I$ are on different sides of $BC$, such that $m(\angle ICB) = 45^{\circ}$. Calculate $m(\angle CAI)$.

Problem 67B. Proposed by Stefan-Ionel Dumitrescu
Consider a cube $ABCDA'B'C'D'$. Point $W$ can be in the interior, on the faces, or on the edges of the cube. Point $X$ can be in the interior of the cube or on the edges of face $ADD'A'$. Point $Y$ can be in the interior of the cube or on the edges of face $ABB'A'$. If $Z$ is the midpoint of $XY$, what is the probability of finding a pair of points ($X$,$Y$), where $W$ is the midpoint of $AZ$?

Problem 68B. Proposed by Vedaant Srivastava
Determine the smallest positive integer $M$ that satisfies the following two conditions:
i) $M$ is a multiple of 2021.
ii) For any positive integer $n$, if $n^n - 1$ is divisible by $M$, then $n-1$ is divisible by $M$.