Year 2: Issue #2 - Nov 25, 2021

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 Monday, January 10, 2022 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 67A. Proposed by Eliza Andreea Radu
Consider the convex quadrilateral $ABCD$ and the parallelograms $ACPD$ and $ABDQ$. Find $m(\measuredangle (AC, BD))$ knowing that $BP=16$, $CQ=12$, $AC=4$, and $BD=7\sqrt{2}$.

Problem 76A. Proposed by Vedaant Srivastava
Find all triples $(x, y, z) \in \mathbb{R}^3$ that satisfy the following system of equations: \[ \begin{cases} x^3 = -3x^2-11y+26\\ y^3 = 3y - 7z + 23\\ z^3 = -9z^2 + 13x - 121 \end{cases} \]

Problem 77A. Proposed by Octavian Tiberiu Bacain
Show that $\frac{2x}{2x+4}+\frac{3y}{3y+9}+\frac{5z}{5z+25} \geq 2$ for any $x$, $y$, and $z$ integers satisfying $2x+3y+5z=12$.

Problem 78A. Proposed by Aurelia Georgescu
Find $x$, $y \in \mathbb{Z}$ such that $\frac{4x^2+10x+7}{4y^3-12y^2+8y-1} \in \mathbb{Z}$.

Problem 79A. Proposed by Octavian Tiberiu Bacain
Prove that $$\frac{2}{3\sqrt{2}}\cdot\frac{2}{5\sqrt{6}} \cdot\frac{2}{7\sqrt{12}}\cdots \frac{2}{201\sqrt{10100}}> \frac{1}{\frac{101!^2}{101}}.$$

Problem 80A. Proposed by Vlad Armeanu
Solve in $\mathbb{R}$ the equation: \[x+ \left[x-\frac{2021}{506}\right]^2 + \left|x^2-9x+20\right| =2\left(\sqrt{x-3}+1\right).\]

Problem 81A. Proposed by Octavian Tiberiu Bacain
Find $x$ and $y$ integers such that \[\left[\frac{y^2(x^2+1)}{x^2+y^2+1}+\frac{x(y^2+1)}{x^2+y^2+1}+\frac{x^4+y^4+2}{x^2+y^2+1}\right] =(x+y)^2-2xy+2.\]

Problem 82A. Proposed by Matei Neacsu
If $a,b,c>0$ and $a+b+c=4$, find the minimum value of the sum $S=\dfrac{a+b+3}{c+5}+\dfrac{b+c+3}{a+5}+\dfrac{c+a+3}{b+5}.$

Problem 83A. Proposed by Gabriel Crisan
Given the points $A(1,4)$, $B(1,0)$, $C(3,2)$, and $M(\frac{\sqrt{10}+2}{2},\frac{\sqrt{6}+4}{2})$, prove that the projections of $M$ on sides $AB$, $BC$, and $AC$ are collinear.

Problem 84A. Proposed by Vlad Armeanu
Solve in $\mathbb{R}$ the equation: \[\sqrt{z-y^2-6x-26}+x^2+6y+z-8=0.\]

Problem 85A. Proposed by Ilinca Maria Popa
Solve in $\mathbb{R}$ the equation: \[\sqrt{x_1\cdot(2-x_1)}+\sqrt{x_2\cdot(4-x_2)}+\sqrt{x_3\cdot(8-x_3)}+\cdots +\] \[+\sqrt{x_{2021}\cdot(2^{2021}-x_{2021})}=2^{2021}-1.\]

Problem 86A. Proposed by Maria Radu
Find how many values $n \in \mathbb{R}\setminus \mathbb{Q}$ satisfy the condition that both $4n^2-6n-2$ and $4n^3-2n^2-1$ are simultaneously rational numbers.

Problem 87A. Proposed by Vlad Armeanu
Solve in $\mathbb{R}$ the equation: \[\left[\frac{3x+11}{15}\right] + \left[\frac{3x+16}{15}\right] =5.\]

 

Problems B (Open to all Grades)

Note 1: Problems are not in order of difficulty.
Note 2: You will have until midnight on Monday, January 10, 2022 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 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 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 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 70B. Proposed by Daisy Sheng
Triangle $ABC$ is obtuse where $\angle C > 90^\circ.$ Show that $$4r^2 \leq \frac{a^2b^2c^2}{(a+b+c)^2(c^2-a^2-b^2)},$$ where $r$ is the inradius of $\triangle ABC$ and $a,b,c$ represent the length of the sides opposite to $\measuredangle A, \measuredangle B, \measuredangle C$, respectively.

Problem 76B. Proposed by Alexander Monteith-Pistor
Let $ABCD$ be a quadrilateral with $\measuredangle ABC = 90^{\circ}$. Points $E$ and $F$ are on $AD$ and $BC$ respectively such that $AB$ is parallel to $EF$. Further, $AC, BD$ and $EF$ intersect at $O$. Given that $BF = 4$, $AB = 9$, $AE = 5$ and $CD = 20$, find a polynomial $p(x)$ such that one of its roots is at $x = \frac{DO}{OB}$.

Problem 77B. Proposed by Andy Kim
(i) Evaluate $${n \choose 0} - 2 {n \choose 1} + \dots \pm 2^n {n \choose n} = \sum_{i=0}^{n} (-1)^i 2^i {n \choose i}$$ for $n\in \mathbb{Z_{+}}$.
(ii) Prove that $$\sum_{i=0}^{n} (-1)^{n-i} i^n {n \choose i} = n!$$ for all $n\in \mathbb{Z_{+}}$.

Problem 78B. Proposed by Pavel Ciurea
Given the positive real numbers $x$, $y$, and $z$, prove that $$2(\sum_{cyc}x) \sqrt{\sum_{cyc}\sqrt{x^2+y^2+z^2}}\geq$$ $$\geq \sum_{cyc} \sqrt{3(x+y)(x+z)(\sqrt{x^2+y^2+xy}+\sqrt{x^2+z^2+xz}-\sqrt{y^2+z^2+yz})}.$$

Problem 79B. Proposed by Alexandru Benescu
Prove that $3S$ has at least 16 natural divisors, where $$S=\left[\sqrt{1\cdot2\cdot3\cdot4}\right]+\left[\sqrt{2\cdot3\cdot4\cdot5}\right]+\cdots+\left[\sqrt{n\cdot(n+1)\cdot(n+2)\cdot(n+3)}\right]$$ and $n>15$, with $n \in \mathbb{N}$.

Problem 80B. Proposed by Pavel Ciurea
Given the positive real numbers $x,y,z$ and $t$, prove that: $$\frac{2xy}{yt}+\frac{2yt}{xz}+\frac{y}{z}+\frac{z}{y}+2\sqrt{(\frac{x}{y}+\frac{y}{x})(\frac{z}{t}+\frac{t}{z})(\frac{y}{z}+\frac{z}{y}-1)(\frac{x}{t}+\frac{t}{x}+1)}\geq $$ $$ \geq \frac{x}{t}+\frac{t}{x}+\sqrt{3}(\frac{x}{z}+\frac{z}{x}+\frac{y}{t}+\frac{t}{y})+4.$$

Problem 81B. Proposed by Alexandru Benescu
Let $ABC$ be a triangle, $H$ its orthocenter and $X,Y,Z$ the circumscribed circles of $\Delta BHC$, $\Delta AHC$, and $\Delta AHB$ respectively. Let $DE$ be the common tangent to $X$ and $Y$, $EF$ to $Y$ and $Z$, and $FD$ to $X$ and $Z$, such that all three circles $X,Y,Z$ are inside $\Delta DEF$. Prove that $AD$, $BE$ and $CF$ are concurrent.

Problem 82B. Proposed by Alexandru Benescu
Prove that: $$(a+b+c+2)^2+\frac{5}{2}(a+b)(b+c)(c+a)+2(a^3+2)(b^3+2)(c^3+2)+1 \geq 100abc$$ where $a$, $b$, $c \in \mathbb{R_{+}}$ and $a^2+b^2+c^2=3$.

Problem 83B. Proposed by Vlad Armeanu
Solve in $\mathbb{R}$ the following equation: $$\frac{10}{x-10}+\frac{11}{x-11}++\frac{12}{x-12}+\frac{13}{x-13}=2x^2-23x-4.$$

Problem 84B. Proposed by Nicholas Sullivan
Let $C$ be a circle of radius $r$ centred at the origin. Consider $n\geq2$ points on $C$, $\{P_k : 1 \leq k \leq n\}$, such that their centroid is at the origin. Show that for any point $Q$ on $C$, the average of the squared lengths $\{\overline{QP_k}\}$ is equal to $2r^2$. That is: \begin{equation*} \begin{split} \frac{1}{n}\sum_{k=1}^{n} (\overline{QP_k})^2 = 2r^2. \end{split} \end{equation*}

Problem 85B. Proposed by Daisy Sheng
We have two sequences of numbers. The first sequence is defined by $a_1 = 4, \text{ } a_2 = 12,$ and $a_{n+2}=2a_{n+1}-a_{n}+4$, where $n\in \mathbb{Z}^+.$ The second sequence is defined by $b_n = 4n^3 + d_1\cdot n^2 +d_2\cdot n + d_3$, where $n\in \mathbb{Z}^+$ and $d_1, d_2, d_3$ represent the coefficients of the polynomial. The polynomial for $b_n$ has roots $r_1, r_2, r_3$ that satisfy $r_1r_2+r_1r_3 = \frac{1}{2},\text{ } r_1r_2r_3 = -\frac{1}{4},$ and $r_2-r_3 = i$. Prove that $a_n$ and $b_n$ are relatively prime for all $n\in \mathbb{Z}^+.$

Problem 86B. Proposed by Nicholas Sullivan
Suppose there are 2021 people sitting at a very large circular table. How many ways are there to give each person a red, blue or green hat such that no two neighbouring people have the same colour hat?

Problem 87B. Proposed by Max Jiang
There is a row of $n$ pies, numbered 1 to $n$ from left to right. You start at pie 1 and go rightward. At each pie, there is a $1/2$ chance that you eat the pie, after which you move on to the next uneaten pie (which depends on the direction you are moving). Upon reaching the last uneaten pie, you change the direction you are going (if the last pie still uneaten you will "repeat" it). You continue until all pies are eaten. What is the probability that pie 1 is the last pie you eat? Express your answer as a finite sum in terms of $n$.

Problem 88B. Proposed by Daisy Sheng
Pierre is coloring a $n\times n$ square grid, where $n$ is even and $n\geq 8$. He chooses to omit the centre $2 \times 2$ square grid (see diagram for an example). In the spirit of the holidays, Pierre is coloring the dots either red or green. He also connects horizontally, vertically, and diagonally adjacent dots with lines using the following color scheme rules:
- 2 red dots are connected by a gold line segment;
- 2 green dots are connected by a blue line segment;
- a red and green dot are connected by a silver line segment.
Let there be: $r_0$ red outside corner dots, $r_1$ red inside corner dots, $r_2$ red edge dots, and $r_3$ red interior dots. If $B$ is the overall number of blue line segments and $G$ is the overall number of gold line segments, find $G$ in terms of $B, n, r_0, r_1, r_2,$ and $r_3$.


Problem 89B. Proposed by Arnab Sanyal
Let $\omega_1$ and $\omega_2$ be two intersecting circles. Let $\omega_1 \cap \omega_2= \{P,Q\}$. Let the tangent to $\omega_1$ and $\omega_2$ at $P$ meet $\omega_1$ and $\omega_2$ respectively at $B$ and $A$. Let the circumcircle of $\Delta PBA$ be $\omega_0$. $AQ$ meets $\omega_0$ (possibly extended) at $Y$ and $BQ$ meets $\omega_0$ (possibly extended) at $X$. Assuming $P \neq Q$, prove or disprove the following statements:
(i) $XABY$ is an isosceles trapezoid;
(ii) $QOBA$ is cyclic, where $O$ is the circumcenter of $\omega_0$.