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.
Note 1: Problems are not in order of difficulty.
Note 2: You will have until midnight on Friday, November 5, 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 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 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 $\cos(\measuredangle AMN) =\frac{3}{4}$.
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 57A. Proposed by Eliza Andreea Radu
Consider the tetrahedron $VABC$ with a volume equal to 4 such that $ m(\measuredangle ACB)= 45^{\circ}$ and $\frac{AC+3\sqrt{2}(BC+VB)}{\sqrt{18}}=6$. Find the distance from $B$ to the plane $(VAC)$.
Problem 66A. Proposed by Adelina Sofian
Let $x, y, z $ be real numbers such that $x^2+y^2+z^2=12$. Find the range of values for $x, y, z $ such that
$$\frac{8+6x^2}{x^2+12}+\frac{8+6y^2}{y^2+12}+\frac{8+6z^2}{z^2+12} \leq 6.$$
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 68A. Proposed by Mihai Teodor Stupariu
In a given triangle AMB whose side lengths AM, MB, and AB are in the ratio of $1:\sqrt{5}:\sqrt{10}$, take $C$ on ray $(AM$ such that $m(\measuredangle (ACB))=135^{\circ}$. Find $\frac{BC}{AM}$.
Problem 69A. Proposed by Luca Vlad Andrei
Solve in $\mathbb{R}$ the equation $ \ \left[\dfrac{2x+5}{6} \right]+\left[\dfrac{2x+7}{6} \right]=\left\lbrace \dfrac{2x+3}{6}\right\rbrace +\dfrac{8}{3}$.
Problem 70A. Proposed by Vlad Nicolae Florescu
Let $H$ be the orthocenter, $I$ the incenter, and $O$ the circumcenter of an acute triangle $ABC$. If $m(\measuredangle AHO)=m(\measuredangle AOH)$, show that $BHIC$
is an inscribable quadrilateral.
Problem 71A. Proposed by Irina Daria Avram Popa
Find the values $\ a,b,c\in \mathbb{R}$ corresponding to the minimum of $E(a,b,c)=3(a^2+b^2+c^2+ab+ac+bc)+4(a+b+c)$ and derive the minimum.
Problem 72A. Proposed by Octavian Tiberiu Bacain
Find the prime numbers $p, q$, and $t$ such that $p^4+q^4+t^4=20151762$.
Problem 73A. Proposed by Ana Boiangiu
Let $ABC$ be a scalene triangle and let $M$ be the midpoint of $BC$. Let the circumcircle of $\Delta AMB$ meet $AC$ at $D$ other than $A$. Similarly, let the circumcircle of $\Delta AMC$ meet $AB$ at $E$ other than $A$. Let $N$ be the midpoint of $DE$. Prove that $MN$ is parallel to the $A$-symmedian of $\Delta ABC$.
Problem 74A. Proposed by Octavian Tiberiu Bacain
Prove that the following relationship is true for any real numbers $a, b, c$, and $d$:
$$(a+b+c)^2+(b+c+d)^2+(a+c+d)^2+(a+b+d)^2+116 \geq 2(13a+9b+17c+15d).$$
Problem 75A. Proposed by Eliza Andreea Radu
Let $a_1$, $a_2$ . . .$a_n$ be a sequence and $n \in$ $\mathbb{N}$. Knowing that $a_1=121$ and
\begin{equation}
a_{n+1}= \cfrac{(n+3)\cdot a_n-27}{n+6}\hspace{20pt}\forall n\geq 1,
\nonumber
\end{equation}
find all numbers $n$ such that $a_n$ $\in$ $\mathbb{N}$.
Problem 76A. Proposed by Vedaant Srivastava
Find all triples $(x, y, z) \in \mathbb{R}^3$ that satisfy the following system of equations:
\[
\begin{cases}
x^3 = -3x^2-11y+26\\
y^3 = 3y - 7z + 23\\
z^3 = -9z^2 + 13x - 121
\end{cases}
\]
Note 1: Problems are not in order of difficulty.
Note 2: You will have until midnight on Friday, November 5, 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 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 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 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 69B. Proposed by Frederick Pu
Consider the following game: The first turn consists of placing one card on the table.
Each turn after the first, insert a new card in a random place in the pile. If the new card you inserted is not at the top of the deck you win. Otherwise, keep playing.
What is the probability that you win eventually?
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 71B. Proposed by Andrew Dong
Let $n \ge 3$ be an integer and let $T_1, T_2, \ldots, T_{n-2}$ be the $n-2$ triangles of a triangulation $\mathcal{T}$ of a convex $n$-gon.
Construct an undirected graph $G = (V, E)$ with $V = [n]$ and $(u, v) \in E$ if and only if $T_u$ and $T_v$ share an edge in $\mathcal{T}$. Show that $G$ is a tree.
Problem 72B. Proposed by Daisy Sheng
For $x\in \mathbb{R}$, the solution to $$5^{2x^4-10x^2+9}+4\cdot45^{x^4-5x^2+4}\geq 3^{4x^4-20x^2+18}$$ is $x \in [-b, -a] \cup [a,b]$, where $a,b \in \mathbb{Z}^+.$ Find $a$ and $b$.
Problem 73B. Proposed by Andrew Dong
Consider $n \ge 3$ distinct points in the plane. Show that there exists a triple of three distinct points $(A, B, C)$ such that $\measuredangle ABC \le \pi/(n-2)$.
Problem 74B. Proposed by Alexander Monteith-Pistor
Prove that, for all positive integers $n$ which are relatively prime to $2021$, $2021$ divides
$$n^{(n^{281} - 281)(n^{280} - 280)...(n^2 - 2)(n - 1)} - 1$$
Problem 75B. Proposed by Max Jiang
Does every point in the unit circle lie on the polar curve $r=\cos(\theta^2)$?
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 Ciurea Pavel
Given the positive real numbers $x$, $y$, and $z$, prove that
$$2(\sum_{cyc}x) \sqrt{\sum_{cyc}\sqrt{x^2+y^2+z^2}}\geq$$ $$\geq \sum_{cyc} \sqrt{3(x+y)(x+z)(\sqrt{x^2+y^2+xy}+\sqrt{x^2+z^2+xy}-\sqrt{y^2+z^2+yz})}.$$