VTMO 2022 — Grade 9 — vtmo

Calculate $S=\sum_{k=2}^{100}\frac{k+(k-1)+\sqrt{k(k-1)}}{\sqrt{k}+\sqrt{k-1}}$.

Increasing both rectangle dimensions by $3$ doubles its area. What is the area after decreasing both original dimensions by $3$?

$P$ has remainders $- 4$ on division by $2 x + 1$ and $5$ by $5 x - 2$; division by $10 x^{2} + x - 2$ has quotient $x$ and linear remainder. Find $P \left(2\right)$.

Tables $I , II , III$ seat $2 , 4 , 8$ respectively. Types $I , II$ have $1$ vase each, type $III$ has $2$. There are $14$ vases and capacity at most $46$ people. What is the fewest tables?

$A , B$ lie on $y = 2 x - 5$ and triangle $OAB$ is right isosceles at $O$. Find its area.

Partition $1 , 2 , \ldots , 9$ into three unlabeled groups of $3$, each with sum divisible by $3$. How many partitions?

How many triangles with vertices of a regular $20$-gon have an angle greater than $100 ^{\circ}$?

Let $\left[a\right]$ denote the floor. How many integers $n$ make $A = | \left[\frac{5 n^{2} + 9 n + 1}{2}\right] |$ prime?

$ABC$ is isosceles at $A , D \in BC , BC = 3 BD$. $CE \perp AD$ at $E \in AD$, and $\angle AEB = 128 ^{\circ}$. Find $\angle BAC$.

Cyclic quadrilateral $ABCD$ has $AB = 3 , AD = 5 , BC = CD = 7$. Find its area.

An and Binh together complete a job in $3$ hours. Each individual time is a positive integer, and Binh is faster. How many hours does Binh need?

Distinct digits $a , b , c$ satisfy the number formed by $2022$ copies of a followed by $2022$ copies of $b$, plus $1$, equals (the number with $2022$ copies of $c$, plus $1 )^{2}$. Maximize $a + b + c$.

Fill $a = 5 \times 5$ grid with $1 , 2 , 3$ so every $2 \times 2$ square contains all three numbers. Find the least total.

Exactly three of these statements about A are true: $A + 19$ is prime; $A + 1$ is a square; $9 A - 14$ is a square; $6 A - 24$ has exactly $8$ positive divisors. Find $A$.

Inside right triangle $ABC$ with $AB = 27 , AC = 36$, four equal circles are tangent to one another and the sides as pictured. Find their radius.

Acute triangle $ABC$ with $AB \lt AC$ lies on $\left(O\right)$. Tangents at $B , C$ meet $M$; $MA$ meets $\left(O\right)$ again at $E$. The line through $E$ parallel to $BC$ meets $\left(O\right)$ again at $F$. Circle $\left(I\right)$ passes through $A$ and is tangent to $BC$ at $C$; it meets $AF$ again at $N$. Prove $BC$ is the perpendicular bisector of $NE$.

$A = 5 \times 5$ board has a light in every cell. Pressing a cell toggles it and its edge-neighbors. Initially only the central light is off and all others are on as pictured. Can all lights be switched off?