VTMO 2013 — Grade 7 — semifinal

$P \left(x\right) = x^{3} - 4 x^{2} - 3 x + 5$. Evaluate $P \left(2\right)$.

Which expression keeps its result when every $7$ is replaced by the same other integer, with division defined?

Triangle $ABC$ has angle $A = 3 B = \frac{C}{2}$. Find $A$.

$AB = AC = 7 \,\mathrm{cm}$ and $BC$ is a positive integer. What maximum triangle perimeter is possible?

Distinct shapes represent distinct digits. In the illustrated addition, square+square+circle-circle=triangle-triangle-triangle. Find square.

For every integer $30$ through $50$, subtract the units digit from the tens digit. Sum all results.

Quadrilateral $ABCD$ has $AB = 11 , BC = 7 , CD = 9 , DA = 3 \,\mathrm{cm}$, and right angles $A , C$. Find its area.

A group’s average age is $15$. In three years the total age will be $198$. How many people are there?

Use digits $1 \text{--} 8$ once to form two four-digit numbers. What is their maximum sum?

A boat takes four hours downstream $A$→$B$ and eight upstream. How long does a drifting raft take $A$→$B$?

Which value makes $P \left(x\right) = \left(x - 2\right) \left(x - 1\right) \left(x + 1\right) \left(x + 2\right) \gt 0$?

A box has $10$ blue, $13$ red, $12$ yellow balls. What minimum blind draw guarantees $10$ of one color?

All balances shown are in equilibrium; ignore the bars and strings. Hanging objects total $112 g$. What does one star weigh?

In the pictured quadrilateral, $BC = AD$, angle $ABC = 75 ^{\circ} , BCA = 30 ^{\circ} , CAD = 50 ^{\circ}$. Find angle $ADC$.

Classify a triangle with sides $7 , 9 , 12$.

Ten distinct positive integers sum to $100$. What is the largest possible maximum?

The $1500$-digit number $20142014 \ldots 2014$ is shortened by deleting digits. What maximum deletions leave digit sum $2014$?

A guard works four days then rests one. Today is a Saturday rest day. After how many days will a rest day next be Saturday?

There are $60$ seats around a circular table. Seat students so each has exactly one occupied neighboring seat. What is the maximum number?

Two parallel lines each contain four marked points. How many triangles have vertices among these points?