UND MATHEMATICS TRACK MEET Individual Test 4

University of North Dakota Grades 11/12
January 12, 2026
School Team Name
Calculators are NOT allowed. Student Name

• 1.

  $x$ and $y$ are non-integers with $3x-7y=10$ and $xy=-1$, find the value of $9x^2+49y^2$.

    (2 pts) 1.

• 2.

  In a room of 20 people, if each each person shakes each other person’s hand exactly one time, how many handshakes occur?

    (3 pts) 2.

• 3.

  Find all positive integers such that $n^2+n+41$ is divisible by 41.

    (3 pts) 3.

• 4.

  Find all real numbers $x,y,z$ satisfying the system of equations.
  

  	$x^2-4y+7=0$
  	$y^2-6z+14=0$
  	$z^2-2x-7=0$

    (3 pts) 4.

• 5.

  The circles below are tangent to the horizontal line and tangent to each other. They have radii of 7 and 10 respectively. Find the distance between $a$ and $b$.
  

    (3 pts) 5.

• 6.

  A standard deck of 52 playing cards is shuffled and laid out in a row. What is the probability that the four aces appear in alphabetical order from left to right? (The aces don’t need to be consecutive; there can be other cards between them. We only care that when looking at the row of cards left to right, we encounter the aces in the order: Ace of Clubs, Ace of Diamonds, Ace of Hearts, Ace of Spades)

    (3 pts) 6.

• 7.

  Find the radius of the semicircle enclosed within the right triangle. The semicircle is tangent to both legs of the right triangle.
  

    (3 pts) 7.

TOTAL POINTS