UND MATHEMATICS TRACK MEET Individual Test 1

University of North Dakota Grades 9/10
January 12, 2026
School Team Name
Calculators are allowed. Solutions Student Name

• 1.

  Roxanna has a collection of silver spoons from all over the world. She finds that she can arrange her spoons in sets of 7 with 6 left over, sets of 8 with 1 left over, or sets of 15 with 3 left over. If Roxanna has fewer than 220 spoons, how many are there?

    (2 pts) 1. $153$

  The numbers 3 greater than a multiple of 15 are: 18, 33, 48, 63, 78, 93, 108, 123, 138, 153, 168, 183, 198, and 213. Of these, the ones that are 1 greater than a multiple of 8 are 33 and 153. Only 153 is also 6 greater than a multiple of 7.

• 2.

  If any two points determine a line, how many lines are determined by seven points in a plane, no three of which are collinear?

    (3 pts) 2. $21$

  $6\cdot 7=42$, but order doesn’t matter, so half of that is 21.

• 3.

  A hockey team played 5 games with an average of 3 goals per game. How many goals do they need to score in their 6th game to increase their average to 4 goals per game?

    (3 pts) 3. $9$

  They’ve scored 15 goals so far. If they score $x$ goals in their 6th game, their average will be $\frac{15+x}{6}$. This is 4 when $x=9$.

• 4.

  Two circular dials are positioned next to each other, and an arrow is drawn on each dial, as shown
  

  The left dial rotates counterclockwise at 20^∘ per second and the right dial rotates clockwise at 8^∘ per second. What is the minimum number of seconds that must pass before the arrows are pointing directly towards each other again?

    (3 pts) 4. $90$

  The left dial completes a rotation every 18 seconds, while the right dial completes a rotation every 45 seconds. The least common multiple of these is 90 seconds.

• 5.

  Two cylinders are standing on a flat table. Cylinder A has radius 2 and height 8. Cylinder B has radius 8 and height 2. Cylinder A is $\frac{3}{4}$ full of water and Cylinder B is empty. If all of the water from Cylinder A is then poured into Cylinder B what fraction of Cylinder B is full of water? (The volume of a cylinder with radius $r$ and height $h$ is $V=\pi r^{2}h$)

    (3 pts) 5. ${\displaystyle \frac{3}{16}}$

  The volume of water in A is $\pi 2^2\cdot 8\cdot \frac{3}{4}=24\pi$. This is $\pi 8^2{h}_B$ when $h_B=\frac{24}{64}=\frac{3}{8}$. Since the height of B is 2, B will be $\frac{3}{16}$ full.

• 6.

  Given that $x$, $y$, and $z$ are natural numbers and $xy=8$ and $yz=4$. What is the smallest possible value of $x+y+z$?

    (3 pts) 6. $7$

  Since $y$ divides 4, it must be 1, 2, or 4. If $y=1$, then $x=8$, $z=4$, and $x+y+z=13$. If $y=2$, then $x=4$, $z=2$, and $x+y+z=8$. If $y=4$, then $x=2$, $z=1$, and $x+y+z=7$. The smallest of these is 7.

• 7.

  Matthew reads 3 more pages each day than the previous day. If he reads 6 pages on the first day, how many days will it take him to finish a 510-page book?

    (3 pts) 7. $17$

  On day $n$ he reads $3+3n$ pages, so after $N$ days he has read ${\sum }_{n=1}^{N}3+3n=3N+3N(N+1)/2$ pages. This equals 510 when $6N+3N(N+1)=1020$, or $0=N^2+3N-340=(N-17)(N+20)$, so that $N=17$.