UND MATHEMATICS TRACK MEET Individual Test 2

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

• 1.

  Let $f$ be a function satisfying

  $$f(f(n))+f(n)=2n+3,$$

  and $f(n)$ is a natural number for all natural numbers $n$. Find $f(2026)$.

  (a) $2025$    (b) $2026$    (c) $2027$    (d) $2028$    (e) $2029$

    (2 pts) 1.

• 2.

  For how many integers $n\ge 0$ is $n^2+6n+5$ a perfect square?

  (a) $0$    (b) $1$    (c) $2$    (d) $3$    (e) infinitely many

    (3 pts) 2.

• 3.

  Let $P(x)$ be a nonzero polynomial satisfying

  $$(x^3+3x^2+3x+2)P(x-1)=(x^3-3x^2+3x-2)P(x)$$

  for all real numbers $x.$ Which is the degree of $P(x)\mathrm{?}$

  (a) $2$    (b) $3$    (c) $4$    (d) $5$    (e) $6$

    (3 pts) 3.

• 4.

  Two triangles have the same perimeter. The first has side ratios $3:4:5$ and the second has ratios $7:24:25$. What is the ratio of their areas?

  (a) $5:7$   (b) $7:5$    (c) $14:9$    (d) $35:32$    (e) $5:4$

    (3 pts) 4.

• 5.

  Triangle $ABC$ has area $1$. A point $M$ moves along side $BC$. Through $M$, draw a line parallel to $AC$ meeting $AB$ at $D$, and a line parallel to $AB$ meeting $AC$ at $E$. The quadrilateral $ADME$ is a parallelogram. Find the maximum possible area of parallelogram $ADME$.

  (a) $1$     (b) $2$     (c) $1/2$     (d) $3/2$     (e) $3$

    (3 pts) 5.

• 6.

  A teacher has 300 identical books and wants to pack them into boxes with a different number of books in each box. What is the greatest possible number of boxes?

  (a) $23$    (b) $24$    (c) $25$    (d) $26$    (e) none of these

    (3 pts) 6.

• 7.

  How many pairs of integers $(x,y)$ satisfy the equation ${\displaystyle \frac{x^2+y^2}{x+y}}={\displaystyle \frac{85}{13}}$?

  (a) $1$    (b) $2$    (c) $3$    (d) $4$    (e) no integer is satisfied

    (3 pts) 7.

TOTAL POINTS