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Stage 6 · Mixed

10 problems · National Olympiad, first round · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

1. Given the triangle $ABC$ right-angled at $A$, we construct on the hypotenuse the square $BCDE$ (with $D, E$ on the opposite side of $A$ with respect to $BC$). Knowing that the areas of triangles $ABE$ and $ACD$ are respectively $6~\mathrm{m}^2$ and $27~\mathrm{m}^2$, what is the area of triangle $ABC$?

   1. A$3 \sqrt{2}~m^2$
   2. B$6~\mathrm{m}^2$
   3. C$12~\mathrm{m}^2$
   4. D$9 \sqrt{2}~\mathrm{m}^2$
   5. E$18~\mathrm{m}^2$

   Progetto Olimpiadi della Matematica

   Geometry Solution and answer checking →

2. A rectangle with side lengths $1$ and $3$, a square with side length $1$, and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
   

   1. A14
   2. B23
   3. C46
   4. D59
   5. E67

   Fall AMC 10 B 2021

   Geometry Solution and answer checking →

3. Let $P$ denote the set of all subsets of $\{1, \ldots, 23\}$. A subset $S \subseteq P$ is called good if whenever $A, B$ are sets in $S$, the set $(A \backslash B) \cup (B \backslash A)$ is also in $S$. (Here, $A \backslash B$ denotes the set of all elements in $A$ that are not in $B$, and $B \backslash A$ denotes the set of all elements in $B$ that are not in $A$.) What fraction of the good subsets of $P$ have between 2015 and 3015 elements, inclusive?
   If your answer is a decimal number or a fraction (of the form $m / n$, where $m$ and $n$ are positive integers), then your score on this problem will be equal to $\max \{0,25-\lfloor 1000|A-N|\rfloor\}$, where $N$ is your answer and $A$ is the actual answer. Otherwise, your score will be zero.

   HMMT February

   Algebra Solution and answer checking →

4. Let $T_n$ denotes the least natural such that
   $$n\mid 1+2+3+\cdots +T_n=\sum_{i=1}^{T_n} i$$
   Find all naturals $m$ such that $m\ge T_m$.

   Iberoamerican Olympiad

   Number theory Solution and answer checking →

5. Let $n$ be square with 4 digits, such that all its digits are less than 6. If we add 1 to each digit the resulting number is another square. Find $n$

   Cono Sur Olympiad

   Number theory Solution and answer checking →

6. Consider the number $N=1000\ldots 0001$ consisting of the digit one followed by 2023 zeros, which are in turn followed by the digit one. How many proper divisors does $N$ have (that is, divisors strictly between 1 and $N$) that are also written as a digit 1 followed by some positive number of zeros, followed in turn by a digit 1?

   1. A3
   2. B5
   3. C12
   4. D14
   5. E16

   Italian Mathematical Olympiad

   Number theory Solution and answer checking →

7. There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds.

   China National Olympiad

   Combinatorics Solution and answer checking →

8. In a game of Riemannian ball two teams face each other; at every moment, each team fields $k>1$ players. At the end of each play a point is awarded to one of the two teams; moreover, each team may make an arbitrary number of substitutions before the next play begins. Alice and Barbara are part of the Geodesics team. At the end of today's match, Alice observes that, while she was on the field, the Geodesics won 7 plays more than they lost. When Barbara was on the field, on the other hand, they lost 2 plays more than they won. Each of the other players of the Geodesics took part in exactly as many winning plays as losing plays. What is the value of $k$?

   1. A3
   2. B5
   3. C6
   4. D9
   5. EThe given data are not sufficient to determine it

   Gara di Febbraio

   Combinatorics Solution and answer checking →

9. Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist?

   Pan African MO

   Combinatorics Solution and answer checking →

10. A $k \times k$ array contains each of the numbers $1, 2, \dots, m$ exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of $m$ if $k = 3^n$ ($n \in \mathbb{N}^+$)?

    KoMaL

    Combinatorics Solution and answer checking →

Answer key — Stage 6 · Mixed

Worked solutions for every problem are on the site, one page per problem.

1. $D$ open
2. $E$ open
3. $18839183877670041942218307147122500601235/47691684840486192422095701784512492731212$ open
4. ${\text{all numbers that are not powers of 2 greater than 1.}}$ open
5. $2025$ open
6. $A$ open
7. $9$ open
8. $B$ open
9. $2^{n-1}$ open
10. $3^{n+1} - 1$ open

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