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Problems

  1. The sum of the first 2005 terms of the sequence 1, 2, 3, 4, 1, 2, 3, 4, $\ldots$ is
    1. $5011$
    2. $5110$
    3. $5020$
    4. $5010$
    5. $501$
  1. Singers in a singing competition are rated by an applause metre. In the diagram, the arrow for one of the contestants is pointing to a rating that is closest to
    The image presents a simple line graph with four data points, each marked by an 'X' symbol. The x-axis is not labeled, but it appears to represent time or another continuous variable. The y-axis is also unlabeled, but it seems to measure some quantity that increases over time. Here are the key features of the image: * **Data Points:** * There are four data points marked on the graph. * Each point is represented by an 'X' symbol. * The x-values (time or continuous variable) for these points are not explicitly labeled but can be inferred from their positions on the x-axis. * **Y-Axis:** * The y-axis measures some quantity that increases over time. * It is unlabeled, making it difficult to determine what specific metric it represents without additional context. In summary, the image displays a line graph with four data points marked by 'X' symbols, indicating changes in an unspecified quantity over time.
    1. $9.4$
    2. $9.3$
    3. $9.7$
    4. $9.9$
    5. $9.5$
  1. If $\sqrt{5+n}=7$, the value of $n$ is
    1. $4$
    2. $9$
    3. $24$
    4. $44$
    5. $74$
  1. A piece of string fits exactly once around the perimeter of a square whose area is 144. Rounded to the nearest whole number, the area of the largest circle that can be formed from the piece of string is
    1. $144$
    2. $733$
    3. $113$
    4. $452$
    5. $183$
  1. Suppose that $k>0$ and that the line with equation $y = 3kx + 4k^2$ intersects the parabola with equation $y = x^2$ at points $P$ and $Q$, as shown. If $O$ is the origin and the area of $\triangle OPQ$ is 80, then the slope of the line is
    The image depicts a graph with two axes, labeled 'x' and 'y', which intersect at a point marked 'O'. The x-axis is horizontal, while the y-axis is vertical. A curved line, resembling an inverted parabola, begins at the origin (0, 0) and extends upwards to the right. It intersects another straight line that originates from the same point as the curve. The graph features several labels: * 'P' is located on the x-axis, to the left of the origin. * 'Q' is situated on the y-axis, above the origin. * An arrow pointing upwards and to the right is positioned at the top-right corner of the graph. The background of the image is white.
    1. $4$
    2. $3$
    3. $\frac{15}{4}$
    4. $6$
    5. $\frac{21}{4}$
  1. There are six identical red balls and three identical green balls in a pail. Four of these balls are selected at random and then these four balls are arranged in a line in some order. How many different-looking arrangements are possible?
    1. 15
    2. 16
    3. 10
    4. 11
    5. 12
  1. Shuxin begins with \(10\) red candies, \(7\) yellow candies, and \(3\) blue candies. After eating some of the candies, there are equal numbers of red, yellow, and blue candies remaining. What is the smallest possible number of candies that Shuxin ate?
    1. \(17\)
    2. \(7\)
    3. \(11\)
    4. \(20\)
    5. \(14\)
  1. A circular spinner is divided into three sections. An arrow is attached to the centre of the spinner. The arrow is spun once. The probability that the arrow stops on the largest section is \(50\%\). The probability it stops on the next largest section is 1 in 3. The probability it stops on the smallest section is
    1. \(\frac14\)
    2. \(\frac25\)
    3. \(\frac16\)
    4. \(\frac27\)
    5. \(\frac{3}{10}\)
  1. There are \(T\) tokens arranged in a circle for some positive integer \(T\). Moving clockwise around the circle, the tokens are labelled, in order, with the integers from 1 to \(T\). Starting from the token labelled 1, Évariste:

    1. Removes the token at the current position.

    2. Moves clockwise to the next remaining token.

    3. Moves clockwise again to the next remaining token.

    4. Repeats steps (i) to (iii) until only one token remains.

    When \(T=337\), the number on the last remaining token is \(L\). There are other integers \(T\) for which the number on the last remaining token is also \(L\). What are the rightmost two digits of the smallest possible value of \(T\)?

  1. In the diagram, each letter from \(A\) to \(H\) is equal to a different integer from \(1\) to \(8\).

    A grid of eight squares arranged into two columns of four squares. From top to bottom, the squares in the first column have the letters A, B, C, and D, and the squares in the second column have the letters E, F, G and H.

    Also,

    • \(H\) is a perfect square and is \(1\) more than \(D\)

    • \(5\) and \(8\) are in the same row

    • \(C\) is a multiple of both \(G\) and \(D\)

    • \(B\) is the largest prime number in the set

    • The value of \(B+G\) is even

    What is the value of \(F\)?
    1. \(2\)
    2. \(6\)
    3. \(1\)
    4. \(7\)
    5. \(8\)