Calculating devices are allowed, provided that they do not have any of the following features: (i) internet access, (ii) the ability to communicate with other devices, (iii) previously stored information such as formulas, programs, notes, etc., (iv) a computer algebra system, (v) dynamic geometry software.
Express answers as simplified exact numbers except where otherwise indicated. For example, \(\pi+1\) and \(1-\sqrt{2}\) are simplified exact numbers.
If \(m=4\), what is the value of \((m+1)(m+2)\)?
In the integer \(n=7 A6\,65B\,2A7\), the digit \(A\) appears twice and the digit \(B\) appears once. The sum of the digits of \(n\) is \(46\) and the sum of the rightmost two digits of \(n\) is \(11\). What is \(A+B\)?
A pot of soup is initially \(\dfrac{2}{3}\) full. After \(3\) L of soup is eaten, the pot is \(\dfrac{1}{2}\) full. What is the capacity of the pot?
What is the smallest positive multiple of \(6\) that can be expressed as the sum of five consecutive positive integers?
In the diagram shown, \(ACEG\) is a rectangle with area \(2420\).
Also, \(\dfrac{AB}{BC}=\dfrac{2}{3}\) and \(ABIH\) and \(DEFJ\) are congruent squares. What is the perimeter of \(ACEG\)?
Ana, Bertrand, and Chi share a collection of stamps.
Ana divides the stamps into piles of \(9\) and notices that there are \(0\) stamps left over.
Bertrand divides the stamps into piles of \(5\) and notices that there are \(2\) stamps left over.
Chi divides the stamps into piles of \(4\) and notices that there is \(1\) stamp left over.
What is the smallest number of stamps they could have in their collection?
Three odd integers between \(0\) and \(10\) are selected randomly, with each odd integer equally likely to be selected. (It is possible for an integer to be selected more than once.) What is the probability that the product of the squares of the three integers is equal to \(2025\)?
The real numbers \(x\), \(y\), and \(z\) simultaneously satisfy the three equations \[\frac{xy}{4x+y}=\frac{3}{10},\qquad\frac{yz}{y+z}=\frac{3}{10},\qquad\frac{xz}{4x+z}=\frac{1}{14}.\] What are all triples \((x,y,z)\) of real numbers that satisfy the three equations?
A flat, circular piece of paper with centre \(O\) and radius \(10\) has sector \(BOC\) with angle \(72\degree\) removed from it. Next, a circle centred at \(O\) of radius \(5\) is removed. (Some of this circle had already been removed when the sector was removed.) The remaining "C-shaped" piece of paper is shown below.
A hollow, 3-dimensional figure can be obtained by folding this piece of paper to connect line segment \(AB\) to line segment \(DC\), then adding circular faces on the top and bottom. This is shown in the diagram below. (This type of figure is called a frustum and is also the figure obtained by removing the top part of a cone by cutting it parallel to its base.)
What is the volume of the resulting hollow figure? Note: The volume of a cone with height \(h\) and a circular base of radius \(r\) is \(\frac{\pi}{3}r^2h\).
\(20\) different students are lined up in a row. \(8\) students wear a red hat and \(12\) students wear a green hat. There are \(20!\) ways to arrange the students. Call a pair of students "complementary" if they are next to each other and have different coloured hats. For example, an arrangement of the students resulting in the following order of hat colours \[GRRGGRRGGGRGGRGGGGRGGGGR\] has \(11\) complementary pairs. Among the \(20!\) ways to arrange the students, what is the average number of complementary pairs of students?
Calculating devices are not permitted.
Express answers as simplified exact numbers except where otherwise indicated. For example, \(\pi+1\) and \(1-\sqrt{2}\) are simplified exact numbers.
What is the integer equal to \(\big(\sqrt{25}+\sqrt{49}\,\big)^2\) ?
How many odd integers are there between \(\dfrac{13}{2}\) and \(\dfrac{37}{2}\)?
If \(3w=x\), \(2x=y\), \(5y=z\), and \(w\neq 0\), what is the value of \(\dfrac{z}{w}\)?
Define a new operation \(\lozenge\) by \(a\lozenge b = ab+a+b\). For example, \(3\lozenge 4=(3)(4)+3+4=12+7=19\). Given that \(5\lozenge x = 41\), what is \(x\)?
In the diagram below, two identical squares overlap to make an \(8\)-sided figure. The region where the squares overlap is shaded and equal to one third of the area of each square. What fraction of the area of the \(8\)-sided figure is shaded?
A car travelled \(360\) km in total. During the first \(180\) km, its average speed was \(45\) km/hr. During the second \(180\) km, its average speed was \(90\) km/hr. What was its average speed for the entire \(360\) km trip?
The letters \(A\), \(B\), \(C\), \(D\), and \(E\) represent the digits \(1\), \(2\), \(3\), \(4\), and \(5\) in some order. \(AB\) is a two-digit integer with tens digit \(A\) and units digit \(B\), \(CD\) is a two-digit integer, and \(BED\) is a three-digit integer. Given that \(AB\times CD = BED\), what are the digits \(A\), \(B\), \(C\), \(D\), and \(E\)?
The circle shown has centre \((0,0)\) and radius \(5\). Two vertical chords are drawn at \(x=-3\) and \(x=3\). Two horizontal chords are drawn at \(y=-3\) and \(y=3\). What is the sum of the lengths of the four chords?
A bag contains eight chips, each labelled with a different prime number greater than \(1\) and less than \(20\). Two different chips are selected from the bag at random and their labels are multiplied. What is the probability that the units digit of the product is \(9\)?
What is the smallest five-digit positive integer with the property that the sum of its digits is one less than the product of its digits?
At some time in the afternoon, Joni set three clocks to the correct time. Clock A is accurate, Clock F is fast, and Clock S is slow. After \(60\) minutes had elapsed on Clock A, Clock S showed that \(54\) minutes had elapsed and Clock F showed that \(64\) minutes had elapsed. At some point later that same afternoon, Clock S reads 7:20 p.m. and Clock F reads 8:10 p.m. At what time did Joni set the three clocks to the correct time?
In a sequence of seven numbers, each number after the first two is the sum of the two previous numbers. If the seventh (last) number in the sequence is \(97\) and the sum of the seven numbers in the sequence is \(245\), what is the third number in the sequence?
If \(f(x)=x^2+9x-n\) for some integer \(n\) and \(f(n)=-16\), what is the value of \(f(-2)\)?
Four dice each have the following three properties.
The number of dots on each face is an integer from \(1\) through \(6\), inclusive.
No two faces on the same die have the same number of dots on them.
Each pair of faces on opposite sides of the same die have a total of \(7\) dots on them.
The dice are arranged as shown below so that each pair of faces from different dice that are touching have a total of \(8\) dots on them. Five of the visible faces, including the one labelled by "\(X\)", have had all of their dots hidden. What is the number of dots on the face labelled by \(X\)?
For a positive integer \(n\), the Anderson number of \(n\) is the integer obtained be writing the integers from \(1\) through \(n\) in order. For example, the Anderson number of \(n=19\) is \[12345678910111213141516171819\] What is the smallest four-digit integer \(n\) for which the Anderson number of \(n\) is divisible by \(9\)?
Given a positive integer with at least two digits, a new integer can be obtained by removing its leading digit. For example, the leading digit can be removed from \(589\) to get the integer \(89\). What is the smallest positive integer \(n\) with the property that \(n\) is \(57\) times the integer obtained by removing its leading digit?
Some devices display digits using an arrangement of segments. For example, the digit "\(8\)" uses seven segments. How many integers from \(100\) through \(999\) inclusive can be displayed using exactly \(17\) segments?
In the diagram, \(OPQ\) is a quarter circle of radius \(1\). Square \(OABC\) is inscribed in \(OPQ\) so that \(A\) is on line segment \(OP\), \(B\) is on arc \(PQ\), and \(C\) is on line segment \(OQ\). Another circular arc is drawn from \(A\) to \(C\) with centre \(O\), and the region bound by this arc, \(AB\), and \(BC\) is shaded.
Inside the quarter circle \(OAC\), this process can be repeated to inscribe a square then draw a circular arc to obtain a smaller shaded region.
If this process is continued indefinitely, what is the total area of the infinitely many shaded regions?
The two angles \(x\) and \(y\), both between \(0\degree\) and \(90\degree\), satisfy both of the equations \[\begin{align*} \sin^2 x + \cos^2 y &= \frac{89}{144} \\ \cos^2 x - \sin^2 y &= \frac{71}{144}\end{align*}\] What is the exact value of \(\sin x + \sin y\)?
In the diagram, \(\triangle ABC\) is right-angled at \(B\). Point \(D\) is on \(BC\) so that \(BD=4\), \(CD=5\), and \(\cos\angle CAD=\dfrac{12}{13}\). What is the length of \(AB\)?
The positive real numbers \(x\) and \(y\) satisfy both of the equations \[\begin{align*} xy &= 144 \\ 3\log_y(x)+3\log_x(y) &= 10\end{align*}\] What is the value of \(x+y\)?
Consider the polynomials \(f(x)=x^2-2x-p\) and \(g(x)=x^3-5x^2-q\). There is an irrational number \(c\) such that \(f(c)=g(c)=0\). Given that \(p\) and \(q\) are rational, what are the three real numbers \(x\) that satisfy the equation \(x^3-5x^2-q=0\)?
In the diagram, \(ABCDEFGHIJKL\) is a regular 12-sided polygon (called a dodecagon). What is the area of \(\triangle ADH\) divided by the area of the dodecagon?
The regular pyramid \(EABCD\) has square base \(ABCD\) and a volume of \(28\). The points \(P\) and \(Q\) are on \(EC\) and \(ED\), respectively, such that \(\dfrac{EP}{EC}=\dfrac{1}{3}\) and \(\dfrac{EQ}{ED}=\dfrac{1}{2}\). Figure \(EABPQ\) has six faces, two of which are \(\triangle ABP\) and \(\triangle AQP\). What is the volume of \(EABPQ\)?
Willow’s job is to write computer programs and Ash’s job is to test them. When Willow finishes a computer program, she adds it to a list of programs that are ready to be tested. Ash always tests the program that was most recently added to the list, and once Ash starts testing a program, he always finishes testing that program before starting the next one. Once Ash is finished testing a program, he removes it from the list. On one particular day, Willow and Ash have 7 programs to write and test, respectively. Willow writes Programs 1 through 7 in order. By the time Ash arrives at work, Willow has already been working for some time. At one moment, Ash tells a colleague that he has already tested Program 6. After this moment, how many possibilities are there for the ordered sequence of programs that Ash will test?
Note: It is possible that all programs were already tested when Ash tells a colleague that he has already tested Program 6. All programs will be tested by the end of the day.
How many integers between \(1\) and \(25\) inclusive are divisible by \(3\) or \(5\) (or both)?
Let \(t\) be TNYWR.
The numbers \(x,t,y,26,\dots\) form an
arithmetic sequence. What is the value of \(x+y\)?
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant, called the common difference. For example, \(3, 5, 7, 9\) are the first four terms of an arithmetic sequence.)
Let \(t\) be TNYWR.
The line with equation \(y=3x+10\)
intersects the parabola with equation \(y=x^2-6x+t\) twice. What is the smallest
value of \(x\) where the curves
intersect?
How many two-digit positive integers are there with two different digits?
Let \(t\) be TNYWR.
The total cost of \(5\) ounces of
coffee and \(6\) ounces of tea is \(t\) cents. The total cost of \(11\) ounces of coffee and \(12\) ounces of tea is \(171\) cents. What is the cost of \(1\) ounce of tea?
Let \(t\) be TNYWR.
After the parabola with equation \(y=x^2+(m-t)x+2m-1\) is translated \(2\) units to the right, what is its new
\(y\)-intercept?
What is the smallest multiple of \(12\) that is greater than \(2025\)?
Let \(t\) be TNYWR.
A green box has a square base with side length \(10\) cm and a height of \(13\) cm. A black box has a volume of \(\dfrac{t}{4}\) cm3. If the
volume of the black box is equal to \(n\)% of the volume of the green box, what
is the value of \(n\)?
Let \(t\) be TNYWR.
In the diagram, \(O\) is the centre of
the circle, and three congruent, isosceles, triangles all have a vertex
at \(O\) and their other two vertices
on the circle.
What is the value of \(x\)?