IMAS 2023–2024 Upper Primary — First Round

What is the sum of all the positive divisors of 2023?

Alice buys 9 souvenirs, each at a price of 1, 3, 5 or 25 dollars. She pays the cashier with several 10-dollar bills. Which of the following is a change that she is impossible to get?

In a certain school, there are less than 1000 students. The principal wants to align all of the students in the schoolyard for a flag ceremony. If the students are arranged in rows, 31 students per row, then there will be 13 students remaining. If the students are arranged 43 per row, then there will be 40 students remaining. How many students are exactly there in the school?

Two candles have different lengths and thickness, but each candle burns at a steady rate. The short candle and the long candle will burn out completely in 280 minutes and 210 minutes, respectively. Both candles were lit at the same time and their lengths were equal after 2 hours. If the initial length of the short candle is 8.4 cm, then what is the initial length, in cm, of the long candle?

Four of the angles of a convex polygon are equal to $120^\circ$ each. What is the minimum possible interior angle, in degrees, of all such polygons?

An urn contains $n$ balls, which are numbered from 1 to $n$ ($n\ge3$). After a ball is drawn from this urn, the average of the numbers written on the remaining balls in the urn is 31.4. What number was written on the drawn ball?

The lower row of shapes, which are all hidden, is a different sequence than the top row. The hidden row does, however, satisfy the following conditions:
- Neither the cross nor the circle is adjacent to the hexagon.
- Neither the cross nor the circle is adjacent to the triangle.
- Neither the circle nor the hexagon is adjacent to the square.
- The triangle is just to the right of the square.
What shape lies in the 4th position from left to right in the lower row?

The diagram below shows a stack of 12 equal cubes of side-length $a$ m each. In each cube, the length of a diagonal of a face is $b$ m. An ant is travelling on the surface of these cubes from point $A$ to point $B$. What is the length, in m, of the shortest path the ant can take?

The product of two positive integers, $m$ and $n$, is 48. Let $x$ be the number of distinct values of $m+n$ and $y$ be the smallest value for $m+n$. What is the value of $x+y$?

In the diagram below, parallelogram $ABCD$ is divided into some parts where only the shaded regions have known areas, in cm$^2$. What is the area, in cm$^2$, of the black triangle?

The average of fifteen distinct positive integers is 23. What is the maximum possible value for the second largest of these integers?

As shown in the diagram below, all three rectangles have integer lengths and widths, in cm. If the area of the largest rectangle is 187 cm$^2$, then what is the area, in cm$^2$, of the second largest rectangle?

The number 8000027 has exactly three prime factors. What is the sum of these three prime factors?

What is the last digit of the sum $1\times2+2\times3+3\times4+\cdots+2022\times2023$?

There are exactly 500 dollars in my bank account. Each time, I may either withdraw 300 dollars from my account, or deposit 198 dollars into my account from money previously withdrawn. What is the minimum amount of money that can be left in my bank account?

Let $S(n)$ be the digit sum of a positive integer. What is the sum of all possible integers $n$ for which $n+S(n)=2023$?

A $16\text{ cm}\times25\text{ cm}$ rectangle piece of paper is folded so that a corner touches the opposite side as shown in the diagram below. If $AB=5$ cm, then what is the length, in cm, of $CD$?

There are four positive numbers $a,b,c,d$ so that $a\lt b\lt c\lt d$. Which of the following inequalities is INCORRECT?

The diagram below on the left shows the closed window of an old bus. All curves forming the round corners are arcs of a quarter circle with radius 13 cm. The two sliding windows on top have equal size and their height is 15 cm. To open the window, the upper parts of the bus window are slid left or right, as shown in the diagram below on the right. If the left window is slid 10 cm to the right, and the right window is slid 8 cm to the left, what is the total area, in cm$^2$, of the opening (the total shaded area)? (Take $\pi=3.14$.)

A basket contains some number of balls, each of them being painted with at least one of the two colours: green and red. It was observed that $\frac35$ of the balls that are painted with green are also painted with red, and $\frac27$ of the balls that are painted with red are also painted with green. What percent of the balls in that basket are painted in both colours?