Online Selection Test

Online Selection Test

English translation by SOTA – AI Community of the Indonesian original, the problem set Seleksi Ekshibisi OSN AI 2025 on TLX (TOKI Learning Center), the online selection round of the OSN 2025 AI Exhibition. The set has 40 questions worth 1,000 points in total; answers are entered on TLX (log-in required). Answers that must be typed as Indonesian words stay in Indonesian, with English glosses. The display maths uses $$ in place of the original \[ \]. Organisers who would like this translation removed can email [email protected].

Seleksi Ekshibisi OSN AI 2025 (OSN 2025 AI Exhibition Selection)

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Problem Solving (Questions 1–2)

[Total 120 points]

Question 1

Multiple choice. [40 points]

A rational number is a number that can be expressed as the quotient of two integers $\tfrac{p}{q}$ with $q \neq 0$. When written in decimal form, a rational number always has a decimal representation that either terminates (terminating) or repeats (repeating). Examples of rational numbers: $-8$, $42.314$ and $42$.

Nesy and Nyoo are playing with a number line. On an inclusive interval $[0,n]$, with an integer $n \geq 4$, they will play in turns. Nesy starts first: she chooses a rational number in the range $[0,n]$. The next player may only choose another rational number whose distance from all the numbers already chosen by either player is greater than 1. The game continues until a player who cannot choose a rational number is declared the loser.

Example: If $n=8$ and Nesy chooses 6 as her first number, then Nyoo may not choose a rational number in the range $[5,7]$. Suppose Nyoo chooses the number 1; as a result, Nesy may not choose a number in the ranges $[0,2]$ and $[5,7]$.

If both players play optimally, which statement is the most correct?

• (a) Nyoo always wins.
• (b) Nesy always wins.
• (c) Nesy wins if and only if $n$ is an odd number.
• (d) Nyoo wins if and only if $n$ is an odd number.
• (e) Nyoo wins if and only if $n \ge 8$.

Question 2

Short answer. [80 points]

You are given a $7 \times 7$ grid board filled with the letters I, O and A. You really like the word IOAI, and you are curious how many IOAI words can be formed on this board.

I	I	I	I	I	I	I
I	O	A	O	O	A	I
I	O	A	A	A	A	I
I	O	O	I	A	O	I
I	O	O	O	O	O	I
I	O	A	A	A	O	I
I	I	I	I	I	I	I

To answer this question, we need to define what is meant by a “4-letter” path on the board:

1. Start from some square.
2. Move to one of the neighbouring squares in any of the 8 directions (up, down, left, right, or diagonally).
3. Move again to one of the 8 neighbours.
4. Move once more to one of the 8 neighbours, and then stop.

So a path produces exactly 4 letters, from the starting square to the final square. Note that a square MAY be visited more than once. As an illustration, an example of forming the word IOAI is marked in red.

Write the answer as a number.

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Mathematical Logic (Questions 3–4)

[Total 120 points]

A research team sampled the body weights of $N$ people from a population of university students. The sample variance of the body weights of these $N$ people is $X$. The next day, the results of further study showed that they needed to take an additional sample of $N$ more people in a second sampling batch. By coincidence, the body-weight samples of all $N$ people in the second batch are exactly the same as the body weights in the first batch. The combined variance of all the new samples (the body weights of $2N$ people) is $Y$.

Question 3

Multiple choice. [40 points]

Which statement is true?

• (a) $X < Y$
• (b) $X = Y$
• (c) $X > Y$
• (d) It cannot be determined whether $X < Y$ or $X > Y$
• (e) $X = 2Y$
• (f) $Y = 2X$

Question 4

Short answer. [80 points]

Suppose the researchers take another data sample, in a third batch, of $M$ people, whose mean is half the mean of the data already collected in the first and second batches ($2N$ people). When combined, i.e. $2N+M$ people, the new mean becomes $\tfrac{3}{4}$ of the mean of the combined data of the first and second batches.

What is the value of $M$?

Express $M$ in terms of $N$ using the following rules:

• If it involves a constant multiplied by the variable $N$, write the answer like $8N$, $9N$, $13N$, etc.
• If it involves addition or subtraction, write the answer like $7N+3$, $4N-5$, $4N+1$, etc.
• The answer must not contain any white space.

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Probability (Questions 5–7)

[Total 145 points]

In a cinema there are 3 kinds of film: action films (🔫), family films (👨‍👩‍👧‍👦) and horror films (👻). Based on visitor data, the probability of watching each kind of film is:

• P(🔫) = 40%
• P(👨‍👩‍👧‍👦) = 35%
• P(👻) = 25%

The cinema also offers 2 kinds of snack: popcorn (🍿) and soft drink (🥤). The snack-purchase data by the kind of film watched are as follows:

Kind of Film	Buys No Snack	Buys Popcorn Only	Buys Soft Drink Only	Buys Popcorn and Soft Drink
Action Film 🔫	30%	20%	10%	40%
Family Film 👨‍👩‍👧‍👦	45%	20%	15%	20%
Horror Film 👻	50%	10%	10%	30%

Answer the following questions.

Write the answer as an integer, without the percent sign. If the probability is a decimal percentage, round it down. For example, 64.71% becomes 64.

Question 5

Short answer. [40 points]

The probability that a visitor buys popcorn. P(🍿)

Question 6

Short answer. [40 points]

The probability that a visitor watches an action film and also buys popcorn or a soft drink. P(🔫 ∧ (🍿 ∨ 🥤))

Question 7

Short answer. [65 points]

Given that a visitor buys only a soft drink, what is the probability that they are watching a horror film? P(👻|🥤)

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Linear Model (Questions 8–9)

[Total 100 points]

A researcher wants to build a model that predicts a senior high school (SMA) student's UTBK score (the Indonesian computer-based university entrance test) from their Mathematics report-card mark (ranging from 0 to 100).

The model used is the linear equation:

$$f(M;\alpha,\beta)=\alpha+\beta\cdot M$$

where:

• $M$ is the Mathematics report-card mark ($0 \le M \le 100$),
• $\alpha,\beta$ are the parameters to be found (optimised).

The student data given are as follows:

Student $i$	Mathematics Mark ($M$)	UTBK Score ($y$)
1	95	785
2	86	790
3	68	600

The criterion used to find $\alpha$ and $\beta$ is to minimise the value of the following loss function $L(\alpha,\beta)$:

$$L(\alpha,\beta) \;=\; \frac{1}{n}\sum_{i=1}^n \big(y_i - (\alpha + \beta M_i)\big)^2,$$

with $n=3$ in the context of this problem.

Question 8

Short answer. [20 points]

Calculate the value of the function $L(\alpha,\beta)$ when $\alpha = 227$ and $\beta = 6.0$.

Write your answer to a precision of 1 digit after the decimal mark (a point).

For example:

• $19.35 \to 19.4$
• $72.71 \to 72.7$
• $0.68 \to 0.7$

Question 9

Short answer. [80 points]

Find the values of $\alpha$ and $\beta$ that minimise the value of the function $L(\alpha,\beta)$.

Write your answer in the format:

$$\text{X.Y; A.B}$$

where $\texttt{X.Y}$ is the value of $\alpha$ to a precision of 1 digit after the decimal mark, and $\texttt{A.B}$ is the value of $\beta$ to a precision of 1 digit after the decimal mark.

For example:

• 19.35→19.4
• 72.71→72.7
• 0.68→0.7

Important note: First find the values of $\alpha$ and $\beta$ that make $L(\alpha,\beta)$ as small as possible, and only then round them as the question requires. Once rounded, the solution is not necessarily the most optimal one at that precision any more.

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Vector Similarity (Questions 10–21)

[Total 155 points]

In modern Natural Language Processing (NLP), a token or word is usually represented as a vector in a high-dimensional space. For example, the vector representations of some words are given below (the words are Indonesian: senang = happy, pria = man, wanita = woman, ayah = father, ibu = mother, raja = king, ratu = queen, sedih = sad, menang = win, kalah = lose):

$$\begin{aligned} \text{senang} &= (21, 10) \\ \text{pria} &= (6, 29) \\ \text{wanita} &= (30, 24) \\ \text{ayah} &= (11, 26) \\ \text{ibu} &= (36, 19) \\ \text{raja} &= (8, 36) \\ \text{ratu} &= (32, 29) \\ \text{sedih} &= (18, 19) \\ \text{menang} &= (26, 8) \\ \text{kalah} &= (23, 16) \end{aligned}$$

These vectors can be manipulated with addition and subtraction.

A well-known example:

$$\text{raja} + \text{wanita} - \text{pria} \approx \text{ratu}, \qquad \text{ayah} + \text{wanita} - \text{pria} \approx \text{ibu}.$$

This means that there is a translation vector $(\text{wanita} - \text{pria})$ that can turn a masculine word into its feminine version.

Question 10

Short answer. [10 points]

What is the Euclidean distance between the vectors $\texttt{raja}$ and $\texttt{kalah}$?

The formula for the Euclidean distance between two vectors $(x_1,y_1)$ and $(x_2,y_2)$:

$$d\big((x_1,y_1),(x_2,y_2)\big) \;=\; \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}.$$

Write the answer as a number.

Question 11

Short answer. [15 points]

What is the vector value of $\text{senang} + \text{kalah} - \text{menang}$?

Write the answer in the format $\texttt{A B}$, where the resulting vector is $(A,B)$.

Questions 12–21 (matching vectors)

Aji, a researcher of multi-galaxy languages, has discovered an ancient alien language with the following 10 words.

$$\begin{aligned} \text{Gohok} &= (34,37) \\ \text{Subi} &= (28,29) \\ \text{Matot} &= (14,35) \\ \text{Befar} &= (38,31) \\ \text{Dursit} &= (38,40) \\ \text{Gien} &= (29,19) \\ \text{Ibakuz} &= (24,20) \\ \text{Gelon} &= (11,46) \\ \text{Wazuke} &= (10,42) \\ \text{Nusgen} &= (22,30) \end{aligned}$$

It is known that these 10 words are exact translations of the 10 Indonesian words above (senang, pria, wanita, …, kalah). However, Aji does not know which word matches which.

Find the word pairs between the alien language and Indonesian, using the fact that the vector structure between words in the alien language is the same as in Indonesian.

Answer format: all in lower case. (Each answer is one of the Indonesian words senang, pria, wanita, ayah, ibu, raja, ratu, sedih, menang, kalah.)

[13 points each]

For each alien word below, give the matching Indonesian word (short answer):

• Question 12: Gohok
• Question 13: Subi
• Question 14: Matot
• Question 15: Befar
• Question 16: Dursit
• Question 17: Gien
• Question 18: Ibakuz
• Question 19: Gelon
• Question 20: Wazuke
• Question 21: Nusgen

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Convolution (Questions 22–25)

[Total 165 points]

A digital image $I$ can be represented as a matrix of size $x \times y$, where each matrix element $I_{i,j}$ gives the pixel value in row $i$ and column $j$.

A kernel is a small matrix (for example $3 \times 3$ or $5 \times 5$) that is slid over the image. The kernel $K$ is a small matrix of size $m \times n$ that serves to extract certain patterns from the image $I$, where the kernel $K$ is usually smaller than the image $I$, i.e. $x \geq m$ and $y \geq n$.

Convolution is the dot product operation (i.e. element-wise multiplication of the two matrices followed by summation) between the kernel $K$ and a piece of the image $I$, namely a patch (sub-matrix) $J$, where the size of the patch $J$ is the same as that of the kernel $K$. If there are two vectors of the same length:

$$a = [a_1, a_2, \dots, a_n], \quad b = [b_1, b_2, \dots, b_n]$$

then the dot product is defined as:

$$a \cdot b = \sum_{i=1}^n a_i b_i = a_1 b_1 + a_2 b_2 + \dots + a_n b_n$$

The convolution process itself can be defined as follows. Convolution over the whole image $I$ using the kernel $K$ can be written as $(K * I)$, and it produces an output image $S$.

From the figure it can be seen that the dot product takes place between the kernel $K$ and the patch $J$: each element of $K$ is multiplied by the corresponding element of $J$, and the results are then summed.

The following is an example of an input image of size $(3 \times 4)$ and a kernel of size $(2 \times 2)$, which produce an output image of size $(2 \times 3)$:

[Figure: see the original statement. (Its image file, conv3.png, returned HTTP 404 on 17 September 2026.)]

The Relation of Convolution to Stride and Padding

The convolution process is closely tied to stride and padding.

Stride: The stride is how far the kernel moves at each step. For example:

• If Stride = 1, the kernel moves 1 pixel.
• If Stride = 2, the kernel moves 2 pixels.
• If Stride = n, the kernel moves n pixels.

The following is an example of Stride = 1 and Stride = 2 with an input image of size ($5 \times 5$) and a kernel of size ($3 \times 3$):

[Figure: see the original statement. (Its image file, conv2.png, returned HTTP 404 on 17 September 2026.)]

Padding: Padding is the addition of layers of pixels at the edges of the image to preserve the output size. For example:

• If padding = $0$, no pixels are added, so the output image is smaller than the input image.
• If padding = "same", the output image is the same size as the input image (assuming Stride = $1$).
• If padding = $n$, $n$ pixels are added on each side of the image (left, right, top, bottom), so the size increases by $2n$ pixels in total. So the total increase in image size is $2n$ pixels (because of left + right and top + bottom).

The following is an example of Padding = “same” and Stride = 1 with an input image of size ($5 \times 5$) and a kernel of size ($3 \times 3$):

[Figure: see the original statement. (Its image file, conv1.png, returned HTTP 404 on 17 September 2026.)]

You are asked to predict the final size of the image after it passes through several convolution layers with different kernel sizes, strides and paddings.

The four questions below have the answer format A x B.

Question 22

Short answer. [20 points]

The initial image $I$ has size $32 \times 32$.

• Convolution 1: Kernel $3 \times 3$, padding = "same", stride = $1$.

What is the size of the final image $S$?

Question 23

Short answer. [30 points]

The initial image $I$ has size $32 \times 32$.

• Convolution 1: Kernel $3 \times 3$, padding = "same", stride = $1$.
• Convolution 2: Kernel $5 \times 5$, padding = $0$, stride = $2$.

What is the size of the final image $S$?

Question 24

Short answer. [40 points]

The initial image $I$ has size $64 \times 64$.

• Convolution 1: Kernel $7 \times 7$, padding = $3$, stride = $2$.
• Convolution 2: Kernel $3 \times 3$, padding = "same", stride = $2$.
• Convolution 3: Kernel $1 \times 1$, padding = $0$, stride = $1$.

What is the size of the final image $S$?

Question 25

Short answer. [75 points]

The initial image $I$ has size $128 \times 64$.

• Convolution 1: Kernel $7 \times 3$, padding = $(3,1)$, stride = $(2,1)$.
• Convolution 2: Kernel $5 \times 5$, padding = $2$, stride = $2$.
• Convolution 3: Kernel $3 \times 3$, padding = "same", stride = $1$.
• Convolution 4: Kernel $1 \times 7$, padding = $(0,3)$, stride = $(1,2)$.
• Convolution 5: Kernel $3 \times 3$, padding = $0$, stride = $2$.

What is the size of the final image $S$?

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Language Modelling (Questions 26–40)

[Total 195 points]

A DNA sequence is made up of a series of nucleotides consisting of four nitrogenous bases: adenine (A), thymine (T), guanine (G) and cytosine (C). DNA samples from 3 different species are given as follows (Kucing = cat, Bebek = duck, Ikan = fish):

DNA Sequence	Species
GATTATAAAGCA	Kucing
AATCTAATTATG	Kucing
ACAAAAAATTTC	Kucing
CATAAAGAAATA	Kucing
ACGACGCGATCG	Bebek
CGCGAGAGAGCG	Bebek
GCTGCGTGTCAG	Bebek
CGAGCTAGCAGC	Bebek
CCAGGGGAGCCC	Ikan
TTAGGTCAAGGC	Ikan
CGGTCCCCAAAG	Ikan
AGCCGGGGCAAG	Ikan

You are asked to build a system that can predict the species from a given DNA sequence, using the available training data.

However, because you are on holiday, you do not have time to build an artificial intelligence system from scratch.

You are also convinced that the patterns in which the nitrogenous bases occur in the DNA strands of each species are different enough that they can actually be classified even without the help of a machine or a computer.

For the following DNA strands, determine their species (Kucing, Bebek or Ikan). Answer format: ONLY Kucing, Bebek or Ikan

[13 points each]

For each DNA strand below, give the species (short answer):

• Question 26: CCGGCTCGTAAG
• Question 27: CATGAAAATATA
• Question 28: TAATAACACATA
• Question 29: TCGCTCGCACGT
• Question 30: GCACGCGCAGAT
• Question 31: TTTCCCGCTGGG
• Question 32: CGGGCTTCCCGG
• Question 33: GTCACTATCACG
• Question 34: CGAAATTTATTA
• Question 35: GCGCTATATGCG
• Question 36: GGCAGGTAGCCC
• Question 37: ACAAATAAACCA
• Question 38: ATGAGGGCCCCG
• Question 39: AAAAACTATTAT
• Question 40: CGCTCTAGCGCA