Prediksi Soal OSN-K Matematika SMP 2026🔥

OSK Discussion and Problem Solving

Introduction to OSK Problems

• The session begins with an overview of the discussion topic: the OSK (Olimpiade Sains Nasional) problems from early 2025.

• Participants are encouraged to attempt the first two problems independently for about 2-3 minutes before discussing solutions.

Problem 1 Breakdown

• The facilitator starts explaining Problem 1, emphasizing the need to calculate a specific value 'a' using given mathematical expressions.

• Detailed calculations are presented, leading to the conclusion that 'a' equals -3. The final expression simplifies down to -4/1, confirming this result.

• Participants are invited to ask questions regarding Problem 1, ensuring clarity before moving on.

Transition to Problem 2

• With no further issues on Problem 1, attention shifts to Problem 2. Key information is highlighted: there are 35 numbers with a maximum of 29 and a median of 22.

Finding Averages in Problem 2

• The goal is to determine both the smallest and largest possible averages based on provided data constraints.

• A systematic approach is suggested: sorting data from smallest to largest helps in understanding how values can be manipulated for average calculations.

Smallest Average Calculation

• To find the smallest average, all values below the median must be minimized while adhering to positivity constraints.

• It’s established that values from positions one through seventeen should be set at their minimum (1), while those above must meet or exceed the median (22).

Largest Average Calculation

• For calculating the largest average, similar logic applies but focuses on maximizing lower values up until they reach limits defined by higher numbers (22 and then maxing out at 29).

Final Results for Problem 2

• After performing necessary calculations, it’s concluded that:

• Smallest average = 12

• Largest average = 25.4

• Combined total yields 37.4 as a final answer.

Questions and Clarifications

• Participants are prompted again for any questions regarding Problem 2's solution process before proceeding further.

Moving On To Problem 3

• As participants prepare for Problem 3, they’re advised that this problem may be more challenging than previous ones. They have a couple of minutes allocated for initial attempts while reviewing prior solutions.

Visual Representation in Geometry Problems

• Emphasis is placed on noting important points within geometric contexts as participants begin tackling visual aspects related to shapes involved in this problem.

Understanding Geometry Concepts in Triangles and Parallels

Key Geometric Relationships

• The discussion begins with the identification of points E, D, F, and H in a geometric configuration where line FK is parallel to line DE. This implies that angle relationships can be inferred.

• Point G is established within triangle ECD, indicating that line GH is perpendicular to GF, reinforcing the concept of right angles in triangles.

• The speaker emphasizes that certain information (like DE = 3x - 7) may not need deep focus as it is visually represented in the diagram provided.

Calculating Perimeter and Area

• The relationship between sides AB and CD is highlighted due to their parallelogram nature; thus, AB equals CD and BC equals AD. This leads to a formula for calculating the perimeter: 2AB + 2BC.

• Further simplification reveals 106 = 16x - 38, leading to the solution x = 9.

• With x determined, lengths of segments AB and BC are calculated as follows:

• AB = 3(9) + 1 = 28

• BC = 5(9) - 20 = 25

Finding Areas of Triangles

• The next step involves determining the area of region DEC minus area GFH. The speaker highlights this area calculation as crucial.

• It’s noted that since GF is parallel to DE, triangles DEC and GFH share similar properties which can be used for further calculations.

• Using known dimensions (base = 28, height = 20), the area of triangle DEC can be computed as 1/2 times base times height = 1/2 times 28 times 20 = 280.

Utilizing Pythagorean Theorem

• To find EC using Pythagorean theorem:

• EC² = DE² + CD² results in EC being calculated as approximately 4√74.

• Acknowledging similarity between triangles FGH and EDC allows for further deductions about side lengths based on angle congruence.

Final Area Calculation Steps

• By establishing ratios from similar triangles (FK/EF = DE/EC), values are substituted leading to finding EF's length.

• HC is derived from EF yielding HC equal to 2EF.

• Finally, substituting back into earlier equations provides an area result for DEC after subtracting FH/EC contributions leading towards a final answer of 262.5.

Conclusion & Questions

• After lengthy calculations resulting in an answer labeled "C," participants are invited to ask questions or clarify any confusion regarding the process discussed.

• (1673)s Participants are encouraged to engage with follow-up questions or move on if they feel confident with the material presented.

Discussion on Problem 4

Overview of the Problem

• The discussion begins with a comparison to a previous problem, indicating that this one is simpler. The focus is on understanding how to manage pocket money among three siblings: Ana, Bona, and Cinta.

• Ana receives an amount of 5,000 as part of their allowance. The speaker suggests that participants read through the problem independently for better comprehension.

Variables and Equations

• The total allowance for the siblings is denoted as A (for Ana), B (for Bona), and C (for Cinta). This sets up a framework for solving the problem mathematically.

• An equation is established where A equals 5x + additional terms representing B and C's allowances in relation to x, y, and z variables respectively. This helps simplify calculations later on.

Total Allowance Calculation

• It’s noted that the total allowance sums up to 700,000 which leads to another equation involving A, B, and C combined: A + B + C = 700,000. This reinforces the need for accurate calculations based on given data.

• Further breakdown shows how each sibling's allowance can be expressed in terms of x, y, and z leading to more complex equations that need verification against initial conditions provided in the problem statement.

Verification Process

• The speaker emphasizes checking answers by verifying values for B and C against known quantities from earlier discussions about allowances; this ensures accuracy before concluding any results.

• There’s uncertainty regarding whether certain assumptions about maximum values are valid due to lack of information about x, y, z which indicates critical thinking in mathematical reasoning is necessary here.

Conclusion on Problem Solving

• Ultimately it’s concluded that only option A can be definitively confirmed as correct based on available information while other options remain uncertain without further data validation or context from the original question prompt. Questions are invited from participants regarding this analysis before moving forward with additional problems or exercises related to similar concepts discussed earlier in class sessions.

Transitioning to Next Problems

Moving Forward with Exercises

• After addressing questions related to Problem 4, participants are encouraged to attempt subsequent problems while waiting for further instructions or clarifications if needed during their practice time allotted until a specified duration has passed (751 seconds).

Techniques for Solving Grid Problems

Introduction of Techniques

• As new problems arise particularly those involving grid patterns or paths taken within them are introduced; specific techniques used by the instructor are shared emphasizing systematic approaches towards finding solutions efficiently rather than trial-and-error methods alone which may lead students astray when faced with complex scenarios ahead!

Path Counting Logic

• Each point within a grid represents potential pathways leading into it; thus establishing foundational logic around counting unique routes becomes essential—this includes recognizing symmetrical properties across various sections within grids allowing simplification during calculations later down line!

Example Calculations

• Through examples presented step-by-step detailing how many ways exist between points using previously established rules allows learners insight into broader applications beyond just singular instances encountered initially throughout course material covered thus far!

Final Thoughts & Questions

Encouragement for Practice

• Students are reminded they should feel free asking questions at any point while also being encouraged practicing independently especially focusing upon newly introduced concepts surrounding grid-based problems ensuring thorough understanding prior tackling future assessments effectively!

Mathematical Concepts and Problem Solving

Discussion on Triangle Properties

• The speaker mentions a previous question about speed, indicating a transition to discussing mathematical problems. They invite questions from the audience.

• The speaker begins drawing clues related to Pythagorean theorem and special triangles, specifically referencing a 30-60-90 triangle.

• They explain that in a 30-60-90 triangle, the sides are in the ratio of 1:√3:2, emphasizing the importance of memorizing this for problem-solving.

Application of Pythagorean Theorem

• Using known values (AC = 14), they calculate BC as half of AC (7) and AB using the ratio derived from the triangle properties (AB = 7√3).

• The speaker advises on simplifying calculations by factoring out common terms instead of performing complex arithmetic.

Power of Points Theorem

• Introduction to the Power of Points theorem is made, explaining relationships between segments created by points on a circle.

• They demonstrate how to apply this theorem with specific segment lengths, leading to further calculations involving CP and PD.

Final Calculations and Conclusions

• After deriving values for CD through addition of previously calculated segments (CD = CP + PD), they conclude with an answer (B).

Addressing Questions and Next Steps

• The speaker checks for understanding among participants before moving on to another problem due to time constraints.

Error Handling During Problem Solving

Technical Difficulties Encountered

• A technical error occurs while screen sharing; the speaker expresses confusion over why it happened but continues engaging with participants.

Exploring Ordered Pairs in Mathematics

• Despite technical issues, they shift focus back to discussing ordered pairs involving integers within specified ranges (-5 ≤ x ≤ 5).

Analyzing Quadratic Equations

• The discussion transitions into quadratic equations where they analyze possible values for x leading up to y² being less than or equal to 30.

Value Derivations

• Various scenarios are explored based on different integer values for x, calculating corresponding valid y-values under given conditions.

Summary of Possible Solutions

• A summary calculation reveals multiple pairs derived from various x-values leading up to a total count across all scenarios presented.