todai-math

Papers (33)
2025
problems1-3 3 todai-engineering-math 5
2024
liberal-arts_official 4 problem1 1 science 6 science_official 6 todai-engineering-math 4
2023
problem1 1 problem2 1 problem3 1 science 5 todai-engineering-math 5
2022
problem1 1 problem2 1 problem3 1 science 6 todai-engineering-math__paper1 2 todai-engineering-math__paper2 2 todai-engineering-math__paper3 7
2021
science 6 todai-engineering-math__paper1 6 todai-engineering-math__paper2 3 todai-engineering-math__paper3 3
2020
science 6 todai-engineering-math 5
2019
todai-engineering-math 5
2018
ist 1 todai-engineering-math 2
2017
ist 2 todai-engineering-math 3
2016
todai-engineering-math 5 translated 3
2015
ist 2
2024 liberal-arts_official

4 maths questions

Q1 Areas by integration View
In the coordinate plane, the parabola $C: y = ax^2 + bx + c$ passes through the two points $\mathrm{P}(\cos\theta,\, \sin\theta)$ and $\mathrm{Q}(-\cos\theta,\, \sin\theta)$, and has a common tangent line with the circle $x^2 + y^2 = 1$ at each of the points $\mathrm{P}$ and $\mathrm{Q}$. Assume $0^\circ < \theta < 90^\circ$.
(1) Express $a$, $b$, $c$ in terms of $s = \sin\theta$.
(2) Express the area $A$ of the region enclosed by the parabola $C$ and the $x$-axis in terms of $s$.
(3) Show that $A \geq \sqrt{3}$.
Q2 Laws of Logarithms Solve a Logarithmic Equation View
Answer the following questions. You may use the fact that $0.3 < \log_{10} 2 < 0.31$ if necessary.
  • [(1)] Find the smallest natural number $n$ such that $5^n > 10^{19}$.
  • [(2)] Find the smallest natural number $m$ such that $5^m + 4^m > 10^{19}$.
Q3 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
In the coordinate plane, let $\mathrm{O}(0,0)$ and $\mathrm{A}(0,1)$ be two points. Suppose two points $\mathrm{P}(p,0)$ and $\mathrm{Q}(q,0)$ on the $x$-axis satisfy both of the following conditions (i) and (ii).
  • [(i)] $0 < p < 1$ and $p < q$
  • [(ii)] Let $\mathrm{M}$ be the midpoint of segment $\mathrm{AP}$; then $\angle \mathrm{OAP} = \angle \mathrm{PMQ}$

(1) Express $q$ in terms of $p$.
(2) Find the value of $p$ such that $q = \dfrac{1}{3}$.
(3) Let $S$ be the area of $\triangle \mathrm{OAP}$ and $T$ be the area of $\triangle \mathrm{PMQ}$. Find the range of $p$ such that $S > T$.
Q4 Combinations & Selection Combinatorial Probability View
Let $n$ be an odd number greater than or equal to 5. Consider a circle centered at point O in the plane, and a regular $n$-gon inscribed in it. Choose 4 distinct points simultaneously from the $n$ vertices. Assume that any 4 points are equally likely to be chosen. Find the probability $p_n$ that the quadrilateral with the chosen 4 points as vertices contains O in its interior.