Problem of the Day
(100 Problems for 100 Days)

Below is the exhaustive list of 100 problems that I explained for the project! The "Video Explanation" button will open an external link to the YouTube video. The "Quick Handwritten Solution" button will open a pdf version of the note that I took while explaining in the respective video. If you are more comfortable with the PDF version of the whole list, here is the link to the file with the entire playlist of the videos available here :) For any corrupted file, feedback, and questions, please do not hesitate to contact me through the contact page!!

Day 1: $36^\text{th}$ KMO High School Division Stage II Problem 1

Suppose three sequences $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ satisfy following properties.

• $a_1 = 2$, $b_1 = 4$, $c_1 = 5$
• $a_{n + 1} = b_n + \frac{1}{c_n}$, $b_{n + 1} = c_n + \frac{1}{a_n}$, and $c_{n + 1} = a_n + \frac{1}{b_n}$ are true for all natural numbers $n$.

For all positive integers $n$, prove that there exists a number greater than $\sqrt{2n + 13}$ from $a_n$, $b_n$, and $c_n$.

Video Explanation Quick Handwritten Solution

Day 2: 2011 USAMO Problem 1

Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \le 4$. Prove that \[ \frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \geq 3. \]
Video Explanation Quick Handwritten Solution

Day 3: 2004 USAMO Problem 5

Let $a$, $b$, and $c$ be positive real numbers. Prove that \[ (a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \geq (a + b + c)^3. \]
Video Explanation Quick Handwritten Solution

Day 4: Deriving Cauchy-Schwarz Inequality from Hölder's Inequality

Let $a_i, b_i, \dots, z_i$ be non-negative real numbers and $\lambda_a, \lambda_b, \dots, \lambda_z$ be positive real numbers such that $\lambda_a + \lambda_b + \dots + \lambda_z = 1$. Then, \[ (a_1 + \dots + a_n)^{\lambda_a} (b_1 + \dots + b_n)^{\lambda_b} \cdots (z_1 + \dots + z_n)^{\lambda_z} \geq a_1^{\lambda_a}b_1^{\lambda_b} \cdots z_1^{\lambda_z} + \dots + a_n^{\lambda_a}b_n^{\lambda_b} \cdots z_n^{\lambda_z}. \] with equality if $a_1 : a_2 : \dots : a_n \equiv b_1 : b_2 : \dots : b_n \equiv \dots \equiv z_1 : z_2 : \dots : z_n$.

Video Explanation Quick Handwritten Solution

Day 5: 2018 AMC 12B Problem 17

Let $p$ and $q$ be positive integers such that \[ \frac{5}{9} < \frac{p}{q} < \frac{4}{7} \] and $q$ is as small as possible. Find $q-p$.

Video Explanation Quick Handwritten Solution

Day 6: 2023 IMO Problem 4

Let $x_1, x_2, \cdots , x_{2023}$ be pairwise different positive real numbers such that \[ a_n = \sqrt{(x_1 + x_2 + \text{···} + x_n) (\frac{1}{x_1} + \frac{1}{x_2} + \text{···} + \frac{1}{x_n})} \] is an integer for every $n = 1, 2, \cdots, 2023$. Prove that $a_{2023} \geq 3034$.

Video Explanation Quick Handwritten Solution

Day 7: 1984 AHSME Problem 30

Given $z = \cos 40^\circ + i \sin 40^\circ$, express $\left| z + 2z^2 + 3z^3 + \cdots + 9z^9 \right|^{-1}$ in the form of $a \sin b$.

Video Explanation Quick Handwritten Solution

Day 8: 2020 HKIMO Prelim Problem 20

Consider the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, \dots$. What are the last three digits (from left to right) of the $2020^{\text{th}}$ term?

Video Explanation Quick Handwritten Solution

Day 9: 2014 Taiwan TST Round 1 Quiz 1

Prove that for positive reals $a$, $b$, $c$ we have \[ 3(a + b + c) \geq 8 \sqrt[3]{abc} + \sqrt[3]{\frac{a^3 + b^3 + c^3}{3}}. \]
Video Explanation Quick Handwritten Solution

Day 10: Enoch's Problem

There are $999$ mysterious Thinking Trees, each labeled uniquely from $1$ to $999$. According to the legends, if a boy waters the tree(s) randomly, and if the sum of the number of watered tree(s) is divisible by $3$, a magical seed will fall from the sky. A young boy Enoch heard about the legend, and wanted to make his own magical farm. How many seeds can Enoch receive by watering the trees? (Enoch Yu)

Video Explanation Quick Handwritten Solution

Day 11: Classic Inequality Problem

Prove that for all positive reals $a, b, c, d$, \[ \frac{1}{a} + \frac{1}{b} + \frac{4}{c} + \frac{16}{d} \geq \frac{64}{a + b + c + d} \]
Video Explanation Quick Handwritten Solution

Day 12: 2000 APMO Problem 1

Compute the sum $\displaystyle S = \sum_{i = 0}^{101} \frac{x_i^3}{1 - 3x_i + 3x_i^2}$ for $\displaystyle x_i = \frac{i}{101}$.

Video Explanation Quick Handwritten Solution

Day 13: 1995 IMO Problem 2

Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that \[ \frac{1}{a^3 (b + c)} + \frac{1}{b^3 (c + a)} + \frac{1}{c^3 (a + b)} \geq \frac{3}{2}. \]
Video Explanation Quick Handwritten Solution

Day 14: 2008 AIME II Problem 12

There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are identical blue flags, and $9$ are identical green flags. Let $N$ be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent. Find the remainder when $N$ is divided by $1000$.

Video Explanation Quick Handwritten Solution

Day 15: Enoch's Problem

Compute $\cos252^\circ$.

Video Explanation Quick Handwritten Solution

Day 16: Proof of Brianchon's Theorem

The principle diagonals of an inscribed hexagon concurs.

Video Explanation Quick Handwritten Solution

Day 17: Enoch's Problem

Express $(6996, 38160)$ as the linear combination of $6996$ and $38160$.

Video Explanation Quick Handwritten Solution

Day 18: Proof of Desargues' Theorem

Two triangles are axially perspective \emph{iff} they are centrally perspective. Two triangles are considered axially perspective if there exists the axis of perspectivity they are centrally perspective if there exists the center of perspectivity.

Video Explanation Quick Handwritten Solution

Day 19: Enoch's Problem

Solve for $x$ modulo 30 that satisfies the following congruences: \begin{align*} x \equiv 3 \pmod{2} \\ x \equiv 5 \pmod{3} \\ x \equiv 2 \pmod{5} \end{align*}
Video Explanation Quick Handwritten Solution

Day 20: Desargues' Theorem Related Problem

$D$ is a point on $\overline{BC}$ in $\triangle{ABC}$. Let $I_1$, $I_2$ be the incenter of $\triangle{ABD}$ and $\triangle{ACD}$ respectively. Moreover, let $I_3$ and $I_4$ be ex-centers in respect to $\angle{BAD}$ and $\angle{CAD}$ respectively. Show that $\overline{I_1 I_2}$, $\overline{I_3 I_4}$, and $\overline{BC}$ concur.

Video Explanation Quick Handwritten Solution

Day 21: Proof of Ceva's Theorem

The product of ratios of lengths in each sides formed by cevians is equal to $1$ if and only if the cevians concur at a point.

Video Explanation Quick Handwritten Solution

Day 22: Proving Pascal's Identity

\[ \binom{n - 1}{k} = \binom{n}{k} + \binom{n}{k - 1} \]
Video Explanation Quick Handwritten Solution

Day 23: Pell's Equation Problem

Solve for $(x, y)$ in each equation. \begin{align*} x^2 - 11y^2 &= 1 \\ x^2 - 11y^2 &= -1 \end{align*}
Video Explanation Quick Handwritten Solution

Day 24: 2001 IMO Problem 2

Let $a,b,c$ be positive real numbers. Prove that $\frac{a}{\sqrt{a^{2} + 8bc}} + \frac{b}{\sqrt{b^{2} + 8ca}} + \frac{c}{\sqrt{c^{2} + 8ab}} \geq 1$.

Video Explanation Quick Handwritten Solution

Day 25: 2003 USAMO Problem 5

Let $a$, $b$, $c$ be positive real numbers. Prove that \[ \frac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \frac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \frac{(2c + a + b)^2}{2c^2 + (a + b)^2} \leq 8. \]
Video Explanation Quick Handwritten Solution

Day 26: Proving Euler's Theorem

\[ a^{\varphi(n)} \equiv 1 \pmod{n} \] is true for $a, n \in \mathbb{N}$ such that $(a, n) = 1$ and Euler's totient function $\varphi(n)$.

Video Explanation Quick Handwritten Solution

Day 27: Wilson's Theorem Example

Compute $(19! + 19, 20! + 19)$.

Video Explanation Quick Handwritten Solution

Day 28: 2020 IMO Shortlist A3

Suppose that $a$, $b$, $c$, $d$ are positive real numbers satisfying $(a + c)(b + d) = ac + bd$. Find the smallest possible value of \[ S = \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}. \]

Video Explanation Quick Handwritten Solution

Day 29: $36^\text{th}$ KMO High School Division Stage II Problem 2

For an acute scalene triangle $ABC$, let $D$ be the intersection of the side $BC$ and the angle bisector of angle $A$. Moreover, let $E$ and $F$ be the circumcenter of $\triangle{ABD}$ and $\triangle{ADC}$ respectively. Assuming that $P (\ne D)$ is the point of intersection of the circumcircle of $\triangle{BDE}$ and $\triangle{DCF}$ and $O$, $X$, $Y$ are the circumcenter of $\triangle{ABC}$, $\triangle{BDE}$, and $\triangle{DCF}$ respectively, show that $OP \parallel XY$.

Video Explanation Quick Handwritten Solution

Day 30: 2010 USA TST Problem 2

Let $a, b, c$ be positive reals such that $abc = 1$. Show that \[ \frac{1}{a^5 (b + 2c)^2} + \frac{1}{b^5 (c + 2a)^2} + \frac{1}{c^5(a + 2b)^2} \geq \frac{1}{3}. \]

Video Explanation Quick Handwritten Solution

Day 31: 2018 USAMO Problem 1

Let $a,b,c$ be positive real numbers such that $a + b + c = 4\sqrt[3]{abc}$. Prove that \[ 2(ab + bc + ca) + 4 \min(a^2, b^2, c^2) \geq a^2 + b^2 + c^2. \]

Video Explanation Quick Handwritten Solution

Day 32: 2009 USAMO Problem 4

For $n \geq 2$ let $a_1$, $a_2$, ..., $a_n$ be positive real numbers such that \[ (a_1 + a_2 + \cdots + a_n) \left( \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n} \right) \leq \left( n + \frac{1}{2} \right)^2. \] Prove that $\text{max}(a_1, a_2, ... ,a_n) \leq 4 \text{ min}(a_1, a_2, ... , a_n)$.

Video Explanation Quick Handwritten Solution

Day 33: 1999 IMO Problem 6

Find all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(x - f(y)) = f(f(y)) + xf(y) + f(x) - 1 \] for all real numbers $x, y$.

Video Explanation Quick Handwritten Solution

Day 34: 2021 USAMO Problem 1

Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that \[ \angle BC_1C + \angle CA_1A + \angle AB_1B = 180^{\circ}. \] Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.

Video Explanation Quick Handwritten Solution

Day 35: Exponential Cauchy Equation

Investigate continuous functions $f: \mathbb{R} \rightarrow (0, \infty)$ that satisfies $f(x + y) = f(x)f(y)$.

Video Explanation Quick Handwritten Solution

Day 36: 2020 IMO Problem 2

The real numbers $a$, $b$, $c$, $d$ are such that $a \geq b \geq c \geq d > 0$ and $a + b + c + d = 1$. Prove that \[ (a + 2b + 3c + 4d) a^a b^b c^c d^d < 1. \]
Video Explanation Quick Handwritten Solution

Day 37: 1991 USAMO Problem 3

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots \pmod{n} \] is eventually constant.
[The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \pmod{n}$ means the remainder which results from dividing $\,a_i\,$ by $\,n$.]

Video Explanation Quick Handwritten Solution

Day 38: 2024 KMO Middle School Division Problem 1

Find all triples $(x, y, z)$ where $x$, $y$, $z$ are distinct positive integer that satisfy the following equation. \[ \frac{1}{x + 1} + \frac{1}{y + 2} + \frac{1}{z + 3} = \frac{1}{2} \left( \frac{1}{1} + \frac{1}{2} + \frac{1}{3} \right) \]

Video Explanation Quick Handwritten Solution

Day 39: 1974 USAMO Problem 2

Prove that if $a$, $b$, and $c$ are positive real numbers, then \[ a^a b^b c^c \geq (abc)^{(a + b + c)/3} \]
Video Explanation Quick Handwritten Solution

Day 40: Enoch Yu

Solve in integers: $3x + 5y + 7z = 11$.

Video Explanation Quick Handwritten Solution

Day 41: 2020 Baltic Way Problem 1

Let $a_0 > 0$ be a real number, and let \[ a_n = \frac{a_{n - 1}}{\sqrt{1 + 2020 \cdot a_{n-1}^2}}, \quad \text{for } n = 1, 2, \ldots, 2020. \] Show that $a_{2020} < \frac{1}{2020}$.

Video Explanation Quick Handwritten Solution

Day 42: 2010 USAMO Problem 4

Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB, AC, BI, ID, CI, IE$ to all have integer lengths.

Video Explanation Quick Handwritten Solution

Day 43: 2021 IMO Shortlist A5

Let $n \geq 2$ be an integer, and let $a_1, a_2, \ldots, a_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = 1$. Prove that \[ \sum_{k=1}^n \frac{a_k}{1 - a_k} (a_1 + a_2 + \cdots + a_{k-1})^2 < \frac{1}{3}. \]
Video Explanation Quick Handwritten Solution

Day 44: 2009 USAMO Problem 1

Given circles $\omega_1$ and $\omega_2$ intersecting at points $X$ and $Y$, let $\ell_1$ be a line through the center of $\omega_1$ intersecting $\omega_2$ at points $P$ and $Q$ and let $\ell_2$ be a line through the center of $\omega_2$ intersecting $\omega_1$ at points $R$ and $S$. Prove that if $P, Q, R$ and $S$ lie on a circle then the center of this circle lies on line $XY$.

Video Explanation Quick Handwritten Solution

Day 45: 2023 IMO Shortlist A1

Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of their bowl). The \textit{dissatisfaction} level of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $|N - C|$. Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute the food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$.

Video Explanation Quick Handwritten Solution

Day 46: Year 11 USAMTS Round 4 Problem 3

Determine the value of \[ S = \sqrt{1 + \frac{1}{1^2} + \frac{1}{2^2}} + \sqrt{1 + \frac{1}{2^2} + \frac{1}{3^2}} + \cdots + \sqrt{1 + \frac{1}{1999^2} + \frac{1}{2000^2}} \]

Video Explanation Quick Handwritten Solution

Day 47: 2018 IMO Problem 2

Find all numbers $n \ge 3$ for which there exists real numbers $a_1, a_2, ..., a_{n+2}$ satisfying $a_{n+1} = a_1, a_{n+2} = a_2$ and \[ a_{i}a_{i + 1} + 1 = a_{i + 2} \] for $i = 1, 2, ..., n$.

Video Explanation Quick Handwritten Solution

Day 48: 2012 IMO Problem 2

Let $a_3, \ldots, a_n$ be $n - 1$ positive real numbers, where $n \geq 3$, such that $a_2 a_3 \cdots a_n = 1$. Prove that \[ (1 + a_2)^2 (1 + a_3)^3 \cdots (1 + a_n)^n > n^n. \]
Video Explanation Quick Handwritten Solution

Day 49: Proof of Existence of Simson Line

The feet of perpendicular lines from a point on the circumcircle of a triangle to its sides are collinear.

Video Explanation Quick Handwritten Solution

Day 50: 2004 IMO Problem 2

Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab + bc + ca = 0$ we have the following relations \[ f(a - b) + f(b - c) + f(c - a) = 2f(a + b + c). \]
Video Explanation Quick Handwritten Solution

Day 51: 2001 USAMO Problem 2

Let $ABC$ be a triangle and let $\omega$ be its incircle. Denote by $D_1$ and $E_1$ the points where $\omega$ is tangent to sides $BC$ and $AC$, respectively. Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$, respectively, such that $CD_2 = BD_1$ and $CE_2 = AE_1$, and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$. Circle $\omega$ intersects segment $AD_2$ at two points, the closer of which to the vertex $A$ is denoted by $Q$. Prove that $AQ = D_2P$.

Video Explanation Quick Handwritten Solution

Day 52: Three Classic Geometry Lemmas

The video includes three common lemmas used in Olympiad geometry.

Video Explanation Quick Handwritten Solution

Day 53: 2001 IMO Shortlist A3

Let $x_1, x_2, \dotsc, x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1 + x_1^2} + \frac{x_2}{1 + x_1^2 + x_2^2} + \dotsb + \frac{x_n}{1 + x_1^2 + \dotsb + x_n^2} < \sqrt{n}. \]
Video Explanation Quick Handwritten Solution

Day 54: Classic Polynomial Problem

Prove that $x^4 + 1$ is irreducible in $\mathbb{Q}[x]$.

Video Explanation Quick Handwritten Solution

Day 55: 2025 USAJMO Problem 5

Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.

Video Explanation Quick Handwritten Solution

Day 56: 1976 USAMO Problem 5

If $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ are all polynomials such that \[ P(x^5) + xQ(x^5) + x^2 R(x^5) = (x^4 + x^3 + x^2 + x +1) S(x), \] prove that $x-1$ is a factor of $P(x)$.

Video Explanation Quick Handwritten Solution

Day 57: 2015 USAMO Problem 1

Solve in integers the equation \[ x^2 + xy + y^2 = \left( \frac{x + y}{3} + 1 \right)^3. \]
Video Explanation Quick Handwritten Solution

Day 58: 2002 USAMO Problem 4

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(x^2 - y^2) = xf(x) - yf(y) \] for all pairs of real numbers $x$ and $y$.

Video Explanation Quick Handwritten Solution

Day 59: 2017 PUMaC Geometry A Problem 6

Triangle $ABC$ has $\angle{A} = 90^\circ$, $AB = 2$, and $AC = 4$. Circle $\omega_1$ has center $C$ and radius $CA$, while circle $\omega_2$ has center $B$ and radius $BA$. The two circles intersect at point $E$, different from point $A$. Point $M$ is on $\omega_2$ and in the interior of $ABC$, such that $BM$ is parallel to $EC$. Suppose $EM$ intersects $\omega_1$ at point $K$ and $AM$ intersects $\omega_1$ at point $Z$. What is the area of quadrilateral $ZEBK$?

Video Explanation Quick Handwritten Solution

Day 60: 2019 IMO Problem 1

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that, for all integers $a$ and $b$, \[ f(2a) + 2f(b) = f(f(a + b)). \]
Video Explanation Quick Handwritten Solution

Day 61: 2008 IMO Problem 4

Find all functions $f: (0, \infty) \mapsto (0, \infty)$ such that \[ \frac {\left( f(w) \right)^2 + \left( f(x) \right)^2} {f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2} \] for all positive real numbers $w,x,y,z,$ satisfying $wx = yz$.

Video Explanation Quick Handwritten Solution

Day 62: 2020 Baltic Way Problem 4

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ so that \[ f(f(x) + x + y) = f(x + y) + yf(y) \] for all real numbers $x, y$.

Video Explanation Quick Handwritten Solution

Day 63: 2023 Baltic Way Problem 2

Let $a_1, a_2, \ldots, a_{2023}$ be positive real numbers with \[ a_1 + a_2^2 + a_3^3 + \cdots + a_{2023}^{2023} = 2023. \] Show that \[ a_1^{2023} + a_2^{2022} + \cdots + a_{2022}^2 + a_{2023} > 1 + \frac{1}{2023}. \]
Video Explanation Quick Handwritten Solution

Day 64: 2020 BMO Problem 1

Let $ABC$ be an acute triangle with $AB=AC$, let $D$ be the midpoint of the side $AC$, and let $\gamma$ be the circumcircle of the triangle $ABD$. The tangent of $\gamma$ at $A$ crosses the line $BC$ at $E$. Let $O$ be the circumcenter of the triangle $ABE$. Prove that midpoint of the segment $AO$ lies on $\gamma$.

Video Explanation Quick Handwritten Solution

Day 65: IrMO 2009 Paper 1 Problem 5

Suppose $a, b, c$ are real numbers such that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 1$. Prove that \[ a^2 b^2 c^2 \leq \frac{1}{54}, \] and determine the cases of equality.

Video Explanation Quick Handwritten Solution

Day 66: 2018 Finnish HSMC Problem 3

The chords $AB$ and $CD$ of a circle intersect at $M$, which is the midpoint of the chord $PQ$. The points $X$ and $Y$ are the intersections of the segments $AD$ and $PQ$, respectively, and $BC$ and $PQ$, respectively. Show that $M$ is the midpoint of $XY$.

Video Explanation Quick Handwritten Solution

Day 67: $38^\text{th}$ KMO High School Division Stage II Problem 1

Three distinct points $A,B,X$ lies on a circle with center $O$ where $A,B,O$ are not collinear. Given $\Omega$ as the circumcircle of $\triangle{ABO}$, segments $AX$ and $BX$ intersects with $\Omega$ at $C(\neq A)$ and $D(\neq B)$ respectively. Show that $O$ is the orthocenter of $\triangle{CXD}$.

Video Explanation Quick Handwritten Solution

Day 68: 2006 IMO Shortlist G1

Let $ABC$ be a triangle with incenter $I$. A point $P$ in the interior of the triangle satisfies \[ \angle{PBA} + \angle{PCA} = \angle{PBC} + \angle{PCB}. \] Show that $AP \geq AI$ and that equality hold if and only if $P$ coincides with $I$.

Video Explanation Quick Handwritten Solution

Day 69: 1976 USAMO Problem 5

If $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ are all polynomials such that \[ P(x^5) + xQ(x^5) + x^2 R(x^5) = (x^4 + x^3 + x^2 + x + 1) S(x), \] prove that $x-1$ is a factor of $P(x)$.

Video Explanation Quick Handwritten Solution

Day 70: Three Tangent Lemma

For an acute triangle $ABC$, let $D$ and $E$ be the foot on $AB$ and $AC$. If $M$ is the midpoint of $BC$ and $H$ is the orthocenter of $\triangle{ABC}$, $DM$, $ME$, and a line through $H$ parallel to BC are tangent to the circumcircle of $\triangle{ADE}$.

Video Explanation Quick Handwritten Solution

Day 71: 2020 KMO Problem 2

$H$ is the orthocenter of an acute triangle $ABC$, and let $M$ be the midpoint of $BC$. Suppose $(AH)$ meets $AB$ and $AC$ at $D,E$ respectively. $AH$ meets $DE$ at $P$, and the line through $H$ perpendicular to $AH$ meets $DM$ at $Q$. Prove that $P,Q,B$ are collinear.

Video Explanation Quick Handwritten Solution

Day 72: 2010 IMO Problem 1

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ the following equality holds \[ f(\left\lfloor x \right\rfloor y) = f(x) \left\lfloor f(y) \right\rfloor \] where $\left\lfloor a \right\rfloor$ is greatest integer not greater than $a$.

Video Explanation Quick Handwritten Solution

Day 73: 2023 IMO Problem 2

Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$.

Video Explanation Quick Handwritten Solution

Day 74: 2025 IMO Problem 1

A line in the plane is called sunny if it is not parallel to any of the $x$–axis, the $y$–axis, and the line $x+y=0$. Let $n \geq 3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both of the following:

• for all positive integers $a$ and $b$ with $a + b \leq n + 1$, the point $(a,b)$ is on at least one of the lines; and
• exactly $k$ of the $n$ lines are sunny.

Video Explanation Quick Handwritten Solution

Day 75: 2024 KMO Problem 6

Find the minimum of a real number $M$ that satisfies the following inequality for positive real numbers $a_1, a_2, \ldots, a_{99}$ where $a_{100} = a_1$ and $a_{101} = a_2$. \[ \sum_{k=1}^{99} \frac{a_{k+1}}{a_k + a_{k+1} + a_{k+2}} < M \]
Video Explanation Quick Handwritten Solution

Day 76: 2000 IMO Problem 2

Let $a, b, c$ be positive real numbers with $abc = 1$. Show that \[ \left( a - 1 + \frac{1}{b} \right) \left( b - 1 + \frac{1}{c} \right) \left( c - 1 + \frac{1}{a} \right) \leq 1 \]
Video Explanation Quick Handwritten Solution

Day 77: 2003 IMO Problem 4

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, and $R$ be the feet of perpendiculars from $D$ to lines $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively. Show that $PQ = QR$ if and only if the bisectors of angles $ABC$ and $ADC$ meet on segment $\overline{AC}$.

Video Explanation Quick Handwritten Solution

Day 78: 2025 KMO Problem 6

For arbitrary real numbers $x_1, x_2, \dots, x_{99}$ such that $0 < x_1 < x_2 < \cdots < x_{99}$, determine the minimum value of positive real number $c$ such that the following inequality always holds. \[ 3 \sqrt{x_1} + 4 \left( \sqrt{x_2 - x_1} + \sqrt{x_3 - x_2} + \cdots + \sqrt{x_{99} - x_{98}} \right) \leq c \left( \sqrt{x_2} + \sqrt{x_3} + \cdots + \sqrt{x_{98}} \right) + 5\sqrt{x_{99}} \]
Video Explanation Quick Handwritten Solution

Day :

Video Explanation Quick Handwritten Solution