Number of lucky numbers

by NO_SQUARES, Apr 23, 2025, 9:00 AM

There are coins in denominations of $a$ and $b$ doubloons, where $a$ and $b$ are given mutually prime natural numbers, with $a < b < 100$ . A non-negative integer $n$ is called lucky if the sum in $n$ doubloons can be scored with using no more than $1000$ coins. Find the number of lucky numbers.
From the folklore

number theory

Interesting inequalities

by sqing, Apr 23, 2025, 6:07 AM

Let $a,b,c\geq 0 ,b+c-ca=1$ and $c+a-ab=3.$ Prove that
$a+\frac{19}{10}b-bc\leq 2-\sqrt 2$ $a+\frac{17}{10}b+c-bc\leq 3$ $a^2+\frac{9}{5}b-bc\leq 6-4\sqrt 2$ $a^2+\frac{8}{5}b^2-bc\leq 6-4\sqrt 2$ $a+1.974873b-bc\leq 2-\sqrt 2$ $a+1.775917b+c-bc\leq 3$

inequalities proposed

inequalities

Continuity of function and line segment of integer length

by egxa, Apr 18, 2025, 5:17 PM

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of $f$ . It is known that the graph of $f$ contains exactly $N$ chords, one of which has length 2025. Find the minimum possible value of $N$ .

continuous function

function

algebra

Woaah a lot of external tangents

by egxa, Apr 18, 2025, 5:14 PM

A quadrilateral $ABCD$ with no parallel sides is inscribed in a circle $\Omega$ . Circles $\omega_a, \omega_b, \omega_c, \omega_d$ are inscribed in triangles $DAB, ABC, BCD, CDA$ , respectively. Common external tangents are drawn between $\omega_a$ and $\omega_b$ , $\omega_b$ and $\omega_c$ , $\omega_c$ and $\omega_d$ , and $\omega_d$ and $\omega_a$ , not containing any sides of quadrilateral $ABCD$ . A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle $\Gamma$ . Prove that the lines joining the centers of $\omega_a$ and $\omega_c$ , $\omega_b$ and $\omega_d$ , and the centers of $\Omega$ and $\Gamma$ all intersect at one point.

geometry

inscribed

A touching question on perpendicular lines

by Tintarn, Mar 17, 2025, 12:23 PM

Let $k$ be a semicircle with diameter $AB$ and midpoint $M$ . Let $P$ be a point on $k$ different from $A$ and $B$ .

The circle $k_A$ touches $k$ in a point $C$ , the segment $MA$ in a point $D$ , and additionally the segment $MP$ . The circle $k_B$ touches $k$ in a point $E$ and additionally the segments $MB$ and $MP$ .

Show that the lines $AE$ and $CD$ are perpendicular.

Help my diagram has too many points

by MarkBcc168, Jul 17, 2024, 12:01 PM

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ . A circle $\Gamma$ is internally tangent to $\omega$ at $A$ and also tangent to $BC$ at $D$ . Let $AB$ and $AC$ intersect $\Gamma$ at $P$ and $Q$ respectively. Let $M$ and $N$ be points on line $BC$ such that $B$ is the midpoint of $DM$ and $C$ is the midpoint of $DN$ . Lines $MP$ and $NQ$ meet at $K$ and intersect $\Gamma$ again at $I$ and $J$ respectively. The ray $KA$ meets the circumcircle of triangle $IJK$ again at $X\neq K$ .

Prove that $\angle BXP = \angle CXQ$ .

Kian Moshiri, United Kingdom

This post has been edited 2 times. Last edited by MarkBcc168, Jul 18, 2024, 8:50 PM

IMO Shortlist

geometry

ISL 2023

Some nice summations

by amitwa.exe, May 24, 2024, 8:52 AM

Problem 1: $\Omega=\left(\sum_{0\le i\le j\le k}^{\infty} \frac{1}{3^i\cdot4^j\cdot5^k}\right)\left(\mathop{{\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}}}_{i\neq j\neq k}\frac{1}{3^i\cdot3^j\cdot3^k}\right)=?$

This post has been edited 4 times. Last edited by amitwa.exe, Aug 6, 2024, 5:43 AM

2023 Hong Kong TST 3 (CHKMO) Problem 4

by PikaNiko, Dec 3, 2022, 12:26 PM

Let $ABCD$ be a quadrilateral inscribed in a circle $\Gamma$ such that $AB=BC=CD$ . Let $M$ and $N$ be the midpoints of $AD$ and $AB$ respectively. The line $CM$ meets $\Gamma$ again at $E$ . Prove that the tangent at $E$ to $\Gamma$ , the line $AD$ and the line $CN$ are concurrent.

geometry

Geometry, SMO 2016, not easy

by Zoom, Apr 1, 2016, 2:43 PM

Let $ABC$ be a triangle and $O$ its circumcentre. A line tangent to the circumcircle of the triangle $BOC$ intersects sides $AB$ at $D$ and $AC$ at $E$ . Let $A'$ be the image of $A$ under $DE$ . Prove that the circumcircle of the triangle $A'DE$ is tangent to the circumcircle of triangle $ABC$ .

geometry

circumcircle

Disjoint Pairs

by MithsApprentice, Oct 9, 2005, 8:47 AM

Suppose that the set $\{1,2,\cdots, 1998\}$ has been partitioned into disjoint pairs $\{a_i,b_i\}$ ( $1\leq i\leq 999$ ) so that for all $i$ , $|a_i-b_i|$ equals $1$ or $6$ . Prove that the sum $|a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}|$ ends in the digit $9$ .

modular arithmetic

number theory proposed

number theory

Art of Problem Solving Blog

by NO_SQUARES, Apr 23, 2025, 9:00 AM

by sqing, Apr 23, 2025, 6:07 AM

by egxa, Apr 18, 2025, 5:17 PM

by egxa, Apr 18, 2025, 5:14 PM

by Tintarn, Mar 17, 2025, 12:23 PM

by MarkBcc168, Jul 17, 2024, 12:01 PM

by amitwa.exe, May 24, 2024, 8:52 AM

by PikaNiko, Dec 3, 2022, 12:26 PM

by Zoom, Apr 1, 2016, 2:43 PM

by MithsApprentice, Oct 9, 2005, 8:47 AM