Olympiad problems

2023 Malaysian Squad Selection Test, 4

Tags: number theory

Find the largest constant $c>0$ such that for every positive integer $n\ge 2$, there always exist a positive divisor $d$ of $n$ such that $$d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)}$$ where $\tau(n)$ is the number of divisors of $n$. [i]Proposed by Mohd. Suhaimi Ramly[/i]

2024 New Zealand MO, 6

Tags: geometry , incircle

Let $\omega$ be the incircle of scalene triangle $ABC$. Let $\omega$ be tangent to $AB$ and $AC$ at points $X$ and $Y$. Construct points $X^\prime$ and $Y^\prime$ on line segments $AB$ and $AC$ respectively such that $AX^\prime=XB$ and $AY^\prime=YC$. Let line $CX^\prime$ intersects $\omega$ at points $P,Q$ such that $P$ is closer to $C$ than $Q$. Also let $R^\prime$ be the intersection of lines $CX^\prime$ and $BY^\prime$. Prove that $CP=RX^\prime$.

1996 Baltic Way, 2

Tags: geometry , geometry proposed

In the figure below, you see three half-circles. The circle $C$ is tangent to two of the half-circles and to the line $PQ$ perpendicular to the diameter $AB$. The area of the shaded region is $39\pi$, and the area of the circle $C$ is $9\pi$. Find the length of the diameter $AB$.

2015 İberoAmerican, 2

Tags: geometry , angle bisector , Inversion

A line $r$ contains the points $A$, $B$, $C$, $D$ in that order. Let $P$ be a point not in $r$ such that $\angle{APB} = \angle{CPD}$. Prove that the angle bisector of $\angle{APD}$ intersects the line $r$ at a point $G$ such that: $\frac{1}{GA} + \frac{1}{GC} = \frac{1}{GB} + \frac{1}{GD}$

1975 Swedish Mathematical Competition, 6

Tags: algebra , analysis

$f(x)$ is defined for $0 \leq x \leq 1$ and has a continuous derivative satisfying $|f'(x)| \leq C|f(x)|$ for some positive constant $C$. Show that if $f(0) = 0$, then $f(x)=0$ for the entire interval.

2014 NZMOC Camp Selection Problems, 10

Tags: Words , combinatorics

In the land of Microbablia the alphabet has only two letters, ‘A’ and ‘B’. Not surprisingly, the inhabitants are obsessed with the band ABBA. Words in the local dialect with a high ABBA-factor are considered particularly lucky. To compute the ABBA-factor of a word you just count the number of occurrences of ABBA within the word (not necessarily consecutively). So for instance AABA has ABBA-factor $0$, ABBA has ABBA-factor $1$, AABBBA has ABBA-factor $6$, and ABBABBA has ABBA factor $8$. What is the greatest possible ABBA-factor for a $100$ letter word?

2010 Junior Balkan Team Selection Tests - Romania, 3

Tags: inequalities , Sum , algebra

We consider the real numbers $a _ 1, a _ 2, a _ 3, a _ 4, a _ 5$ with the zero sum and the property that $| a _ i - a _ j | \le 1$ , whatever it may be $i,j \in \{1, 2, 3, 4, 5 \} $. Show that $a _ 1 ^ 2 + a _ 2 ^ 2 + a _ 3 ^ 2 + a _ 4 ^ 2 + a _ 5 ^ 2 \le \frac {6} {5}$ .

2005 Taiwan TST Round 2, 2

Tags: inequalities , inequalities unsolved

Find all positive integers $n \ge 3$ such that there exists a positive constant $M_n$ satisfying the following inequality for any $n$ positive reals $a_1, a_2,\dots\>,a_n$: \[\displaystyle \frac{a_1+a_2+\cdots\>+a_n}{\sqrt[n]{a_1a_2\cdots\>a_n}} \le M_n \biggl( \frac{a_2}{a_1} + \frac{a_3}{a_2} +\cdots\>+ \frac{a_n}{a_{n-1}} + \frac {a_1}{a_n} \biggr).\] Moreover, find the minimum value of $M_n$ for such $n$. The difficulty is finding $M_n$...

1959 Poland - Second Round, 2

Tags: geometry , similar , Medians

What relationship between the sides of a triangle makes it similar to the triangle formed by its medians?

2024 Bundeswettbewerb Mathematik, 2

Tags: number theory , number theory proposed , Digits , Perfect Squares , Perfect Square

Can a number of the form $44\dots 41$, with an odd number of decimal digits $4$ followed by a digit $1$, be a perfect square?

2025 Kyiv City MO Round 1, Problem 3

Tags: geometry , combinatorial geometry

What's the smallest positive integer $ n > 3 $, for which there does [b]not[/b] exist a (not necessarily convex) $ n $-gon such that all its diagonals have equal lengths? A diagonal of any polygon is defined as a segment connecting any two non-adjacent vertices of the polygon. [i]Proposed by Anton Trygub[/i]

2002 IMO Shortlist, 5

Tags: combinatorics , IMO Shortlist , Set systems

Let $r\geq2$ be a fixed positive integer, and let $F$ be an infinite family of sets, each of size $r$, no two of which are disjoint. Prove that there exists a set of size $r-1$ that meets each set in $F$.

2015 Dutch BxMO/EGMO TST, 5

Tags: algebra , functional equation

Find all functions $f : R \to R$ satisfying $(x^2 + y^2)f(xy) = f(x)f(y)f(x^2 + y^2)$ for all real numbers $x$ and $y$.

Kvant 2024, M2787

Tags: geometry

Let $XY$ be a segment, which is a diameter of a semi-circle. Let $Z$ be a point on $XY$ and 9 rays from $Z$ are drawn that divide $\angle XZY=180^{\circ}$ into $10$ equal angles. These rays meet the semi-circle at $A_1, A_2, \ldots, A_9$ in this order in the direction from $X$ to $Y$. Prove that the sum of the areas of triangles $ZA_2A_3$ and $ZA_7A_8$ equals the area of the quadrilateral $A_2A_3A_7A_8$.

1941 Moscow Mathematical Olympiad, 075

Tags: number theory , Perfect Square , consecutive

Prove that $1$ plus the product of any four consecutive integers is a perfect square.

2020 ELMO Problems, P5

Tags: number theory , algebra , Polynomials

Let $m$ and $n$ be positive integers. Find the smallest positive integer $s$ for which there exists an $m \times n$ rectangular array of positive integers such that [list] [*]each row contains $n$ distinct consecutive integers in some order, [*]each column contains $m$ distinct consecutive integers in some order, and [*]each entry is less than or equal to $s$. [/list] [i]Proposed by Ankan Bhattacharya.[/i]

2018 CHMMC (Fall), 1

Tags: combinatorics , CHMMC

A large pond contains infinitely many lily pads labelled $1$, $2$, $3$,$ ... $, placed in a line, where for each $k$, lily pad $k + 1$ is one unit to the right of lily pad $k$. A frog starts at lily pad $100$. Each minute, if the frog is at lily pad $n$, it hops to lily pad $n + 1$ with probability $\frac{n-1}{n}$ , and hops all the way back to lily pad $1$ with probability $\frac{1}{n}$. Let $N$ be the position of the frog after $1000$ minutes. What is the expected value of $N$?

1976 IMO Longlists, 20

Tags: inequalities , induction , number theory unsolved , number theory

Let $(a_n), n = 0, 1, . . .,$ be a sequence of real numbers such that $a_0 = 0$ and \[a^3_{n+1} = \frac{1}{2} a^2_n -1, n= 0, 1,\cdots\] Prove that there exists a positive number $q, q < 1$, such that for all $n = 1, 2, \ldots ,$ \[|a_{n+1} - a_n| \leq q|a_n - a_{n-1}|,\] and give one such $q$ explicitly.

2011 Sharygin Geometry Olympiad, 5

Tags: geometry unsolved , geometry

Given triangle $ABC$. The midperpendicular of side $AB$ meets one of the remaining sides at point $C'$. Points $A'$ and $B'$ are defined similarly. Find all triangles $ABC$ such that triangle $A'B'C'$ is regular.

1985 Miklós Schweitzer, 2

Tags: college contests

[b]2.[/b] Let $S$ be a given finite set of hyperplanes in $\mathbb{R}^n$, and let $O$ be a point. Show that there exists a compact set $K \subseteq \mathbb{R}^n$ containing $O$ such that the orthogonal projection of any point of $K$ onto any hyperplane in $S$ is also in $K$. ([b]G.37[/b]) [Gy. Pap]

2023 Harvard-MIT Mathematics Tournament, 3

Tags:

Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle BCD = \theta$ for some acute angle $\theta$. Point $X$ lies inside the quadrilateral such that $\angle XAD = \angle XDA = 90^{\circ}-\theta$. Prove that $BX = XC$.

1993 AMC 12/AHSME, 5

Tags: percent , percent increase , AMC

Last year a bicycle cost $\$160$ and a cycling helmet cost $ \$ 40$. This year the cost of the bicycle increased by $5\%$, and the cost of the helmet increased by $10\%$. The percent increase in the combined cost of the bicycle and the helmet is $ \textbf{(A)}\ 6\% \qquad\textbf{(B)}\ 7\% \qquad\textbf{(C)}\ 7.5\% \qquad\textbf{(D)}\ 8\% \qquad\textbf{(E)}\ 15\% $

2023 USA TSTST, 6

Tags:

Let $ABC$ be a scalene triangle and let $P$ and $Q$ be two distinct points in its interior. Suppose that the angle bisectors of $\angle PAQ,\,\angle PBQ,$ and $\angle PCQ$ are the altitudes of triangle $ABC$. Prove that the midpoint of $\overline{PQ}$ lies on the Euler line of $ABC$. (The Euler line is the line through the circumcenter and orthocenter of a triangle.) [i]Proposed by Holden Mui[/i]

2008 Bundeswettbewerb Mathematik, 1

Tags: inequalities , algebra unsolved , algebra

Determine all real $ x$ satisfying the equation \[ \sqrt[5]{x^3 \plus{} 2x} \equal{} \sqrt[3]{x^5\minus{}2x}.\] Odd roots for negative radicands shall be included in the discussion.

2025 CMIMC Team, 10

Tags: team

In a $2024 \times 2024$ grid of squares, each square is colored either black or white. An ant starts at some black square in the grid and starts walking parallel to the sides of the grid. During this walk, it can choose (not required) to turn $90^\circ$ clockwise or counterclockwise if it is currently on a black square, otherwise it must continue walking in the same direction. A coloring of the grid is called [i]simple[/i] if it is [b]not[/b] possible for the ant to arrive back at its starting location after some time. How many simple colorings of the grid are maximal, in the sense that adding any black square results in a coloring that is not simple?