Olympiad problems

2015 APMO, 1

Tags: geometry , circumcircle

Let $ABC$ be a triangle, and let $D$ be a point on side $BC$. A line through $D$ intersects side $AB$ at $X$ and ray $AC$ at $Y$ . The circumcircle of triangle $BXD$ intersects the circumcircle $\omega$ of triangle $ABC$ again at point $Z$ distinct from point $B$. The lines $ZD$ and $ZY$ intersect $\omega$ again at $V$ and $W$ respectively. Prove that $AB = V W$ [i]Proposed by Warut Suksompong, Thailand[/i]

2023 Indonesia TST, 3

Tags: function , combinatorics

Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$. A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called [i]nice[/i] if there exists an index $k$ such that \[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\] Prove that the number of nice functions is at least $n^n$.

2010 Saudi Arabia Pre-TST, 3.1

Tags: algebra , inequalities

Let $a \ge b \ge c > 0$. Prove that $$(a-b+c)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right) \ge 1$$

2019 Latvia Baltic Way TST, 14

Tags: number theory

Let $m$ be a positive integer and $p$ be a prime, such that $m^2 - 2$ is divisible by $p$. Suppose that there exists positive integer $a$ such that $a^2+m-2$ is divisible by $p$. Prove that there exists positive integer $b$ such that $b^2- m -2$ is divisible by $p$.

1966 Miklós Schweitzer, 5

Tags: geometry , rectangle , real analysis

A "letter $ T$" erected at point $ A$ of the $ x$-axis in the $ xy$-plane is the union of a segment $ AB$ in the upper half-plane perpendicular to the $ x$-axis and a segment $ CD$ containing $ B$ in its interior and parallel to the $ x$-axis. Show that it is impossible to erect a letter $ T$ at every point of the $ x$-axis so that the union of those erected at rational points is disjoint from the union of those erected at irrational points. [i]A.Csaszar[/i]

LMT Team Rounds 2021+, 2

Tags: combinatorics

Ada rolls a standard $4$-sided die $5$ times. The probability that the die lands on at most two distinct sides can be written as $ \frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$

2008 Harvard-MIT Mathematics Tournament, 13

Tags: algebra , polynomial , factorial

Let $ P(x)$ be a polynomial with degree 2008 and leading coefficient 1 such that \[ P(0) \equal{} 2007, P(1) \equal{} 2006, P(2) \equal{} 2005, \dots, P(2007) \equal{} 0. \]Determine the value of $ P(2008)$. You may use factorials in your answer.

2018 Iran Team Selection Test, 5

Tags: combinatorics

$2n-1$ distinct positive real numbers with sum $S $ are given. Prove that there are at least $\binom {2n-2}{n-1}$ different ways to choose $n $ numbers among them such that their sum is at least $\frac {S}{2}$. [i]Proposed by Amirhossein Gorzi[/i]

2013 AMC 12/AHSME, 6

Tags:

In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on $20\%$ of her three-point shots and $30\%$ of her two-point shots. Shenille attempted $30$ shots. How many points did she score? $ \textbf{(A)}\ 12\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 36 $

1979 IMO Longlists, 34

Tags: sfft , number theory

Notice that in the fraction $\frac{16}{64}$ we can perform a simplification as $\cancel{\frac{16}{64}}=\frac 14$ obtaining a correct equality. Find all fractions whose numerators and denominators are two-digit positive integers for which such a simplification is correct.

2025 Caucasus Mathematical Olympiad, 3

Tags: combinatorics

Let $K$ be a positive integer. Egor has $100$ cards with the number “$2$” written on them, and $100$ cards with the number “$3$” written on them. Egor wants to paint each card red or blue so that no subset of cards of the same color has the sum of the numbers equal to $K$. Find the greatest $K$ such that Egor will not be able to paint the cards in such a way.

2012 Denmark MO - Mohr Contest, 2

Tags: square , geometry , rectangle , combinatorial geometry

It is known about a given rectangle that it can be divided into nine squares which are situated relative to each other as shown. The black rectangle has side length $1$. Are there more than one possibility for the side lengths of the rectangle? [img]https://cdn.artofproblemsolving.com/attachments/1/0/af6bc5b867541c04586e4b03db0a7f97f8fe87.png[/img]

2009 Harvard-MIT Mathematics Tournament, 2

Tags: algorithm , number theory , euclidean algorithm , greatest common divisor

Suppose N is a $6$-digit number having base-$10$ representation $\underline{a}\text{ }\underline{b}\text{ }\underline{c}\text{ }\underline{d}\text{ }\underline{e}\text{ }\underline{f}$. If $N$ is $6/7$ of the number having base-$10$ representation $\underline{d}\text{ }\underline{e}\text{ }\underline{f}\text{ }\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, find $N$.

2000 Taiwan National Olympiad, 2

Tags: floor function , ceiling function , combinatorics

Let $n$ be a positive integer and $A=\{ 1,2,\ldots ,n\}$. A subset of $A$ is said to be connected if it consists of one element or several consecutive elements. Determine the maximum $k$ for which there exist $k$ distinct subsets of $A$ such that the intersection of any two of them is connected.

2005 Iran Team Selection Test, 3

Tags: induction , combinatorics

Suppose there are 18 lighthouses on the Persian Gulf. Each of the lighthouses lightens an angle with size 20 degrees. Prove that we can choose the directions of the lighthouses such that whole of the blue Persian (always Persian) Gulf is lightened.

2010 Indonesia TST, 2

Tags: combinatorics

Given an equilateral triangle, all points on its sides are colored in one of two given colors. Prove that the is a right-angled triangle such that its three vertices are in the same color and on the sides of the equilateral triangle. [i]Alhaji Akbar, Jakarta[/i]

1993 Iran MO (3rd Round), 6

Tags: algebra

Let $x_1, x_2, \ldots, x_{12}$ be twelve real numbers such that for each $1 \leq i \leq 12$, we have $|x_i| \geq 1$. Let $I=[a,b]$ be an interval such that $b-a \leq 2$. Prove that number of the numbers of the form $t= \sum_{i=1}^{12} r_ix_i$, where $r_i=\pm 1$, which lie inside the interval $I$, is less than $1000$.

2023 Peru MO (ONEM), 3

Tags: geometry , combinatorics , combinatorial geometry

Prove that, for every integer $n \ge 2$, it is possible to divide a regular hexagon into $n$ quadrilaterals such that any two of them are similar. Clarification: Two quadrilaterals are similar if they have their corresponding sides proportional and their corresponding angles are equal, that is, the quadrilaterals $ABCD$ and $EFGH$ are similar if $\frac{AB}{EF}= \frac{BC}{FG}= \frac{CD}{GH} = \frac{DA}{HE}$, $\angle ABC = \angle EFG$, $\angle BCD = \angle FGH$, $\angle CDA = \angle GHE$ and $\angle DAB = \angle HEF$.

2016 Sharygin Geometry Olympiad, P13

Tags: geometry , circumcircle

$L$ is a Line that intersect with the side $AB,BC,AC$ of triangle $ABC$ at $F,D,E$ The line perpendicular to $BC$ from $D$ intersect $AB,AC$ at $A_{1},A_{2}$ respectively Name $B_{1},B_{2},C_{1},C_{2}$ similarly Prove that the circumcenters of $AA_{1}A_{2},BB_{1}B_{2},CC_{1}C_{2}$ are collinear

Geometry Mathley 2011-12, 1.1

Tags: geometry , geometric inequality , hexagon , perimeter

Let $ABCDEF$ be a hexagon having all interior angles equal to $120^o$ each. Let $P,Q,R, S, T, V$ be the midpoints of the sides of the hexagon $ABCDEF$. Prove the inequality $$p(PQRSTV ) \ge \frac{\sqrt3}{2} p(ABCDEF)$$, where $p(.)$ denotes the perimeter of the polygon. Nguyễn Tiến Lâm

1990 Vietnam Team Selection Test, 2

Tags: 3d geometry , tetrahedron , trigonometry , geometry , inequalities

Given a tetrahedron such that product of the opposite edges is $ 1$. Let the angle between the opposite edges be $ \alpha$, $ \beta$, $ \gamma$, and circumradii of four faces be $ R_1$, $ R_2$, $ R_3$, $ R_4$. Prove that \[ \sin^2\alpha \plus{} \sin^2\beta \plus{} \sin^2\gamma\ge\frac {1}{\sqrt {R_1R_2R_3R_4}} \]

2020 Azerbaijan IZHO TST, 5

Tags: algebra

Let $x,y,z$ be positive real numbers such that $x^4+y^4+z^4=1$ . Determine with proof the minimum value of $\frac{x^3}{1-x^8}+\frac{y^3}{1-y^8}+\frac{z^3}{1-z^8}$

1998 Moldova Team Selection Test, 3

Tags: geometry

Prove that in a triangle $Sum of medians >\frac{3}{4}(perimeter of triangle )$

2020 HMNT (HMMO), 3

Tags: geometry

Harvard has recently built a new house for its students consisting of $n$ levels, where the $k$th level from the top can be modeled as a $1$-meter-tall cylinder with radius $k$ meters. Given that the area of all the lateral surfaces (i.e. the surfaces of the external vertical walls) of the building is $35$ percent of the total surface area of the building (including the bottom), compute $n$.

2011 Hanoi Open Mathematics Competitions, 1

Tags: number theory , divisible

An integer is called "octal" if it is divisible by $8$ or if at least one of its digits is $8$. How many integers between $1$ and $100$ are octal? (A): $22$, (B): $24$, (C): $27$, (D): $30$, (E): $33$