Olympiad problems

2019 China Team Selection Test, 4

Tags: number theory

Prove that there exist a subset $A$ of $\{1,2,\cdots,2^n\}$ with $n$ elements, such that for any two different non-empty subset of $A$, the sum of elements of one subset doesn't divide another's.

2023 AMC 10, 8

Tags: AMC , AMC 10 , AMC 10 A

Barb the baker creates a new temperature system for baking bread, Breadus, which is linearly based on Fahrenheit. Bread rises at $110$ F$^\circ$, which is $0$ on the Breadus scale. Bread bakes at $350$ F$^\circ$, which is $100$ on the Breadus scale. Bread is done when it’s internal temperature is $200$ F$^\circ.$ What is this temperature on the Breadus scale? $\textbf{(A) }33\qquad\textbf{(B) }34.5\qquad\textbf{(C) }36\qquad\textbf{(D) }37.5\qquad\textbf{(E) }39$

2008 Postal Coaching, 1

Tags: floor function , number theory , Fraction , natural , algebra , irrational

For each positive $ x \in \mathbb{R}$, define $ E(x)=\{[nx]: n\in \mathbb{N}\}$ Find all irrational $ \alpha >1$ with the following property: If a positive real $ \beta$ satisfies $ E(\beta) \subset E(\alpha)$. then $ \frac{\beta}{\alpha}$ is a natural number.

2010 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: geometry , identity

It is given acute triangle $ABC$ with orthocenter at point $H$. Prove that $$AH \cdot h_a+BH \cdot h_b+CH \cdot h_c=\frac{a^2+b^2+c^2}{2}$$ where $a$, $b$ and $c$ are sides of a triangle, and $h_a$, $h_b$ and $h_c$ altitudes of $ABC$

2011 JBMO Shortlist, 9

Tags: JBMO , combinatorics , combinatorial geometry

Decide if it is possible to consider $2011$ points in a plane such that the distance between every two of these points is different from $1$ and each unit circle centered at one of these points leaves exactly $1005$ points outside the circle.

2007 Hanoi Open Mathematics Competitions, 7

Tags: geometry , combinatorial geometry , Equilateral

Nine points, no three of which lie on the same straight line, are located inside an equilateral triangle of side $4$. Prove that some three of these points are vertices of a triangle whose area is not greater than $\sqrt3$.

2006 QEDMO 3rd, 1

Tags: geometry , circumcircle , geometry proposed

1998 Vietnam Team Selection Test, 3

Tags: combinatorics unsolved , combinatorics

In a conference there are $n \geq 10$ people. It is known that: [b]I.[/b] Each person knows at least $\left[\frac{n+2}{3}\right]$ other people. [b]II.[/b] For each pair of person $A$ and $B$ who don't know each other, there exist some people $A(1), A(2), \ldots, A(k)$ such that $A$ knows $A(1)$, $A(i)$ knows $A(i+1)$ and $A(k)$ knows $B$. [b]III.[/b] There doesn't exist a Hamilton path. Prove that: We can divide those people into 2 groups: $A$ group has a Hamilton cycle, and the other contains of people who don't know each other.

1969 IMO Shortlist, 58

Tags: pigeonhole principle , geometry , 3D geometry , tetrahedron , IMO Shortlist , IMO Longlist

$(SWE 1)$ Six points $P_1, . . . , P_6$ are given in $3-$dimensional space such that no four of them lie in the same plane. Each of the line segments $P_jP_k$ is colored black or white. Prove that there exists one triangle $P_jP_kP_l$ whose edges are of the same color.

2020 Azerbaijan National Olympiad, 4

Tags: geometry , AZE JUNIOR NMO

There is a non-equilateral triangle $ABC$.Let $ABC$'s Incentri $I$.Point $D$ is on the $BC$ side.The circle drawn outside the triangle $IBD$ and $ICD$ intersects the sides $AB$ and $AC$ at points $E$ and $F.$The circle drawn outside the triangle $DEF$ intersects the sides $AB$ and $AC$ at $N$ and $M$.Prove that $EM\parallel FN $.

2023 LMT Spring, 9

Tags: algebra

Evin’s calculator is broken and can only perform $3$ operations: Operation $1$: Given a number $x$, output $2x$. Operation $2$: Given a number $x$, output $4x +1$. Operation $3$: Given a number $x$, output $8x +3$. After initially given the number $0$, how many numbers at most $128$ can he make?

2019 Sharygin Geometry Olympiad, 3

Tags: geometry , Sharygin Geometry Olympiad , moving points

Let $ABCD$ be a cyclic quadrilateral such that $AD=BD=AC$. A point $P$ moves along the circumcircle $\omega$ of triangle $ABCD$. The lined $AP$ and $DP$ meet the lines $CD$ and $AB$ at points $E$ and $F$ respectively. The lines $BE$ and $CF$ meet point $Q$. Find the locus of $Q$.

2006 Stanford Mathematics Tournament, 14

Tags: floor function , factorial , function , number theory , modular arithmetic

Find the smallest nonnegative integer $n$ for which $\binom{2006}{n}$ is divisible by $7^3$.

2000 Harvard-MIT Mathematics Tournament, 18

Tags: trigonometry , limit

What is the value of $ \sum_{n=1}^\infty (\tan^{-1}\sqrt{n}-\tan^{-1}\sqrt{n+1})$?

2012 Online Math Open Problems, 18

Tags: Online Math Open

There are 32 people at a conference. Initially nobody at the conference knows the name of anyone else. The conference holds several 16-person meetings in succession, in which each person at the meeting learns (or relearns) the name of the other fifteen people. What is the minimum number of meetings needed until every person knows everyone else's name? [i]David Yang, Victor Wang.[/i] [size=85][i]See the "odd version" [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=810&t=500914]here[/url].[/i][/size]

2000 Miklós Schweitzer, 1

Tags: college contests , Miklos Schweitzer , function , ordinals

Prove that there exists a function $f\colon [\omega_1]^2 \rightarrow \omega _1$ such that (i) $f(\alpha, \beta)< \mathrm{min}(\alpha, \beta)$ whenever $\mathrm{min}(\alpha,\beta)>0$; and (ii) if $\alpha_0<\alpha_1<\ldots<\alpha_i<\ldots<\omega_1$ then $\sup\left\{ a_i \colon i<\omega \right\} =\sup \left\{ f(\alpha_i, \alpha_j)\colon i,j<\omega\right\}$.

1998 Dutch Mathematical Olympiad, 4

Tags: geometry , rhombus , vector

Let $ABCD$ be a convex quadrilateral such that $AC \perp BD$. (a) Prove that $AB^2 + CD^2 = BC^2 + DA^2$. (b) Let $PQRS$ be a convex quadrilateral such that $PQ = AB$, $QR = BC$, $RS = CD$ and $SP = DA$. Prove that $PR \perp QS$.

2019 Dutch BxMO TST, 5

Tags: combinatorics

In a country, there are $2018$ cities, some of which are connected by roads. Each city is connected to at least three other cities. It is possible to travel from any city to any other city using one or more roads. For each pair of cities, consider the shortest route between these two cities. What is the greatest number of roads that can be on such a shortest route?

2008 Iran MO (3rd Round), 6

Tags: geometry , 3D geometry , sphere

There are five research labs on Mars. Is it always possible to divide Mars to five connected congruent regions such that each region contains exactly on research lab. [img]http://i37.tinypic.com/f2iq8g.png[/img]

2022 Taiwan TST Round 1, C

Tags: IMO Shortlist , pigeonhole principle , ISL 2021 , c2

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

1967 Spain Mathematical Olympiad, 5

Tags: geometry , equal angles

Let $\gamma$ be a semicircle with diameter $AB$ . A creek is built with origin in $A$ , which has its vertices alternately in the diameter $AB$ and in the semicircle $\gamma$ , so that its sides make equal angles $\alpha$ with the diameter (but alternately in either direction). It is requested: a) Values of the angle $\alpha$ for the ravine to pass through the other end $B$ of the diameter. b) The total length of the ravine, in the case that it ends in $B$ , as a function of the length $d$ of the diameter and of the angle $\alpha$ . [img]https://cdn.artofproblemsolving.com/attachments/3/c/54c71cdf1bf8fbb3bdfa38f4b14a1dc961c5fe.png[/img]

2012 Mathcenter Contest + Longlist, 5

Tags: algebra , inequalities , Hi

Let $a,b,c>0$ and $a+b+c+abc=4$. Prove that $$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{c+a}}+\frac{c}{\sqrt{a+b} }\ge \frac{1}{\sqrt{2}}(a+b+c).$$ [i](Zhuge Liang)[/i]

2010 Vietnam Team Selection Test, 2

Tags: geometry , circumcircle , power of a point , radical axis , geometry unsolved

Let $ABC$ be a triangle with $ \widehat{BAC}\neq 90^\circ $. Let $M$ be the midpoint of $BC$. We choose a variable point $D$ on $AM$. Let $(O_1)$ and $(O_2)$ be two circle pass through $ D$ and tangent to $BC$ at $B$ and $C$. The line $BA$ and $CA$ intersect $(O_1),(O_2)$ at $ P,Q$ respectively. [b]a)[/b] Prove that tangent line at $P$ on $(O_1)$ and $Q$ on $(O_2)$ must intersect at $S$. [b]b)[/b] Prove that $S$ lies on a fix line.

2019 USMCA, 4

Tags:

How many six-letter words formed from the letters of AMC do not contain the substring AMC? (For example, AMAMMC has this property, but AAMCCC does not.)

2005 Georgia Team Selection Test, 5

Tags: geometry , circumcircle , trigonometry , geometry proposed

Let $ ABCD$ be a convex quadrilateral. Points $ P,Q$ and $ R$ are the feets of the perpendiculars from point $ D$ to lines $ BC, CA$ and $ AB$, respectively. Prove that $ PQ\equal{}QR$ if and only if the bisectors of the angles $ ABC$ and $ ADC$ meet on segment $ AC$.