Olympiad problems

2013 Online Math Open Problems, 8

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How many ways are there to choose (not necessarily distinct) integers $a,b,c$ from the set $\{1,2,3,4\}$ such that $a^{(b^c)}$ is divisible by $4$? [i]Ray Li[/i]

2010 Baltic Way, 8

Tags: combinatorics

In a club with $30$ members, every member initially had a hat. One day each member sent his hat to a different member (a member could have received more than one hat). Prove that there exists a group of $10$ members such that no one in the group has received a hat from another one in the group.

2014 Brazil Team Selection Test, 4

Tags: function , algebra , functional equation

Let $\mathbb{Z}_{\ge 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} $ satisfying the relation \[ f(f(f(n))) = f(n+1 ) +1 \] for all $ n\in \mathbb{Z}_{\ge 0}$.

2003 Czech-Polish-Slovak Match, 2

Tags: geometric transformation , reflection , geometry

In an acute-angled triangle $ABC$ the angle at $B$ is greater than $45^\circ$. Points $D,E, F$ are the feet of the altitudes from $A,B,C$ respectively, and $K$ is the point on segment $AF$ such that $\angle DKF = \angle KEF$. (a) Show that such a point $K$ always exists. (b) Prove that $KD^2 = FD^2 + AF \cdot BF$.

2010 Today's Calculation Of Integral, 603

Tags: calculus , integration , derivative , calculus computations

Find the minimum value of $\int_0^1 \{\sqrt{x}-(a+bx)\}^2dx$. Please solve the problem without using partial differentiation for those who don't learn it. 1961 Waseda University entrance exam/Science and Technology

2014 BMO TST, 1

Tags: inequalities , induction , algebra

Prove that for $n\ge 2$ the following inequality holds: $$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$

2016 NIMO Problems, 2

Tags: function , algebra

For real numbers $x$ and $y$, define \[\nabla(x,y)=x-\dfrac1y.\] If \[\underbrace{\nabla(2, \nabla(2, \nabla(2, \ldots \nabla(2,\nabla(2, 2)) \ldots)))}_{2016 \,\nabla\text{s}} = \dfrac{m}{n}\] for relatively prime positive integers $m$, $n$, compute $100m + n$. [i] Proposed by David Altizio [/i]

2003 All-Russian Olympiad, 4

Tags: induction , combinatorics

Ana and Bora are each given a sufficiently long paper strip, one with letter $A$ written , and the other with letter $B$. Every minute, one of them (not necessarily one after another) writes either on the left or on the right to the word on his/her strip the word written on the other strip. Prove that the day after, one will be able to cut word on Ana's strip into two words and exchange their places, obtaining a palindromic word.

2019 Estonia Team Selection Test, 5

Tags: combinatorics

Boeotia is comprised of $3$ islands which are home to $2019$ towns in total. Each flight route connects three towns, each on a different island, providing connections between any two of them in both directions. Any two towns in the country are connected by at most one flight route. Find the maximal number of flight routes in the country

2024 AMC 12/AHSME, 19

Tags: geometry , cyclic quadrilateral

Cyclic quadrilateral $ABCD$ has lengths $BC=CD=3$ and $DA=5$ with $\angle CDA=120^\circ$. What is the length of the shorter diagonal of $ABCD$? $ \textbf{(A) }\frac{31}7 \qquad \textbf{(B) }\frac{33}7 \qquad \textbf{(C) }5 \qquad \textbf{(D) }\frac{39}7 \qquad \textbf{(E) }\frac{41}7 \qquad $

1993 IMO Shortlist, 8

Tags: geometry , ratio , circumcircle , trigonometry , inequalities , geometric inequality

The vertices $D,E,F$ of an equilateral triangle lie on the sides $BC,CA,AB$ respectively of a triangle $ABC.$ If $a,b,c$ are the respective lengths of these sides, and $S$ the area of $ABC,$ prove that \[ DE \geq \frac{2 \cdot \sqrt{2} \cdot S}{\sqrt{a^2 + b^2 + c^2 + 4 \cdot \sqrt{3} \cdot S}}. \]

1977 Putnam, B6

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Let $H$ be a subgroup with $h$ elements in a group $G.$ Suppose that $G$ has an element $a$ such that for all $x$ in $H,$ $(xa)^3=1,$ the identity. In $G$, let $P$ be the subset of all products $x_1ax_2a\dots x_na,$ with $n$ a positive integer and the $x_i$ in $H.$ (a) Show that $P$ is a finite set. (b) Show that, in fact, $P$ has no more that $3h^2$ elements.

1995 Vietnam National Olympiad, 3

Tags: geometry

Let a non-equilateral triangle $ ABC$ and $ AD,BE,CF$ are its altitudes. On the rays $ AD,BE,CF,$ respectively, let $ A',B',C'$ such that $ \frac {AA'}{AD} \equal{} \frac {BB'}{BE} \equal{} \frac {CC'}{CF} \equal{} k$. Find all values of $ k$ such that $ \triangle A'B'C'\sim\triangle ABC$ for any non-triangle $ ABC.$

2010 Iran MO (2nd Round), 3

Tags: geometric transformation , geometry

Circles $W_1,W_2$ meet at $D$and $P$. $A$ and $B$ are on $W_1,W_2$ respectively, such that $AB$ is tangent to $W_1$ and $W_2$. Suppose $D$ is closer than $P$ to the line $AB$. $AD$ meet circle $W_2$ for second time at $C$. Let $M$ be the midpoint of $BC$. Prove that $\angle{DPM}=\angle{BDC}$.

2010 SEEMOUS, Problem 1

Tags: function , convergence , integration , calculus , sequence

Let $f_0:[0,1]\to\mathbb R$ be a continuous function. Define the sequence of functions $f_n:[0,1]\to\mathbb R$ by $$f_n(x)=\int^x_0f_{n-1}(t)dt$$ for all integers $n\ge1$. a) Prove that the series $\sum_{n=1}^\infty f_n(x)$ is convergent for every $x\in[0,1]$. b) Find an explicit formula for the sum of the series $\sum_{n=1}^\infty f_n(x),x\in[0,1]$.

2016 CMIMC, 1

Tags: team

Construction Mayhem University has been on a mission to expand and improve its campus! The university has recently adopted a new construction schedule where a new project begins every two days. Each project will take exactly one more day than the previous one to complete (so the first project takes 3, the second takes 4, and so on.) Suppose the new schedule starts on Day 1. On which day will there first be at least $10$ projects in place at the same time?

2006 Sharygin Geometry Olympiad, 15

Tags: geometry , circumcircle , equal segments , incircle

A circle is circumscribed around triangle $ABC$ and a circle is inscribed in it, which touches the sides of the triangle $BC,CA,AB$ at points $A_1,B_1,C_1$, respectively. The line $B_1C_1$ intersects the line $BC$ at the point $P$, and $M$ is the midpoint of the segment $PA_1$. Prove that the segments of the tangents drawn from the point $M$ to the inscribed and circumscribed circle are equal.

2017 Turkey Team Selection Test, 9

Tags: combinatorics

Let $S$ be a set of finite number of points in the plane any 3 of which are not linear and any 4 of which are not concyclic. A coloring of all the points in $S$ to red and white is called [i]discrete coloring[/i] if there exists a circle which encloses all red points and excludes all white points. Determine the number of [i]discrete colorings[/i] for each set $S$.

2019 Canada National Olympiad, 2

Tags: number theory , cube of a natural number.

Let $a,b$ be positive integers such that $a+b^3$ is divisible by $a^2+3ab+3b^2-1$. Prove that $a^2+3ab+3b^2-1$ is divisible by the cube of an integer greater than 1.

2013 China Northern MO, 6

Tags: perpendicular , geometry

As shown in figure , it is known that $M$ is the midpoint of side $BC$ of $\vartriangle ABC$. $\odot O$ passes through points $A, C$ and is tangent to $AM$. The extension of the segment $BA$ intersects $\odot O$ at point $D$. The lines $CD$ and $MA$ intersect at the point $P$. Prove that $PO \perp BC$. [img]https://cdn.artofproblemsolving.com/attachments/8/a/da3570ec7eb0833c7a396e22ffac2bd8902186.png[/img]

2020 Online Math Open Problems, 8

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Let $\lambda$ be a real number. Suppose that if $ABCD$ is any convex cyclic quadrilateral such that $AC=4$, $BD=5$, and $\overline{AB}\perp\overline{CD}$, then the area of $ABCD$ is at least $\lambda$. Then the greatest possible value of $\lambda$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$. [i]Proposed by Eric Shen[/i]

2021 China Team Selection Test, 1

Tags: combinatorial geometry , combinatorics , graph theory

Given positive integer $ n \ge 5 $ and a convex polygon $P$, namely $ A_1A_2...A_n $. No diagonals of $P$ are concurrent. Proof that it is possible to choose a point inside every quadrilateral $ A_iA_jA_kA_l (1\le i<j<k<l\le n) $ not on diagonals of $P$, such that the $ \tbinom{n}{4} $ points chosen are distinct, and any segment connecting these points intersect with some diagonal of P.

1998 National Olympiad First Round, 26

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How many ordered integer pairs $ \left(x,y\right)$ are there satisfying following equation: \[ y \equal{} \sqrt{x\plus{}1998\plus{}\sqrt{x\plus{}1998\plus{}\sqrt{x\plus{}1997\plus{}\sqrt{x\plus{}1997\plus{}\ldots \plus{}\sqrt{x\plus{}1\plus{}\sqrt{x\plus{}1\plus{}\sqrt{x\plus{}\sqrt{x} } } } } } } }.\] $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 1998 \qquad\textbf{(D)}\ 3996 \qquad\textbf{(E)}\ \text{Infinitely many}$

the 12th XMO, Problem 2

Tags: inequalities , algebra

Let $a_1,a_2,\cdots,a_{22}\in [1,2],$ find the maximum value of $$\dfrac{\sum\limits_{i=1}^{22}a_ia_{i+1}}{\left( \sum\limits_{i=1}^{22}a_i\right) ^2}$$where $a_{23}=a_1.$

2019 CCA Math Bonanza, I10

Tags: absolute value