Olympiad problems

2004 Miklós Schweitzer, 6

Is is true that if the perfect set $F\subseteq [0,1]$ is of zero Lebesgue measure then those functions in $C^1[0,1]$ which are one-to-one on $F$ form a dense subset of $C^1[0,1]$? (We use the metric $$d(f,g)=\sup_{x\in[0,1]} |f(x)-g(x)| + \sup_{x\in[0,1]} |f'(x)-g'(x)|$$ to define the topology in the space $C^1[0,1]$ of continuously differentiable real functions on $[0,1]$.)

2004 Unirea, 2

Tags: group theory

Consider a group $ G $ which has the property that any element of it, with the exception of the identity, has order $ p\ge 2. $ Prove that [b]a)[/b] $ p $ is prime. [b]b)[/b] $ G $ is commutative if any subset of $ G $ having $ p^2-1 $ elements contains at least $ p $ elements that commute between themselves pairwise.

2004 Alexandru Myller, 2

Tags: geometry , area

Let $ M,N,P,Q $ be points on the sides $ AB,BC,CD,DA $ (respectively) of a convex quadrilateral $ ABCD $ so that: $$ \frac{MA}{MB} =\frac{NB}{NC} =\frac{PD}{PC} =\frac{QA}{QD}\neq 1 $$ Show that the area of $ MNPQ $ is half the area of $ ABCD $ if and only if $ ABD $ and $ BCD $ have equal areas. [i]Petre Asaftei[/i]

2000 Brazil Team Selection Test, Problem 1

Tags: geometry , circumcircle , trigonometry , calculus , trig identities , Law of Sines , geometry solved

Consider a triangle $ABC$ and $I$ its incenter. The line $(AI)$ meets the circumcircle of $ABC$ in $D$. Let $E$ and $F$ be the orthogonal projections of $I$ on $(BD)$ and $(CD)$ respectively. Assume that $IE+IF=\frac{1}{2}AD$. Calculate $\angle{BAC}$. [color=red][Moderator edited: Also discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=5088 .][/color]

2015 IFYM, Sozopol, 3

Tags: game strategy , combinatorics

A cube 10x10x10 is constructed from 1000 white unit cubes. Polly and Velly play the following game: Velly chooses a certain amount of parallelepipeds 1x1x10, no two of which have a common vertex or an edge, and repaints them in black. Polly can choose an arbitrary number of unit cubes and ask Velly for their color. What’s the least amount of unit cubes she has to choose so that she can determine the color of each unit cube?

2020 Regional Olympiad of Mexico West, 2

Tags: geometry , equal areas

Let $ L $, $ M $ and $ N $ be the midpoints on the sides $ BC $, $ AC $ and $ AB$ of a triangle $ ABC $. Points $ D $, $ E $ and $ F $ are taken on the circle circumscribed to the triangle $ LMN $ so that the segments $ LD $, $ ME $ and $ NF $ are diameters of said circumference. Prove that the area of the hexagon $ LENDMF $ is equal to half the area of the triangle $ ABC $

2006 Canada National Olympiad, 1

Tags: combinatorics proposed , combinatorics

Let $ f(n,k)$ be the number of ways of distributing $ k$ candies to $ n$ children so that each child receives at most $ 2$ candies. For example $ f(3,7) \equal{} 0,f(3,6) \equal{} 1,f(3,4) \equal{} 6$. Determine the value of $ f(2006,1) \plus{} f(2006,4) \plus{} \ldots \plus{} f(2006,1000) \plus{} f(2006,1003) \plus{} \ldots \plus{} f(2006,4012)$.

1907 Eotvos Mathematical Competition, 2

Tags: geometry , parallelogram , geometric inequality

Let $P$ be any point inside the parallelogram $ABCD$ and let $R$ be the radius of the circle through $A$, $B$, and $C$. Show that the distance from $P$ to the nearest vertex is not greater than $R$.

1970 Miklós Schweitzer, 6

Tags: real analysis , topology , advanced fields , advanced fields unsolved

Let a neighborhood basis of a point $ x$ of the real line consist of all Lebesgue-measurable sets containing $ x$ whose density at $ x$ equals $ 1$. Show that this requirement defines a topology that is regular but not normal. [i]A. Csaszar[/i]

1967 IMO Longlists, 42

Tags: trigonometry , algebra , Trigonometric Equations , factorization , IMO Shortlist , IMO Longlist

Decompose the expression into real factors: \[E = 1 - \sin^5(x) - \cos^5(x).\]

2019 Auckland Mathematical Olympiad, 4

Tags: algebra , Sequence

Suppose $a_1 =\frac16$ and $a_n = a_{n-1} - \frac{1}{n}+ \frac{2}{n + 1} - \frac{1}{n + 2}$ for $n > 1$. Find $a_{100}$.

1964 All Russian Mathematical Olympiad, 054

Tags: number theory , Perfect Square

Find the smallest exact square with last digit not $0$, such that after deleting its last two digits we shall obtain another exact square.

2025 AMC 8, 19

Tags: AMC 8 , 2025 AMC 8

Two towns, $A$ and $B$, are connected by a straight road, $15$ miles long. Traveling from town $A$ to town $B$, the speed limit changes every $5$ miles: from $25$ to $40$ to $20$ miles per hour (mph). Two cars, one at town $A$ and one at town $B$, start moving toward each other at the same time. They drive exactly the speed limit in each portion of the road. How far from town $A$, in miles, will the two cars meet? $\textbf{(A) }7.75 \qquad\textbf{(B) }8 \qquad\textbf{(C) }8.25\qquad\textbf{(D) }8.5 \qquad\textbf{(E) }8.75$

PEN P Problems, 20

Tags: Additive Number Theory

If an integer $n$ is such that $7n$ is the form $a^2 +3b^2$, prove that $n$ is also of that form.

2015 Puerto Rico Team Selection Test, 8

Tags: multiple , divisible , divides , number theory

Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.

2019 India IMO Training Camp, P2

Tags: number theory

Show that there do not exist natural numbers $a_1, a_2, \dots, a_{2018}$ such that the numbers \[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \] are all powers of $5$ [i]Proposed by Tejaswi Navilarekallu[/i]

2020 Purple Comet Problems, 23

Tags: trigonometry

There is a real number $x$ between $0$ and $\frac{\pi}{2}$ such that $$\frac{\sin^3 x + \cos^3 x}{\sin^5 x + \cos^5 x}=\frac{12}{11}$$ and $\sin x + \cos x =\frac{\sqrt{m}}{n}$ , where $m$ and $n$ are positive integers, and $m$ is not divisible by the square of any prime. Find $m + n$.

1989 China National Olympiad, 2

Tags: inequalities

Let $x_1, x_2, \dots ,x_n$ ($n\ge 2$) be positive real numbers satisfying $\sum^{n}_{i=1}x_i=1$. Prove that:\[\sum^{n}_{i=1}\dfrac{x_i}{\sqrt{1-x_i}}\ge \dfrac{\sum_{i=1}^{n}\sqrt{x_i}}{\sqrt{n-1}}.\]

2023 239 Open Mathematical Olympiad, 2

Tags: number theory , Divisibility

Let $1 < a_1 < a_2 < \cdots < a_N$ be natural numbers. It is known that for any $1\leqslant i\leqslant N$ the product of all these numbers except $a_i$ increased by one, is a multiple of $a_i$. Prove that $a_1\leqslant N$.

2006 Alexandru Myller, 2

Tags: geometry

$ ABC $ is a triangle with $ \angle BCA= 90^{\circ } $ and $ D,E $ on sides $ BC,CA, $ rspectively, so that $ \frac{BD}{AC}=\frac{AE}{CD}=k. $ The line $ BE $ meets $ AD $ at $ O. $ Show that $ \angle BOD =60^{\circ } $ if and only if $ k=\sqrt 3. $

2011 Romania National Olympiad, 1

Tags: inequalities , formulas , algebra , factorization , sum of cubes

Let be a natural number $ n $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n $ such that $$ a_m+a_{m+1} +\cdots +a_n\ge \frac{(m+n)(n-m+1)}{2} ,\quad\forall m\in\{ 1,2,\ldots ,n \} . $$ Prove that $ a_1^2+a_2^2+\cdots +a_n^2\ge\frac{n(n+1)(2n+1)}{6} . $

2007 Princeton University Math Competition, 10

Tags: probability , expected value

Bob, having little else to do, rolls a fair $6$-sided die until the sum of his rolls is greater than or equal to $700$. What is the expected number of rolls needed? Any answer within $.0001$ of the correct answer will be accepted.

2013 Vietnam Team Selection Test, 6

Tags: combinatorics , combinatorics unsolved

A cube with size $10\times 10\times 10$ consists of $1000$ unit cubes, all colored white. $A$ and $B$ play a game on this cube. $A$ chooses some pillars with size $1\times 10\times 10$ such that no two pillars share a vertex or side, and turns all chosen unit cubes to black. $B$ is allowed to choose some unit cubes and ask $A$ their colors. How many unit cubes, at least, that $B$ need to choose so that for any answer from $A$, $B$ can always determine the black unit cubes?

2007 Today's Calculation Of Integral, 201

Tags: calculus , integration , limit , function , calculus computations

Evaluate the following definite integral. \[\int_{-1}^{1}\frac{e^{2x}+1-(x+1)(e^{x}+e^{-x})}{x(e^{x}-1)}dx\]

1990 Turkey Team Selection Test, 4

Tags: geometry , analytic geometry , linear algebra , matrix , rotation , geometry proposed

Let $ABCD$ be a convex quadrilateral such that \[\begin{array}{rl} E,F \in [AB],& AE = EF = FB \\ G,H \in [BC],& BG = GH = HC \\ K,L \in [CD],& CK = KL = LD \\ M,N \in [DA],& DM = MN = NA \end{array}\] Let \[[NG] \cap [LE] = \{P\}, [NG]\cap [KF] = \{Q\},\] \[{[}MH] \cap [KF] = \{R\}, [MH]\cap [LE]=\{S\}\] Prove that [list=a][*]$Area(ABCD) = 9 \cdot Area(PQRS)$ [*] $NP=PQ=QG$ [/list]