Olympiad problems

2004 Estonia National Olympiad, 5

Tags: alphabet , combinatorics

The alphabet of language $BAU$ consists of letters $B, A$, and $U$. Independently of the choice of the $BAU$ word of length n from which to start, one can construct all the $BAU$ words with length n using iteratively the following rules: (1) invert the order of the letters in the word; (2) replace two consecutive letters: $BA \to UU, AU \to BB, UB \to AA, UU \to BA, BB \to AU$ or $AA \to UB$. Given that $BBAUABAUUABAUUUABAUUUUABB$ is a $BAU$ word, does $BAU$ have a) the word $BUABUABUABUABAUBAUBAUBAUB$ ? b) the word $ABUABUABUABUAUBAUBAUBAUBA$ ?

2023 Stanford Mathematics Tournament, 2

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Every cell in a $5\times5$ grid of paper is to be painted either red or white with equal probability. An edge of the paper is said to have a "tree" if the set of cells depicted in the diagram below are all painted red when the paper is rotated so that the edge lies at the bottom. Given that at least one edge of the paper has a tree, what is the expected number of edges that have a tree? [center][img]https://cdn.artofproblemsolving.com/attachments/1/2/f81d8da53d7bc6819fc1dfe4acb9567d545856.png[/img][/center]

MathLinks Contest 6th, 6.1

Tags: algebra , inequalities

Let $p > 1$ and let $a, b, c, d$ be positive numbers such that $$(a + b + c + d) \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)= 16p^2.$$ Find all values of the ratio $ R =\frac{\max \{a, b, c, d\}}{\min \{a, b, c, d\}}$ (depending on the parameter $p$)

1977 Germany Team Selection Test, 1

Tags: rearrangement inequality , inequality , permutation , algebra

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

Kyiv City MO Seniors Round2 2010+ geometry, 2019.10.3.1

Tags: geometry , equal segments , pentagon , regular pentagon

Let $ABCDE$ be a regular pentagon with center $M$. Point $P \ne M$ is selected on segment $MD$. The circumscribed circle of triangle $ABP$ intersects the line $AE$ for second time at point $Q$, and a line that is perpendicular to the $CD$ and passes through $P$, for second time at the point $R$. Prove that $AR = QR$.

2015 Greece Team Selection Test, 1

Tags: number theory

Solve in positive integers the following equation; $xy(x+y-10)-3x^2-2y^2+21x+16y=60$

1983 National High School Mathematics League, 7

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$P$ is a point on the plane which square $ABCD$ belongs to, satisfying that $\triangle PAB,\triangle PBC,\triangle PCD,\triangle PDA$ are isosceles triangles. What's the number of such points? $\text{(A)}9\qquad\text{(B)}17\qquad\text{(C)}1\qquad\text{(D)}5$

2024 Korea Junior Math Olympiad (First Round), 12.

Tags: algebra , maximum value

For reals $x,y$, find the maximum of A. $ A=\frac{-x^2-y^2-2xy+30x+30y+75}{3x^2-12xy+12y^2+12} $

PEN R Problems, 6

Tags: the geometry of numbers , geometry

Let $R$ be a convex region symmetrical about the origin with area greater than $4$. Show that $R$ must contain a lattice point different from the origin.

2014 Online Math Open Problems, 8

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Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers satisfying \begin{align*} 2a_1+a_2+a_3+a_4+a_5 &= 1 + \tfrac{1}{8}a_4 \\ 2a_2+a_3+a_4+a_5 &= 2 + \tfrac{1}{4}a_3 \\ 2a_3+a_4+a_5 &= 4 + \tfrac{1}{2}a_2 \\ 2a_4+a_5 &= 6 + a_1 \end{align*} Compute $a_1+a_2+a_3+a_4+a_5$. [i]Proposed by Evan Chen[/i]

2019 Istmo Centroamericano MO, 3

Tags: equal angles , tangent circle , geometry

Let $ABC$ be an acute triangle, with $AB <AC$. Let $M$ be the midpoint of $AB$, $H$ the foot of the altitude from $A$, and $Q$ be point on side $AC$ such that $\angle ABQ = \angle BCA$. Show that the circumcircles of the triangles $ABQ$ and $BHM$ are tangent.

1997 All-Russian Olympiad, 3

Tags: geometry , rectangle , combinatorics

The lateral sides of a box with base $a\times b$ and height $c$ (where $a$; $b$;$ c$ are natural numbers) are completely covered without overlap by rectangles whose edges are parallel to the edges of the box, each containing an even number of unit squares. (Rectangles may cross the lateral edges of the box.) Prove that if $c$ is odd, then the number of possible coverings is even. [i]D. Karpov, C. Gukshin, D. Fon-der-Flaas[/i]

2006 CHKMO, 1

Tags: pigeonhole principle , induction , combinatorics

On a planet there are $3\times2005!$ aliens and $2005$ languages. Each pair of aliens communicates with each other in exactly one language. Show that there are $3$ aliens who communicate with each other in one common language.

2023 China Girls Math Olympiad, 2

Tags: combinatorics

On an $8\times 8$ chessboard, place a stick on each edge of each grid (on a common edge of two grid only one stick will be placed). What is the minimum number of sticks to be deleted so that the remaining sticks do not form any rectangle?

1999 Mexico National Olympiad, 2

Tags: arithmetic sequence , number theory , prime

Prove that there are no $1999$ primes in an arithmetic progression that are all less than $12345$.

1955 AMC 12/AHSME, 38

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Four positive integers are given. Select any three of these integers, find their arithmetic average, and add this result to the fourth integer. Thus the numbers $ 29$, $ 23$, $ 21$, and $ 17$ are obtained. One of the original integers is: $ \textbf{(A)}\ 19 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 29 \qquad \textbf{(E)}\ 17$

2015 Iran MO (3rd round), 1

Tags: geometry , trapezoid , circumcircle

Let $ABCD$ be the trapezoid such that $AB\parallel CD$. Let $E$ be an arbitrary point on $AC$. point $F$ lies on $BD$ such that $BE\parallel CF$. Prove that circumcircles of $\triangle ABF,\triangle BED$ and the line $AC$ are concurrent.

1975 Chisinau City MO, 94

Tags: locus , geometry

A straight line $\ell$ and a point $A$ outside of it are given on the plane. Find the locus of the vertices $C$ of the equilateral triangle $ABC$, the vertex $B$ of which lies on the straight line $\ell$.

2013 AMC 10, 10

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A basketball team's players were successful on $50\%$ of their two-point shots and $40\%$ of their three-point shots, which resulted in $54$ points. They attempted $50\%$ more two-point shots than three-point shots. How many three-point shots did they attempt? $ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }30 $

2003 India IMO Training Camp, 9

Tags: inequalities , induction , number theory , combinatorics

Let $n$ be a positive integer and $\{A,B,C\}$ a partition of $\{1,2,\ldots,3n\}$ such that $|A|=|B|=|C|=n$. Prove that there exist $x \in A$, $y \in B$, $z \in C$ such that one of $x,y,z$ is the sum of the other two.

2000 Belarus Team Selection Test, 1.4

Tags: trigonometry , inequalities , geometric inequalities , sphere , geometry , 3d geometry

A closed pentagonal line is inscribed in a sphere of the diameter $1$, and has all edges of length $\ell$. Prove that $\ell \le \sin \frac{2\pi}{5}$ .

2023 Belarusian National Olympiad, 10.8

Tags: graph , combinatorics , turan

On the Alphamegacentavra planet there are $2023$ cities, some of which are connected by non-directed flights. It turned out that among any $4$ cities one can find two with no flight between them. Find the maximum number of triples of cities such that between any two of them there is a flight.

1989 Vietnam National Olympiad, 2

Tags: algebra , polynomial , inequalities

The sequence of polynomials $ \left\{P_n(x)\right\}_{n\equal{}0}^{\plus{}\infty}$ is defined inductively by $ P_0(x) \equal{} 0$ and $ P_{n\plus{}1}(x) \equal{} P_n(x)\plus{}\frac{x \minus{} P_n^2(x)}{2}$. Prove that for any $ x \in [0, 1]$ and any natural number $ n$ it holds that $ 0\le\sqrt x\minus{} P_n(x)\le\frac{2}{n \plus{} 1}$.

2010 Today's Calculation Of Integral, 601

Tags: calculus , integration , trigonometry , calculus computations , logarithm

Evaluate $\int_0^{\frac{\pi}{4}} (\tan x)^{\frac{3}{2}}dx$. created by kunny

1979 IMO Longlists, 62

Tags: combinatorics

$T$ is a given triangle with vertices $P_1,P_2,P_3$. Consider an arbitrary subdivision of $T$ into finitely many subtriangles such that no vertex of a subtriangle lies strictly between two vertices of another subtriangle. To each vertex $V$ of the subtriangles there is assigned a number $n(V)$ according to the following rules: $(\text{i})$ If $V$ = $P_i$, then $n(V) = i$. $(\text{ii})$ If $V$ lies on the side $P_i P_j$ of $T$, then $n(V) = i$ or $j$. $(\text{iii})$ If $V$ lies inside the triangle $T$, then $n(V)$ is any of the numbers $1,2,3$. Prove that there exists at least one subtriangle whose vertices are numbered $1, 2, 3$.