Olympiad problems

2017 BMT Spring, 3

Compute $\int^9_{-9}17x^3 \cos (x^2) dx.$

2007 Princeton University Math Competition, 10

Tags: modular arithmetic

Find all primes $p$ such that there exist positive integers $q$ and $r$ such that $p \nmid q$, $3 \nmid q$, $p^3 = r^3 - q^2$.

Let $n$ ($n \ge 1$) be an integer. Consider the equation $2\cdot \lfloor{\frac{1}{2x}}\rfloor - n + 1 = (n + 1)(1 - nx)$, where $x$ is the unknown real variable. (a) Solve the equation for $n = 8$. (b) Prove that there exists an integer $n$ for which the equation has at least $2021$ solutions. (For any real number $y$ by $\lfloor{y} \rfloor$ we denote the largest integer $m$ such that $m \le y$.)

2001 Tuymaada Olympiad, 4

Tags: function , combinatorics

Is it possible to colour all positive real numbers by 10 colours so that every two numbers with decimal representations differing in one place only are of different colours? (We suppose that there is no place in a decimal representations such that all digits starting from that place are 9's.) [i]Proposed by A. Golovanov[/i]

1966 IMO Shortlist, 57

Tags: geometry , 3d geometry , symmetry , rotation , cube

Is it possible to choose a set of $100$ (or $200$) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.

1984 All Soviet Union Mathematical Olympiad, 390

Tags: chessboard , combinatorics

The white fields of $1983\times 1984 $1983x1984 are filled with either $+1$ or $-1$. For every black field, the product of neighbouring numbers is $+1$. Prove that all the numbers are $+1$.

2017 Baltic Way, 18

Tags: number theory , abstract algebra

Let $p>3$ be a prime and let $a_1,a_2,...,a_{\frac{p-1}{2}}$ be a permutation of $1,2,...,\frac{p-1}{2}$. For which $p$ is it always possible to determine the sequence $a_1,a_2,...,a_{\frac{p-1}{2}}$ if it for all $i,j\in\{1,2,...,\frac{p-1}{2}\}$ with $i\not=j$ the residue of $a_ia_j$ modulo $p$ is known?

2020/2021 Tournament of Towns, P2

Tags: number theory

Let us say that a pair of distinct positive integers is nice if their arithmetic mean and their geometric mean are both integer. Is it true that for each nice pair there is another nice pair with the same arithmetic mean? (The pairs $(a, b)$ and $(b, a)$ are considered to be the same pair.) [i]Boris Frenkin[/i]

2004 AMC 10, 16

Tags: induction

The $ 5\times 5$ grid shown contains a collection of squares with sizes from $ 1\times 1$ to $ 5\times 5$. How many of these squares contain the black center square? [asy]unitsize(6mm); defaultpen(linewidth(.8pt)); for(int i=0; i<=5; ++i) { draw((0,i)--(5,i)); draw((i,0)--(i,5)); } fill((2,2)--(2,3)--(3,3)--(3,2)--cycle);[/asy]$ \textbf{(A)}\ 12\qquad \textbf{(B)}\ 15\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 19\qquad \textbf{(E)}\ 20$

2020 Ukrainian Geometry Olympiad - December, 5

Tags: geometry , circumcircle , altitude , angle

In an acute triangle $ABC$ with an angle $\angle ACB =75^o$, altitudes $AA_3,BB_3$ intersect the circumscribed circle at points $A_1,B_1$ respectively. On the lines $BC$ and $CA$ select points $A_2$ and $B_2$ respectively suchthat the line $B_1B_2$ is parallel to the line $BC$ and the line $A_1A_2$ is parallel to the line $AC$ . Let $M$ be the midpoint of the segment $A_2B_2$. Find in degrees the measure of the angle $\angle B_3MA_3$.

2013 AMC 8, 8

Tags: probability , complementary counting

A fair coin is tossed 3 times. What is the probability of at least two consecutive heads? $\textbf{(A)}\ \frac18 \qquad \textbf{(B)}\ \frac14 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac34$

2011 Switzerland - Final Round, 8

Tags: parallelogram , ratio , similar triangles , power of a point , geometry

Let $ABCD$ be a parallelogram and $H$ the Orthocentre of $\triangle{ABC}$. The line parallel to $AB$ through $H$ intersects $BC$ at $P$ and $AD$ at $Q$ while the line parallel to $BC$ through $H$ intersects $AB$ at $R$ and $CD$ at $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic. [i](Swiss Mathematical Olympiad 2011, Final round, problem 8)[/i]

2023 Euler Olympiad, Round 1, 5

Tags: euler , combinatorics

Consider a 3 × 4 rectangular table where each cell can be colored using one of three available colors. Determine the number of different ways the table can be colored such that no two cells sharing a common side have the same color. It is not necessary to use all three colors in each coloring. [i]Proposed by Prudencio Guerrero Fernández, Cuba[/i]

2023 USA TSTST, 4

Tags: graph theory , binomial

Let $n\ge 3$ be an integer and let $K_n$ be the complete graph on $n$ vertices. Each edge of $K_n$ is colored either red, green, or blue. Let $A$ denote the number of triangles in $K_n$ with all edges of the same color, and let $B$ denote the number of triangles in $K_n$ with all edges of different colors. Prove \[ B\le 2A+\frac{n(n-1)}{3}.\] (The [i]complete graph[/i] on $n$ vertices is the graph on $n$ vertices with $\tbinom n2$ edges, with exactly one edge joining every pair of vertices. A [i]triangle[/i] consists of the set of $\tbinom 32=3$ edges between $3$ of these $n$ vertices.) [i]Proposed by Ankan Bhattacharya[/i]

2005 Estonia National Olympiad, 1

Tags: geometry

Seven brothers bought a round pizza and cut it $12$ piece as shown in the figure. Of the six elder brothers, each took one piece of the shape of an equilateral triangle, the remaining $6$ edge pieces by the older brothers did not want, was given to the youngest brother. Did the youngest brother get it more or less a seal than his every older brother? [img]https://cdn.artofproblemsolving.com/attachments/0/7/2efaec7dab171b8bb239dc8eb282947a5c44b0.png[/img]

Russian TST 2015, P3

Tags: combinatorics , geometry

Let $0<\alpha<1$ be a fixed number. On a lake shaped like a convex polygon, at some point there is a duck and at another point a water lily grows. If the duck is at point $X{}$, then in one move it can swim towards one of the vertices $Y$ of the polygon a distance equal to a $\alpha\cdot XY$. Find all $\alpha{}$ for which, regardless of the shape of the lake and the initial positions of the duck and the lily, after a sequence of adequate moves, the distance between the duck and the lily will be at most one meter.

2013 IFYM, Sozopol, 4

Tags: number theory , algebra , sum

Let $a_i$, $i=1,2,...,n$ be non-negative real numbers and $\sum_{i=1}^na_i =1$. Find $\max S=\sum_{i\mid j}a_i a_j $.

Taiwan TST 2015 Round 1, 1

Tags: inequalities , algebra

Let $a,b,c,d$ be any real numbers such that $a+b+c+d=0$, prove that \[1296(a^7+b^7+c^7+d^7)^2\le637(a^2+b^2+c^2+d^2)^7\]

2017 Turkey EGMO TST, 3

Tags: algebra , inequalities

For all positive real numbers $x,y,z$ satisfying the inequality $$\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}\leq 3,$$ prove that $$\frac{x^2}{y^3}+\frac{y^2}{z^3}+\frac{z^2}{x^3}\geq \frac{x}{y}+\frac{y}{z}+\frac{z}{x}.$$

1986 IMO Longlists, 39

Tags: floor function , combinatorics

Let $S$ be a $k$-element set. [i](a)[/i] Find the number of mappings $f : S \to S$ such that \[\text{(i) } f(x) \neq x \text{ for } x \in S, \quad \text{(ii) } f(f(x)) = x \text{ for }x \in S.\] [i](b)[/i] The same with the condition $\text{(i)}$ left out.

2021 Azerbaijan IMO TST, 2

Tags: coloring , quadrilateral , partition , combinatorics

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

1987 Tournament Of Towns, (156) 7

Tags: coloring , combinatorics , combinatorial geometry

Three triangles (blue, green and red) have a common interior point $M$. Prove that it is possible to choose one vertex from each triangle so that point $M$ is located inside the triangle formed by these selected vertices. (Imre Barani, Hungary)

2015 Costa Rica - Final Round, 3

Tags: algebra , fractional part , functional equation , functional

Indicate (justifying your answer) if there exists a function $f: R \to R$ such that for all $x \in R$ fulfills that i) $\{f(x))\} \sin^2 x + \{x\} cos (f(x)) cosx =f (x)$ ii) $f (f(x)) = f(x)$ where $\{m\}$ denotes the fractional part of $m$. That is, $\{2.657\} = 0.657$, and $\{-1.75\} = 0.25$.

2014 ASDAN Math Tournament, 1

Tags:

Points $A$, $B$, $C$, and $D$ lie in the plane with $AB=AD=7$, $CB=CD=4$, and $BD=6$. Compute the sum of all possible values of $AC$.

2014 ELMO Shortlist, 9

Tags: number theory , logarithm

Let $d$ be a positive integer and let $\varepsilon$ be any positive real. Prove that for all sufficiently large primes $p$ with $\gcd(p-1,d) \neq 1$, there exists an positive integer less than $p^r$ which is not a $d$th power modulo $p$, where $r$ is defined by \[ \log r = \varepsilon - \frac{1}{\gcd(d,p-1)}. \][i]Proposed by Shashwat Kishore[/i]