Olympiad problems

1988 Austrian-Polish Competition, 7

Each side of a regular octagon is colored blue or yellow. In each step, the sides are simultaneously recolored as follows: if the two neighbors of a side have different colors, the side will be recolored blue, otherwise it will be recolored yellow. Show that after a finite number of moves all sides will be colored yellow. What is the least value of the number $N$ of moves that always lead to all sides being yellow?

1996 All-Russian Olympiad Regional Round, 9.1

Tags: algebra , trinomial

Find all pairs of square trinomials $x^2 + ax + b$, $ x^2 + cx + d$ such that $a$ and $b$ are the roots of the second trinomial, $c$ and $d$ are the roots of the first.

1999 Estonia National Olympiad, 5

Tags: tiling , combinatorics

There is a hole in the roof with dimensions $23 \times 19$ cm. Can August fill the the roof with tiles of dimensions $5 \times 24 \times 30$ cm?

2021 Yasinsky Geometry Olympiad, 3

Tags: equal angles , geometry , equal segments , angle

The segments $AC$ and $BD$ are perpendicular, and $AC$ is twice as large as $BD$ and intersects $BD$ in it in the midpoint. Find the value of the angle $BAD$, if we know that $\angle CAD = \angle CDB$. (Gregory Filippovsky)

1980 AMC 12/AHSME, 16

Tags: geometry , ratio , 3d geometry , tetrahedron , congruent triangles

Four of the eight vertices of a cube are the vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron. $\text{(A)} \ \sqrt 2 \qquad \text{(B)} \ \sqrt 3 \qquad \text{(C)} \ \sqrt{\frac{3}{2}} \qquad \text{(D)} \ \frac{2}{\sqrt{3}} \qquad \text{(E)} \ 2$

2006 Pre-Preparation Course Examination, 1

Tags: function , combinatorics

a) Find the value of $\sum_{n=1}^{\infty}\frac{\phi(n)}{2^n-1}$; b) Show that $\sum_k {m\choose k}{{n+k}\choose m}=\sum_k {m\choose k} {n\choose k} 2^k$ for $m,n\geq 0$; c) Using the identity $(1-x)^{-\frac 12}(1-x)^{-\frac 12}=(1-x)^{-1}$ derive a combinatorial identity! d) Express the value of $\sum (2^{a_1}-1)\ldots (2^{a_k}-1)$ where the sum is over all $2^{n-1}$ ways of choosing $(a_1,a_2,\ldots,a_k)$ such that $a_1+a_2+\ldots +a_k=n$, as a function of some Fibonacci term.

2022 VIASM Summer Challenge, Problem 2

Tags: algebra , inequalities

Let $S$ be the set of real numbers $k$ with the following property: for all set of real numbers $(a,b,c)$ satisfying $ab+bc+ca=1$, we always have the inequality:$$\frac{a}{{\sqrt {{a^2} + ab + {b^2} + k} }} + \frac{b}{{\sqrt {{b^2} + bc + {c^2} + k} }} + \frac{c}{{\sqrt {{c^2} + ca + {a^2} + k} }} \ge \sqrt {\frac{3}{{k + 1}}} .$$ a) Assume that $k\in S$. Prove that: $k\ge 2$. b) Prove that: $2\in S$.

2004 Czech and Slovak Olympiad III A, 5

Tags: circumcircle , geometry

Let $L$ be an arbitrary point on the minor arc $CD$ of the circumcircle of square $ABCD$. Let $K,M,N$ be the intersection points of $AL,CD$; $CL,AD$; and $MK,BC$ respectively. Prove that $B,M,L,N$ are concyclic.

1956 AMC 12/AHSME, 49

Tags:

Triangle $ PAB$ is formed by three tangents to circle $ O$ and $ < APB \equal{} 40^{\circ}$; then angle $ AOB$ equals: $ \textbf{(A)}\ 45^{\circ} \qquad\textbf{(B)}\ 50^{\circ} \qquad\textbf{(C)}\ 55^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 70^{\circ}$

2014 ASDAN Math Tournament, 14

Tags: team test

Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated $2$ seats away from the person at the head of the table.

2019 Purple Comet Problems, 15

Tags: equation , number theory

Let $a, b, c$, and $d$ be prime numbers with $a \le b \le c \le d > 0$. Suppose $a^2 + 2b^2 + c^2 + 2d^2 = 2(ab + bc - cd + da)$. Find $4a + 3b + 2c + d$.

2008 Czech and Slovak Olympiad III A, 1

Tags: algebra

Find all pairs of real numbers $(x,y)$ satisfying: \[x+y^2=y^3,\]\[y+x^2=x^3.\]

1969 AMC 12/AHSME, 22

Tags: geometry , trapezoid

Let $K$ be the measure of the area bounded by the $x$-axis, the line $x=8$, and the curve defined by \[f=\{(x,y)\,|\, y=x\text{ when }0\leq x\leq 5,\,y=2x-5\text{ when }5\leq x\leq 8\}.\] Then $K$ is: $\textbf{(A) }21.5\qquad \textbf{(B) }36.4\qquad \textbf{(C) }36.5\qquad \textbf{(D) }44\qquad$ $\textbf{ (E) }\text{less than 44 but arbitrarily close to it.}$

1993 Bulgaria National Olympiad, 4

Tags: number theory

Find all natural numbers $n > 1$ for which there exists such natural numbers $a_1,a_2,...,a_n$ for which the numbers $\{a_i +a_j | 1 \le i \le j \le n \}$ form a full system modulo $\frac{n(n+1)}{2}$.

2023 VIASM Summer Challenge, Problem 4

Tags: geometry

Let $ABCD$ be a parallelogram and $P$ be an arbitrary point in the plane. Let $O$ be the intersection of two diagonals $AC$ and $BD.$ The circumcircles of triangles $POB$ and $POC$ intersect the circumcircles of triangle $OAD$ at $Q$ and $R,$ respectively $(Q,R \ne O).$ Construct the parallelograms $PQAM$ and $PRDN.$ Prove that: the circumcircle of triangle $MNP$ passes through $O.$ [i]Proposed by Tran Quang Hung ([url=https://artofproblemsolving.com/community/user/68918]buratinogigle[/url])[/i]

1967 IMO Shortlist, 3

Tags: trigonometric equation , trigonometry , number theory , diophantine equation

Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.

MOAA Accuracy Rounds, 2023.3

Tags:

Ms. Raina's math class has 6 students, including the troublemakers Andy and Harry. For a group project, Ms. Raina randomly divides the students into three groups containing 1, 2, and 3 people. The probability that Andy and Harry unfortunately end up in the same group can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andy Xu[/i]

2004 Switzerland - Final Round, 1

Tags: geometry , isosceles , tangent , secant

Let $\Gamma$ be a circle and $P$ a point outside of $\Gamma$ . A tangent from $P$ to the circle intersects it in $A$. Another line through $P$ intersects $\Gamma$ at the points $B$ and $C$. The bisector of $\angle APB$ intersects $AB$ at $D$ and $AC$ at $E$. Prove that the triangle $ADE$ is isosceles.

1953 Putnam, A4

Tags: trigonometry , logarithm

From the identity $$ \int_{0}^{\pi \slash 2} \log \sin 2x \, dx = \int_{0}^{\pi \slash 2} \log \sin x \, dx + \int_{0}^{\pi \slash 2} \log \cos x \, dx +\int_{0}^{\pi \slash 2} \log 2 \, dx, $$ deduce the value of $\int_{0}^{\pi \slash 2} \log \sin x \, dx.$

2022 USAJMO, 4

Tags: coordinate geometry , geometry

Let $ABCD$ be a rhombus, and let $K$ and $L$ be points such that $K$ lies inside the rhombus, $L$ lies outside the rhombus, and $KA = KB = LC = LD$. Prove that there exist points $X$ and $Y$ on lines $AC$ and $BD$ such that $KXLY$ is also a rhombus. [i]Proposed by Ankan Bhattacharya[/i]

1968 Bulgaria National Olympiad, Problem 5

Tags: geometry , 3d geometry , tetrahedron , ratio

The point $M$ is inside the tetrahedron $ABCD$ and the intersection points of the lines $AM,BM,CM$ and $DM$ with the opposite walls are denoted with $A_1,B_1,C_1,D_1$ respectively. It is given also that the ratios $\frac{MA}{MA_1}$, $\frac{MB}{MB_1}$, $\frac{MC}{MC_1}$, and $\frac{MD}{MD_1}$ are equal to the same number $k$. Find all possible values of $k$. [i]K. Petrov[/i]

1990 IMO Longlists, 85

Tags: geometry theorems , geometry

Let $A_1, A_2, \ldots, A_n (n \geq 4)$ be $n$ convex sets in plane. Knowing that every three convex sets have a common point. Prove that there exists a point belonging to all the sets.

2018 Romania National Olympiad, 3

Tags: function , inequalities , calculus , real analysis , integral

Let $f:[a,b] \to \mathbb{R}$ be an integrable function and $(a_n) \subset \mathbb{R}$ such that $a_n \to 0.$ $\textbf{a) }$ If $A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \},$ prove that every open interval of strictly positive real numbers contains elements from $A.$ $\textbf{b) }$ If, for any $n \in \mathbb{N}^*$ and for any $x,y \in [a,b]$ with $|x-y|=a_n,$ the inequality $\left| \int_x^yf(t)dt \right| \leq |x-y|$ is true, prove that $$\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]$$ [i]Nicolae Bourbacut[/i]

2019 IFYM, Sozopol, 2

Tags: geometry

$\Delta ABC$ is a triangle with center $I$ of its inscribed circle and $B_1$ and $C_1$ are feet of its angle bisectors through $B$ and $C$. Let $S$ be the middle point on the arc $\widehat{BAC}$ of the circumscribed circle of $\Delta ABC$ (denoted with $\Omega$) and let $\omega_a$ be the excircle of $\Delta ABC$ opposite to $A$. Let $\omega_a (I_a)$ be tangent to $AB$ and $AC$ in points $D$ and $E$ respectively and $SI\cap \Omega=\{S,P\}$. Let $M$ be the middle point of $DE$ and $N$ be the middle point of $SI$. If $MN\cap AP=K$, prove that $KI_a\perp B_1 C_1$.

2017-2018 SDPC, 6

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle with circumcenter $O$. Let the parallel to $BC$ through $A$ intersect line $BO$ at $B_A$ and $CO$ at $C_A$. Lines $B_AC$ and $BC_A$ intersect at $A'$. Define $B'$ and $C'$ similarly. (a) Prove that the the perpendicular from $A'$ to $BC$, the perpendicular from $B'$ to $AC$, and $C'$ to $AB$ are concurrent. (b) Prove that likes $AA'$, $BB'$, and $CC'$ are concurrent.