Olympiad problems

2002 Vietnam National Olympiad, 1

Solve the equation $ \sqrt{4 \minus{} 3\sqrt{10 \minus{} 3x}} \equal{} x \minus{} 2$.

2008 Tournament Of Towns, 3

Tags: geometry , perimeter , perpendicular bisector , geometry unsolved

A $30$-gon $A_1A_2\cdots A_{30}$ is inscribed in a circle of radius $2$. Prove that one can choose a point $B_k$ on the arc $A_kA_{k+1}$ for $1 \leq k \leq 29$ and a point $B_{30}$ on the arc $A_{30}A_1$, such that the numerical value of the area of the $60$-gon $A_1B_1A_2B_2 \dots A_{30}B_{30}$ is equal to the numerical value of the perimeter of the original $30$-gon.

2017 CMIMC Combinatorics, 3

Tags: 2017 , combinatorics

Annie stands at one vertex of a regular hexagon. Every second, she moves independently to one of the two vertices adjacent to her, each with equal probability. Determine the probability that she is at her starting position after ten seconds.

2023 Indonesia TST, 1

Tags: combinatorics , Integer sequence , ISL 2022

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

2014 Contests, 3a

Tags: analytic geometry , combinatorics proposed , combinatorics

A grasshopper is jumping about in a grid. From the point with coordinates $(a, b)$ it can jump to either $(a + 1, b),(a + 2, b),(a + 1, b + 1),(a, b + 2)$ or $(a, b + 1)$. In how many ways can it reach the line $x + y = 2014?$ Where the grasshopper starts in $(0, 0)$.

2024-25 IOQM India, 22

Tags: geometry , quadratic formula , trigonometry

In a triangle $ABC$, $\angle BAC = 90^{\circ}$. Let $D$ be the point on $BC$ such that $AB + BD = AC + CD$. Suppose $BD : DC = 2:1$. if $\frac{AC}{AB} = \frac{m + \sqrt{p}}{n}$, Where $m,n$ are relatively prime positive integers and $p$ is a prime number, determine the value of $m+n+p$.

2006 Baltic Way, 19

Tags: modular arithmetic , induction , number theory proposed , number theory

Does there exist a sequence $a_1,a_2,a_3,\ldots $ of positive integers such that the sum of every $n$ consecutive elements is divisible by $n^2$ for every positive integer $n$?

2016 Mexico National Olmypiad, 3

Tags: inequalities , floor function , algebra , national olympiad

Find the minimum real $x$ that satisfies $$\lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots$$

1931 Eotvos Mathematical Competition, 3

Tags: geometry , inequalities , geometric inequality

Let $A$ and $B$ be two given points, distance $1 $ apart. Determine a point $P$ on the line $AB$ such that $$\frac{1}{1 + AP}+\frac{1}{1 + BP}$$ is a maximum.

2016 Saudi Arabia IMO TST, 3

Tags: combinatorics , number theory , divisible

Let $n \ge 4$ be a positive integer and there exist $n$ positive integers that are arranged on a circle such that: $\bullet$ The product of each pair of two non-adjacent numbers is divisible by $2015 \cdot 2016$. $\bullet$ The product of each pair of two adjacent numbers is not divisible by $2015 \cdot 2016$. Find the maximum value of $n$

2015 India Regional MathematicaI Olympiad, 1

Tags: geometry , circumcircle , incenter , right angle , cyclic quadrilateral

In a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ intersect at $X$. Let the circumcircles of triangles $AXD$ and $BXC$ intersect again at $Y$ . If $X$ is the incentre of triangle $ABY$ , show that $\angle CAD = 90^o$.

2024 Mozambique National Olympiad, P5

Tags: algebra , number theory , Diophantine equation

Find all pairs of positive integers $x,y$ such that $\frac{4}{x}+\frac{2}{y}=1$

2025 AMC 8, 10

Tags: AMC 8 , 2025 AMC 8

In the figure below, $ABCD$ is a rectangle with sides of length $AB = 5$ inches and $AD = 3$ inches. Rectangle $ABCD$ is rotated $90^{\circ}$ clockwise about the midpoint of side $\overline{DC}$ to give a second rectangle. What is the total area, in square inches, covered by the two overlapping rectangles? [img]https://i.imgur.com/NyhZpL6.png[/img] $\textbf{(A) }21 \qquad\textbf{(B) }22.25 \qquad\textbf{(C) }23\qquad\textbf{(D) }23.75 \qquad\textbf{(E) }25$

2018 India IMO Training Camp, 2

Tags: IMO Shortlist , function , algebra

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

2022 Philippine MO, 4

Tags: geometry , incenter , exterior angle , angle bisector

Let $\triangle ABC$ have incenter $I$ and centroid $G$. Suppose that $P_A$ is the foot of the perpendicular from $C$ to the exterior angle bisector of $B$, and $Q_A$ is the foot of the perpendicular from $B$ to the exterior angle bisector of $C$. Define $P_B$, $P_C$, $Q_B$, and $Q_C$ similarly. Show that $P_A, P_B, P_C, Q_A, Q_B,$ and $Q_C$ lie on a circle whose center is on line $IG$.

1991 Arnold's Trivium, 47

Tags:

Map the exterior of the disc conformally onto the exterior of a given ellipse.

Durer Math Competition CD 1st Round - geometry, 2017.D+2

Tags: geometry , trapezoid , Durer , collinear

Let the trapezoids $A_iB_iC_iD_i$ ($i = 1, 2, 3$) be similar and have the same clockwise direction. Their angles at $A_i$ and $B_i$ are $60^o$ and the sides $A_1B_1$, $B_2C_2$ and $A_3D_3$ are parallel. The lines $B_iD_{i+1}$ and $C_iA_{i+1}$ intersect at the point $P_i$ (the indices are understood cyclically, i.e. $A_4 = A_1$ and $D_4 = D_1$). Prove that the points $P_1$, $P_2$ and $P_3$ lie on a line.

2021 Saudi Arabia Training Tests, 10

Tags: geometry , equal segments , tangent circles

Let $AB$ be a chord of the circle $(O)$. Denote M as the midpoint of the minor arc $AB$. A circle $(O')$ tangent to segment $AB$ and internally tangent to $(O)$. A line passes through $M$, perpendicular to $O'A$, $O'B$ and cuts $AB$ respectively at $C, D$. Prove that $AB = 2CD$.

LMT Accuracy Rounds, 2023 S5

Tags: number theory

Let $$N = \sum^{512}_{i=0}i {512 \choose i}.$$ What is the greatest integer $a$ such that $2^a$ is a divisor of $N$?

1967 IMO Shortlist, 5

Tags: geometry , 3D geometry , sphere , polyhedron , IMO Shortlist , IMO Longlist

Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.

2020 Centroamerican and Caribbean Math Olympiad, 3

Tags: algebra , functional equation , function

Find all the functions $f: \mathbb{Z}\to \mathbb{Z}$ satisfying the following property: if $a$, $b$ and $c$ are integers such that $a+b+c=0$, then $$f(a)+f(b)+f(c)=a^2+b^2+c^2.$$

2003 All-Russian Olympiad, 1

Tags: graph theory , combinatorics unsolved , combinatorics

There are $N$ cities in a country. Any two of them are connected either by a road or by an airway. A tourist wants to visit every city exactly once and return to the city at which he started the trip. Prove that he can choose a starting city and make a path, changing means of transportation at most once.

1975 Bundeswettbewerb Mathematik, 4

Tags: combinatorics , graph , cycle

In the country of Sikinia there are finitely many cities. From each city, exactly three roads go out and each road goes to another Sikinian city. A tourist starts a trip from city $A$ and drives according to the following rule: he turns left at the first city, then right at the next city, and so on, alternately. Show that he will eventually return to $A.$

1992 IberoAmerican, 3

Tags: geometry , circumcircle , trigonometry , inradius , inequalities , incenter , geometry proposed

Let $ABC$ be an equilateral triangle of sidelength 2 and let $\omega$ be its incircle. a) Show that for every point $P$ on $\omega$ the sum of the squares of its distances to $A$, $B$, $C$ is 5. b) Show that for every point $P$ on $\omega$ it is possible to construct a triangle of sidelengths $AP$, $BP$, $CP$. Also, the area of such triangle is $\frac{\sqrt{3}}{4}$.

1984 AMC 12/AHSME, 5

Tags:

The largest integer $n$ for which $n^{200} < 5^{300}$ is $\textbf{(A) }8\qquad \textbf{(B) }9\qquad \textbf{(C) }10\qquad \textbf{(D) }11\qquad \textbf{(E) }12$