Olympiad problems

2013 Stanford Mathematics Tournament, 8

Tags: geometry

Let equilateral triangle $ABC$ with side length $6$ be inscribed in a circle and let $P$ be on arc $AC$ such that $AP \cdot P C = 10$. Find the length of $BP$.

2019 Brazil Team Selection Test, 1

Tags: geometry , orthocenter , circumcircle , equilateral triangle , collinear

Let $ABC$ be an acute triangle, with $\angle A > 60^\circ$, and let $H$ be it's orthocenter. Let $M$ and $N$ be points on $AB$ and $AC$, respectively, such that $\angle HMB = \angle HNC = 60^\circ$. Also, let $O$ be the circuncenter of $HMN$ and $D$ be a point on the semiplane determined by $BC$ that contains $A$ in such a way that $DBC$ is equilateral. Prove that $H$, $O$ and $D$ are collinear.

2003 Mexico National Olympiad, 2

Tags: parallelogram , power of a point , radical axis , geometry

$A, B, C$ are collinear with $B$ betweeen $A$ and $C$. $K_{1}$ is the circle with diameter $AB$, and $K_{2}$ is the circle with diameter $BC$. Another circle touches $AC$ at $B$ and meets $K_{1}$ again at $P$ and $K_{2}$ again at $Q$. The line $PQ$ meets $K_{1}$ again at $R$ and $K_{2}$ again at $S$. Show that the lines $AR$ and $CS$ meet on the perpendicular to $AC$ at $B$.

2009 HMNT, 1-3

Tags: geometry

[u]Down the Infinite Corridor[/u] Consider an isosceles triangle $T$ with base $10$ and height $12$. Define a sequence $\omega_1$, $\omega_2$,$...$of circles such that $\omega_1$ is the incircle of $T$ and $\omega_{i+1}$ is tangent to $\omega_i$ and both legs of the isosceles triangle for $i > 1$. [b]p1.[/b] Find the radius of $\omega_1$. [b]p2.[/b] Find the ratio of the radius of $\omega_{i+1}$ to the radius of $\omega_i$. [b]p3.[/b] Find the total area contained in all the circles.

2011 Hanoi Open Mathematics Competitions, 7

Tags: algebra , system of equations

Find all pairs $(x, y)$ of real numbers satisfying the system : $\begin{cases} x + y = 3 \\ x^4 - y^4 = 8x - y \end{cases}$

2010 LMT, 10

Tags:

How many integers less than $2502$ are equal to the square of a prime number?

1978 Dutch Mathematical Olympiad, 4

Tags: combinatorics , combinatorial geometry , lattice

On the plane with a rectangular coordinate system, a set of infinitely many rectangles is given. Every rectangle has the origin as one of its vertices. The sides of all rectangles are parallel to the coordinate axes, and all sides have integer lengths. Prove that there are at least two rectangles in the set, one of which completely covers the other.

2024 HMNT, 7

Tags:

Let triangle $ABC$ have $AB = 5, BC = 8,$ and $\angle ABC = 60^\circ.$ A circle $\omega$ tangent to segments $AB$ and $BC$ intersects segment $CA$ at points $X$ and $Y$ such that points $C, Y , X,$ and $A$ lie along $CA$ in this order. If $\omega$ is tangent to $AB$ at point $Z$ and $ZY \parallel BC,$ compute the radius of $\omega.$

PEN B Problems, 6

Tags: primitive root , modular arithmetic , relatively prime , number theory

Suppose that $m$ does not have a primitive root. Show that \[a^{ \frac{\phi(m)}{2}}\equiv 1 \; \pmod{m}\] for every $a$ relatively prime $m$.

2018 Latvia Baltic Way TST, P3

Tags: sequence , algebra

Let $a_1,a_2,...$ be an infinite sequence of integers that satisfies $a_{n+2}=a_{n+1}+a_n$ for all $n \ge 1$. There exists a positive integer $k$ such that $a_k=a_{k+2018}$. Prove that there exists a term of the sequence which is equal to zero.

2015 ASDAN Math Tournament, 5

Tags: geometry test

The eight corners of a cube are cut off, yielding a polyhedron with $6$ octagonal faces and $8$ triangular faces. Given that all polyhedron's edges have length $2$, compute the volume of the polyhedron.

2004 Federal Competition For Advanced Students, P2, 4

Tags: perfect square , sequence , recurrence relation , number theory

Show that there is an infinite sequence $a_1,a_2,...$ of natural numbers such that $a^2_1+a^2_2+ ...+a^2_N$ is a perfect square for all $N$. Give a recurrent formula for one such sequence.

1989 IMO Shortlist, 2

Tags: algebra , area , rectangle , equation , geometry

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?

1987 IMO Shortlist, 4

Tags: geometry , geometric inequality , 3d geometry , polyhedron

Let $ABCDEFGH$ be a parallelepiped with $AE \parallel BF \parallel CG \parallel DH$. Prove the inequality \[AF + AH + AC \leq AB + AD + AE + AG.\] In what cases does equality hold? [i]Proposed by France.[/i]

2023 Flanders Math Olympiad, 4

Tags: combinatorics

There are $12$ mathematicians living in a village, each of whom belongs to the $\sqrt2$-clan or belong to the $\pi$-clan. Moreover every mathematician's birthday is in a different month and every mathematician has an odd number of friends among them the mathematicians. We agree that if mathematician $A$ is a friend of mathematician $B$, then so is $B$ is a friend of $A$. On his birthday, every mathematician looks at which clan the majority of his friends belong to, and decides to join that clan until his next birthday. Prove that the mathematicians no longer change clans after a certain point.

2024 Austrian MO National Competition, 1

Tags: algebra , inequalities

Determine the smallest real constant $C$ such that the inequality \[(X+Y)^2(X^2+Y^2+C)+(1-XY)^2 \ge 0\] holds for all real numbers $X$ and $Y$. For which values of $X$ and $Y$ does equality hold for this smallest constant $C$? [i](Walther Janous)[/i]

2011 Morocco National Olympiad, 1

Tags: linear algebra , algebra , system of equations

Solve the following equation in $\mathbb{R}^+$ : \[\left\{\begin{matrix} \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2010\\ x+y+z=\frac{3}{670} \end{matrix}\right.\]

2015 Purple Comet Problems, 18

Tags:

Deﬁne the determinant $D_1$ = $|1|$, the determinant $D_2$ = $|1 1|$ $|1 3|$ , and the determinant $D_3=$ |1 1 1| |1 3 3| |1 3 5| . In general, for positive integer n, let the determinant $D_n$ have 1s in every position of its ﬁrst row and ﬁrst column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that $D_n$ $\geq$ 2015.

Today's calculation of integrals, 863

Tags: calculus , integration , trigonometry , limit , derivative , inequalities , calculus computations

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

2005 Cuba MO, 1

Tags: geometry , perimeter , equal area

Determine all the quadrilaterals that can be divided by a diagonal into two triangles of equal area and equal perimeter.

2004 USA Team Selection Test, 6

Tags: function , modular arithmetic , induction , algebra

Define the function $f: \mathbb N \cup \{0\} \to \mathbb{Q}$ as follows: $f(0) = 0$ and \[ f(3n+k) = -\frac{3f(n)}{2} + k , \] for $k = 0, 1, 2$. Show that $f$ is one-to-one and determine the range of $f$.

PEN N Problems, 12

Tags: modular arithmetic , more sequences

The sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{n}= 1+2^{2}+3^{3}+\cdots+n^{n}.\] Prove that there are infinitely many $n$ such that $a_{n}$ is composite.

1999 Harvard-MIT Mathematics Tournament, 5

Tags:

You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out you must open all the doors and disarm all the mines. In the room is a panel with $3$ buttons, which conveniently contains an instruction manual. The red button arms a mine, the yellow button disarms two mines and closes a door, and the green button opens two doors. Initially $3$ doors are closed and $3$ mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?

1997 Brazil Team Selection Test, Problem 5

Tags: geometry , geometric inequality , inequalities , geometric inequalities , triangle , area

Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly. (a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$. (b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.

2016 India National Olympiad, P2

Tags: algebra , inequalities

For positive real numbers $a,b,c$ which of the following statements necessarily implies $a=b=c$: (I) $a(b^3+c^3)=b(c^3+a^3)=c(a^3+b^3)$, (II) $a(a^3+b^3)=b(b^3+c^3)=c(c^3+a^3)$ ? Justify your answer.