Olympiad problems

2003 India Regional Mathematical Olympiad, 1

Tags: trigonometry , geometry

Let $ABC$ be a triangle in which $AB =AC$ and $\angle CAB = 90^{\circ}$. Suppose that $M$ and $N$ are points on the hypotenuse $BC$ such that $BM^2 + CN^2 = MN^2$. Prove that $\angle MAN = 45^{\circ}$.

1967 IMO Longlists, 45

Tags: trigonometry , algebra , trigonometric equation , geometry

[b](i)[/b] Solve the equation: \[ \sin^3(x) + \sin^3\left( \frac{2 \pi}{3} + x\right) + \sin^3\left( \frac{4 \pi}{3} + x\right) + \frac{3}{4} \cos {2x} = 0.\] [b](ii)[/b] Supposing the solutions are in the form of arcs $AB$ with one end at the point $A$, the beginning of the arcs of the trigonometric circle, and $P$ a regular polygon inscribed in the circle with one vertex in $A$, find: 1) The subsets of arcs having the other end in $B$ in one of the vertices of the regular dodecagon. 2) Prove that no solution can have the end $B$ in one of the vertices of polygon $P$ whose number of sides is prime or having factors other than 2 or 3.

2022 Princeton University Math Competition, A1 / B3

Tags: number theory

Find the sum of all prime numbers $p$ such that $p$ divides $$(p^2+p+20)^{p^2+p+2}+4(p^2+p+22)^{p^2-p+4}.$$

2020 Thailand TSTST, 6

Tags: combinatorics , set

A nonempty set $S$ is called [i]Bally[/i] if for every $m\in S$, there are fewer than $\frac{1}{2}m$ elements of $S$ which are less than $m$. Determine the number of Bally subsets of $\{1, 2, . . . , 2020\}$.

STEMS 2024 Math Cat B, P1

Tags: combinatorics

Let $S = \mathbb Z \times \mathbb Z$. A subset $P$ of $S$ is called [i]nice[/i] if [list] [*] $(a, b) \in P \implies (b, a) \in P$ [*] $(a, b)$, $(c, d)\in P \implies (a + c, b - d) \in P$ [/list] Find all $(p, q) \in S$ so that if $(p, q) \in P$ for some [i]nice[/i] set $P$ then $P = S$.

2020 Azerbaijan IMO TST, 2

Tags: combinatorics , induction , local argument

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

2011 USA TSTST, 8

Tags: floor function , number theory

Let $x_0, x_1, \dots , x_{n_0-1}$ be integers, and let $d_1, d_2, \dots, d_k$ be positive integers with $n_0 = d_1 > d_2 > \cdots > d_k$ and $\gcd (d_1, d_2, \dots , d_k) = 1$. For every integer $n \ge n_0$, define \[ x_n = \left\lfloor{\frac{x_{n-d_1} + x_{n-d_2} + \cdots + x_{n-d_k}}{k}}\right\rfloor. \] Show that the sequence $\{x_n\}$ is eventually constant.

2015 Baltic Way, 8

Tags: combinatorics

With inspiration drawn from the rectilinear network of streets in [i]New York[/i] , the [i]Manhattan distance[/i] between two points $(a,b)$ and $(c,d)$ in the plane is defined to be \[|a-c|+|b-d|\] Suppose only two distinct [i]Manhattan distance[/i] occur between all pairs of distinct points of some point set. What is the maximal number of points in such a set?

2021 LMT Fall, Tie

Tags: algebra

Estimate the value of $e^f$ , where $f = e^e$ .

PEN K Problems, 32

Tags: functional equation , function

Find all functions $f: \mathbb{Z}^{2}\to \mathbb{R}^{+}$ such that for all $i, j \in \mathbb{Z}$: \[f(i,j)=\frac{f(i+1, j)+f(i,j+1)+f(i-1,j)+f(i,j-1)}{4}.\]

2005 AMC 12/AHSME, 9

Tags:

On a certain math exam, $ 10 \%$ of the students got 70 points, $ 25 \%$ got 80 points, $ 20 \%$ got 85 points, $ 15 \%$ got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 5$

1953 Putnam, A7

Tags: trigonometry , polynomial , root

Assuming that the roots of $x^3 +px^2 +qx +r=0$ are all real and positive, find the relation between $p,q,r$ which is a necessary and sufficient condition that the roots are the cosines of the angles of a triangle.

1994 Romania TST for IMO, 3:

Tags: number theory

Prove that the sequence $a_n = 3^n- 2^n$ contains no three numbers in geometric progression.

1991 AMC 8, 12

Tags:

If $\frac{2+3+4}{3}=\frac{1990+1991+1992}{N}$, then $N=$ $\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 1990 \qquad \text{(D)}\ 1991 \qquad \text{(E)}\ 1992$

I Soros Olympiad 1994-95 (Rus + Ukr), 10.2

Tags: geometry , geometric inequality

Given a triangle $ABC$ and a point $O$ inside it, it is known that $AB\le BC\le CA$. Prove that $$OA+OB+OC<BC+CA.$$

2018 Yasinsky Geometry Olympiad, 3

Tags: geometry , altitude , midpoint

In the triangle $ABC$, $\angle B = 2 \angle C$, $AD$ is altitude, $M$ is the midpoint of the side $BC$. Prove that $AB = 2DM$.

2010 Princeton University Math Competition, 4

Tags:

Let $S$ be the sum of all real $x$ such that $4^x = x^4$. Find the nearest integer to $S$.

PEN E Problems, 23

Tags: induction , number theory , prime number

Let $p_{1}=2, p_{2}={3}, p_{3}=5, \cdots, p_{n}$ be the first $n$ prime numbers, where $n \ge 3$. Prove that \[\frac{1}{{p_{1}}^{2}}+\frac{1}{{p_{2}}^{2}}+\cdots+\frac{1}{{p_{n}}^{2}}+\frac{1}{p_{1}p_{2}\cdots p_{n}}< \frac{1}{2}.\]

2015 Iran Team Selection Test, 3

Tags: combinatorics

Find the maximum number of rectangles with sides equal to 1 and 2 and parallel to the coordinate axes such that each two have an area equal to 1 in common.

1998 Tournament Of Towns, 3

Tags: tangent , geometry , tangential , circles

Segment $AB$ intersects two equal circles, is parallel to the line joining their centres, and all the points of intersection of the segment and the circles lie between $A$ and $B$. From the point $A$ tangents to the circle nearest to $A$ are drawn, and from the point $B$ tangents to the circle nearest to $B$ are also drawn. It turns out that the quadrilateral formed by the four tangents extended contains both circles. Prove that a circle can be drawn so that it touches all four sides of the quadrilateral. (P Kozhevnikov)

1973 Chisinau City MO, 68

Tags: geometry , equilateral , similar

Inside the triangle $ABC$, point $O$ was chosen so that the triangles $AOB, BOC, COA$ turned out to be similar. Prove that triangle $ABC$ is equilateral.

2005 All-Russian Olympiad Regional Round, 8.6

Tags: geometry , angle bisector

In quadrilateral $ABCD$, angles $A$ and $C$ are equal. Angle bisector of $B$ intersects line $AD$ at point $P$. Perpendicular on $BP$ passing through point $A$ intersects line $BC$ at point $Q$. Prove that the lines $PQ$ and $CD$ are parallel.

2007 Grigore Moisil Intercounty, 4

Tags: function , group theory , complex number

Consider the group $ \{f:\mathbb{C}\setminus\mathbb{Q}\longrightarrow\mathbb{C}\setminus\mathbb{Q} | f\text{ is bijective}\} $ under the composition of functions. Find the order of the smallest subgroup of it that: $ \text{(1)} $ contains the function $ z\mapsto \frac{z-1}{z+1} . $ $ \text{(2)} $ contains the function $ z\mapsto \frac{z-3}{z+1} . $ $ \text{(3)} $ contain both of the above functions.

2010 AMC 12/AHSME, 10

Tags:

The average of the numbers $ 1,2,3,...,98,99$, and $ x$ is $ 100x$. What is $ x$? $ \textbf{(A)}\ \frac{49}{101} \qquad\textbf{(B)}\ \frac{50}{101} \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ \frac{51}{101} \qquad\textbf{(E)}\ \frac{50}{99}$

2002 Germany Team Selection Test, 1

Tags: quadratic , number theory

Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121.