Olympiad problems

2013 Brazil Team Selection Test, 3

In a $999 \times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $(C_1,C_2,C_3)$ of cells, the first two in the same row and the last two in the same column, with $C_1,C_3$ white and $C_2$ red. Find the maximum value $T$ can attain. [i]Proposed by Merlijn Staps, The Netherlands[/i]

2014 Contests, 3

Tags: real analysis

Consider $f(x)=x^4+ax^3+bx^2+cx+d\; (a,b,c,d\in\mathbb{R})$. It is known that $f$ intersects X-axis in at least $3$ (distinct) points. Show either $f$ has $4$ $\mathbf{distinct}$ real roots or it has $3$ $\mathbf{distinct}$ real roots and one of them is a point of local maxima or minima.

2021 Brazil National Olympiad, 2

Tags: combinatorics , board

Let $n$ be a positive integer. On a $2 \times 3 n$ board, we mark some squares, so that any square (marked or not) is adjacent to at most two other distinct marked squares (two squares are adjacent when they are distinct and have at least one vertex in common, i.e. they are horizontal, vertical or diagonal neighbors; a square is not adjacent to itself). (a) What is the greatest possible number of marked square? (b) For this maximum number, in how many ways can we mark the squares? configurations that can be achieved through rotation or reflection are considered distinct.

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.