Olympiad problems

1984 AMC 12/AHSME, 13

Tags:

$\frac{2 \sqrt 6}{\sqrt 2 + \sqrt 3 + \sqrt 5}$ equals A. $\sqrt 2 + \sqrt 3 - \sqrt 5$ B. $4 - \sqrt 2 - \sqrt 3$ C. $\sqrt 2 + \sqrt 3 + \sqrt 6 - 5$ D. $\frac{1}{2} (\sqrt 2 + \sqrt 5 - \sqrt 3)$ E. $\frac{1}{3} (\sqrt 3 + \sqrt 5 - \sqrt 2)$

2005 Romania National Olympiad, 3

Let the $ABCA'B'C'$ be a regular prism. The points $M$ and $N$ are the midpoints of the sides $BB'$, respectively $BC$, and the angle between the lines $AB'$ and $BC'$ is of $60^\circ$. Let $O$ and $P$ be the intersection of the lines $A'C$ and $AC'$, with respectively $B'C$ and $C'N$. a) Prove that $AC' \perp (OPM)$; b) Find the measure of the angle between the line $AP$ and the plane $(OPM)$. [i]Mircea Fianu[/i]

2005 China Western Mathematical Olympiad, 3

Tags: number theory , prime number , relatively prime

Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.

2014 Peru Iberoamerican Team Selection Test, P3

Tags: number theory

A positive integer $n$ is called $special$ if there exist integers $a > 1$ and $b > 1$ such that $n=a^b + b$. Is there a set of $2014$ consecutive positive integers that contains exactly $2012$ $special$ numbers?

2021 BMT, 10

Tags: combinatorics , algebra

Compute the number of nonempty subsets $S$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ such that $\frac{\max \,\, S + \min \,\,S}{2}$ is an element of $S$.

1984 IMO Longlists, 21

Tags: probability , limit , combinatorics

$(1)$ Start with $a$ white balls and $b$ black balls. $(2)$ Draw one ball at random. $(3)$ If the ball is white, then stop. Otherwise, add two black balls and go to step $2$. Let $S$ be the number of draws before the process terminates. For the cases $a = b = 1$ and $a = b = 2$ only, find $a_n = P(S = n), b_n = P(S \le n), \lim_{n\to\infty} b_n$, and the expectation value of the number of balls drawn: $E(S) =\displaystyle\sum_{n\ge1} na_n.$

2021 Israel National Olympiad, P3

Tags: geometric inequalities , inequalities , geometry

Let $ABC$ be a triangle. Let $X$ be the tangency point of the incircle with $BC$. Let $Y$ be the second intersection point of segment $AX$ with the incircle. Prove that \[AX+AY+BC>AB+AC\]

2000 Harvard-MIT Mathematics Tournament, 5

Tags: geometry

Side $\overline{AB} = 3$. $\vartriangle ABF$ is an equilateral triangle. Side $\overline{DE} =\overline{ AB} = \overline{AF} = \overline{GE}$, $\angle FED = 60^o$, $FG = 1$. Calculate the area of $ABCDE$. [img]https://cdn.artofproblemsolving.com/attachments/e/9/0ac1a88b4a83cdf3d562af0ce11b5ddbc5b8bc.png[/img]

2022 Grosman Mathematical Olympiad, P6

Tags: coloring , combinatorics

In the following image is a beehive lattice of hexagons. Each cell is colored in one of three colors Red, Blue, or Green (denoted by the letters $R, B, G$). The frame is colored according to the instructions in the image, and the rest of the hexagons are colored however one wants. Is there necessarily a point where three hexagons of different colors meet?

1995 Denmark MO - Mohr Contest, 3

Tags: ratio , geometry , median

From the vertex $C$ in triangle $ABC$, draw a straight line that bisects the median from $A$. In what ratio does this line divide the segment $AB$? [img]https://1.bp.blogspot.com/-SxWIQ12DIvs/XzcJv5xoV0I/AAAAAAAAMY4/Ezfe8bd7W-Mfp2Qi4qE_gppbh9Fzvb4XwCLcBGAsYHQ/s0/1995%2BMohr%2Bp3.png[/img]

2019 Slovenia Team Selection Test, 3

Tags: geometry

Let $ABC$ be a non-right triangle and let $M$ be the midpoint of $BC$. Let $D$ be a point on $AM$ (D≠A, D≠M). Let ω1 be a circle through $D$ that intersects $BC$ at $B$ and let ω2 be a circle through $D$ that intersects $BC$ at $C$. Let $AB$ intersect ω1 at $B$ and $E$, and let $AC$ intersect ω2 at $C$ and $F$. Prove, that the tangent on ω1 at $E$ and the tangent on ω2 at $F$ intersect on $AM$.

2008 Tournament Of Towns, 4

Tags: geometry , angle

Each of Peter and Basil draws a convex quadrilateral with no parallel sides. The angles between a diagonal and the four sides of Peter's quadrilateral are $\alpha, \alpha, \beta$ and $\gamma$ in some order. The angles between a diagonal and the four sides of Basil's quadrilateral are also $\alpha, \alpha, \beta$ and $\gamma$ in some order. Prove that the acute angle between the diagonals of Peter's quadrilateral is equal to the acute angle between the diagonals of Basil's quadrilateral.

2018 Thailand TSTST, 1

Tags: functional equation , algebra

Let $P$ be a given quadratic polynomial. Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that $$f(x+y)=f(x)+f(y)\text{ and } f(P(x))=f(x)\text{ for all }x,y\in\mathbb{R}.$$

1971 IMO, 1

Tags: 3d geometry , tetrahedron , trigonometry , distance , geometry

All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$. [b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length; [b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.

1981 AMC 12/AHSME, 12

Tags: inequalities

If $p$, $q$ and $M$ are positive numbers and $q<100$, then the number obtained by increasing $M$ by $p\%$ and decreasing the result by $q\%$ exceeds $M$ if and only if $\text{(A)}\ p>q ~~ \text{(B)}\ p>\frac{q}{100-q} ~~ \text{(C)}\ p>\frac{q}{1-q} ~~ \text{(D)}\ p>\frac{100q}{100+q} ~~ \text{(E)}\ p>\frac{100q}{100-q}$

1985 National High School Mathematics League, 8

Tags:

The number of nonnegative solutions to the equation $2x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8+x_9+x_{10}=3$ is________.

2003 Bundeswettbewerb Mathematik, 2

Tags: algebra

The sequence $\{a_1,a_2,\ldots\}$ is recursively defined by $a_1 = 1$, $a_2 = 1$, $a_3 = 2$, and \[ a_{n+3} = \frac 1{a_n}\cdot (a_{n+1}a_{n+2}+7), \ \forall \ n > 0. \] Prove that all elements of the sequence are integers.

1989 Putnam, A2

Tags: integration

Evaluate $\int^{a}_{0}{\int^{b}_{0}{e^{max(b^{2}x^{2},a^{2}y^{2})}dy dx}}$

1999 Gauss, 5

Tags: gauss

Which one of the following gives an odd integer? $\textbf{(A)}\ 6^2 \qquad \textbf{(B)}\ 23-17 \qquad \textbf{(C)}\ 9\times24 \qquad \textbf{(D)}\ 96\div8 \qquad \textbf{(E)}\ 9\times41$

May Olympiad L1 - geometry, 1995.5

Tags: geometry , algebra , rectangle

A tortoise walks $60$ meters per hour and a lizard walks at $240$ meters per hour. There is a rectangle $ABCD$ where $AB =60$ and $AD =120$. Both start from the vertex $A$ and in the same direction ($A \to B \to D \to A$), crossing the edge of the rectangle. The lizard has the habit of advancing two consecutive sides of the rectangle, turning to go back one, turning to go forward two, turning to go back one and so on. How many times and in what places do the tortoise and the lizard meet when the tortoise completes its third turn?

2018 Bosnia and Herzegovina Junior BMO TST, 1

Tags: combinatorics , counting

Students are in classroom with $n$ rows. In each row there are $m$ tables. It's given that $m,n \geq 3$. At each table there is exactly one student. We call neighbours of the student students sitting one place right, left to him, in front of him and behind him. Each student shook hands with his neighbours. In the end there were $252$ handshakes. How many students were in the classroom?

2022 CMIMC, 2.1

Tags: algebra , number theory

Alice and Bob live on the same road. At time $t$, they both decide to walk to each other's houses at constant speed. However, they were busy thinking about math so that they didn't realize passing each other. Alice arrived at Bob's house at $3:19\text{pm}$, and Bob arrived at Alice's house at $3:29\text{pm}$. Charlie, who was driving by, noted that Alice and Bob passed each other at $3:11\text{pm}$. Find the difference in minutes between the time Alice and Bob left their own houses and noon on that day. [i]Proposed by Kevin You[/i]

2023 China Northern MO, 3

Tags: equation , floor function , algebra

Find all solutions of the equation $$sin\pi \sqrt x+cos\pi \sqrt x=(-1)^{\lfloor \sqrt x \rfloor }$$

2003 BAMO, 3

Tags: lattice points , combinatorial geometry , integer , area , geometry

A lattice point is a point $(x, y)$ with both $x$ and $y$ integers. Find, with proof, the smallest $n$ such that every set of $n$ lattice points contains three points that are the vertices of a triangle with integer area. (The triangle may be degenerate, in other words, the three points may lie on a straight line and hence form a triangle with area zero.)

1970 AMC 12/AHSME, 9

Tags: ratio

Points $P$ and $Q$ are on line segment $AB$, and both points are on the same side of the midpoint of $AB$. Point $P$ divides $AB$ in the ratio $2:3$ and $Q$ divides $AB$ in the ratio $3:4$. If $PQ=2$, then the length of segment $AB$ is $\textbf{(A) }12\qquad\textbf{(B) }28\qquad\textbf{(C) }70\qquad\textbf{(D) }75\qquad \textbf{(E) }105$