Olympiad problems

Estonia Open Senior - geometry, 2003.1.2

Four rays spread out from point $O$ in a $3$-dimensional space in a way that the angle between every two rays is $a$. Find $\cos a$.

2019 Iran Team Selection Test, 5

Tags: combinatorics , perimeter , combinatorial geometry

Let $P$ be a simple polygon completely in $C$, a circle with radius $1$, such that $P$ does not pass through the center of $C$. The perimeter of $P$ is $36$. Prove that there is a radius of $C$ that intersects $P$ at least $6$ times, or there is a circle which is concentric with $C$ and have at least $6$ common points with $P$. [i]Proposed by Seyed Reza Hosseini[/i]

1987 IMO Longlists, 64

Tags: algebra

Let $r > 1$ be a real number, and let $n$ be the largest integer smaller than $r$. Consider an arbitrary real number $x$ with $0 \leq x \leq \frac{n}{r-1}.$ By a [i]base-$r$ expansion[/i] of $x$ we mean a representation of $x$ in the form \[x=\frac{a_1}{r} + \frac{a_2}{r^2}+\frac{a_3}{r^3}+\cdots\] where the $a_i$ are integers with $0 \leq a_i < r.$ You may assume without proof that every number $x$ with $0 \leq x \leq \frac{n}{r-1}$ has at least one [i]base-$r$ expansion[/i]. Prove that if $r$ is not an integer, then there exists a number $p$, $0 \leq p \leq \frac{n}{r-1}$, which has infinitely many distinct [i]base-$r$ expansions[/i].

Kyiv City MO Seniors 2003+ geometry, 2014.10.4

Tags: equal segments , altitude , geometry

The altitueds $A {{A} _ {1}} $, $B {{B} _ {1}}$ and $C {C} _ 1$ are drawn in the acute triangle $ABC$. . The perpendicular $AK$ is drawn from the vertex $A$ on the line ${{A} _ {1}} {{B} _ {1}}$, and the perpendicular $BL$ is drawn from the vertex $B$ on the line ${{C} _ {1}} {{B} _ {1}}$. Prove that ${{A} _ {1}} K = {{B} _ {1}} L$. (Maria Rozhkova)

2005 Bulgaria Team Selection Test, 4

Tags: inequalities , quadratic , function

Let $a_{i}$ and $b_{i}$, where $i \in \{1,2, \dots, 2005 \}$, be real numbers such that the inequality $(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})$ holds for all $x \in \mathbb{R}$ and all $i \in \{1,2, \dots, 2005 \}$. Find the maximum possible number of positive numbers amongst $a_{i}$ and $b_{i}$, $i \in \{1,2, \dots, 2005 \}$.

1999 Brazil Team Selection Test, Problem 4

Tags: logarithm , triangle inequality , ring theory , function , number theory , induction , inequalities

Let Q+ and Z denote the set of positive rationals and the set of inte- gers, respectively. Find all functions f : Q+ → Z satisfying the following conditions: (i) f(1999) = 1; (ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+; (iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.

2023 Belarusian National Olympiad, 8.6

Tags: geometry

On the side $BC$ of a triangle $ABC$ the midpoint $M$ and arbitrary point $K$ is marked. Lines that pass through $K$ parallel to the sides of the triangle intersect the line $AM$ at $L$ and $N$. Prove that $ML=MN$.

2012 Tournament of Towns, 3

Tags: combinatorics

In a team of guards, each is assigned a different positive integer. For any two guards, the ratio of the two numbers assigned to them is at least $3:1$. A guard assigned the number $n$ is on duty for $n$ days in a row, off duty for $n$ days in a row, back on duty for $n$ days in a row, and so on. The guards need not start their duties on the same day. Is it possible that on any day, at least one in such a team of guards is on duty?

2022 Korea -Final Round, P4

Tags: concyclic , concentric , geometry

Let $ABC$ be a scalene triangle with incenter $I$ and let $AI$ meet the circumcircle of triangle $ABC$ again at $M$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $AB, AC$ at $D, E$, respectively. Let $O$ be the circumcenter of triangle $BDE$ and let $L$ be the intersection of $\omega$ and the altitude from $A$ to $BC$ so that $A$ and $L$ lie on the same side with respect to $DE$. Denote by $\Omega$ a circle centered at $O$ and passing through $L$, and let $AL$ meet $\Omega$ again at $N$. Prove that the lines $LD$ and $MB$ meet on the circumcircle of triangle $LNE$.

2011 IFYM, Sozopol, 8

Tags: number theory , rational number

Let $a$ and $b$ be some rational numbers and there exist $n$, such that $\sqrt[n]{a}+\sqrt[b]{b}$ is also a rational number. Prove that $\sqrt[n]{a}$ is a rational number.

2013 IPhOO, 6

Tags: trigonometry , mechanics

A fancy bathroom scale is calibrated in Newtons. This scale is put on a ramp, which is at a $40^\circ$ angle to the horizontal. A box is then put on the scale and the box-scale system is then pushed up the ramp by a horizontal force $F$. The system slides up the ramp at a constant speed. If the bathroom scale reads $R$ and the coefficient of static friction between the system and the ramp is $0.40$, what is $\frac{F}{R}$? Round to the nearest thousandth. [i](Proposed by Ahaan Rungta)[/i]

2017 Brazil Undergrad MO, 4

Tags: sequence , real analysis

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers in which $\lim_{n\to\infty} a_n = 0$ such that there is a constant $c >0$ so that for all $n \geq 1$, $|a_{n+1}-a_n| \leq c\cdot a_n^2$. Show that exists $d>0$ with $na_n \geq d, \forall n \geq 1$.

Ukrainian TYM Qualifying - geometry, 2020.11

Tags: geometry , angle bisector , perpendicular

In the acute-angled triangle $ABC$, the segment $AP$ was drawn and the center was marked $O$ of the circumscribed circle. The circumcircle of triangle $ABP$ intersects the line $AC$ for the second time at point $X$, the circumcircle of the triangle $ACP$ intersects the line $AB$ for the second time at the point $Y$. Prove that the lines $XY$ and $PO$ are perpendicular if and only if $P$ is the foor of the bisector of the triangle $ABC$.

2012 Olympic Revenge, 1

Tags: real analysis , inequalities

Let $a$ and $b$ real numbers. Let $f:[a,b] \rightarrow \mathbb{R}$ a continuous function. We say that f is "smp" if $[a,b]=[c_0,c_1]\cup[c_1,c_2]...\cup[c_{n-1},c_n]$ satisfying $c_0<c_1...<c_n$ and for each $i\in\{0,1,2...n-1\}$: $c_i<x<c_{i+1} \Rightarrow f(c_i)<f(x)<f(c_{i+1})$ or $c_i>x>c_{i+1} \Rightarrow f(c_i)>f(x)>f(c_{i+1})$ Prove that if $f:[a,b] \rightarrow \mathbb{R}$ is continuous such that for each $v\in\mathbb{R}$ there are only finitely many $x$ satisfying $f(x)=v$, then $f$ is "smp".

2022 AIME Problems, 8

Tags:

Find the number of positive integers $n \le 600$ whose value can be uniquely determined when the values of $\left\lfloor \frac n4\right\rfloor$, $\left\lfloor\frac n5\right\rfloor$, and $\left\lfloor\frac n6\right\rfloor$ are given, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to the real number $x$.

1970 Yugoslav Team Selection Test, Problem 3

Tags: 3d geometry , sphere , geometry

If all edges of a non-planar quadrilateral tangent the faces of a sphere, prove that all of the points of tangency belong to a plane.

2006 Estonia Team Selection Test, 4

Tags: circles , concyclic , cyclic , altitude , cyclic quadrilateral , geometry

The side $AC$ of an acute triangle $ABC$ is the diameter of the circle $c_1$ and side $BC$ is the diameter of the circle $c_2$. Let $E$ be the foot of the altitude drawn from the vertex $B$ of the triangle and $F$ the foot of the altitude drawn from the vertex $A$. In addition, let $L$ and $N$ be the points of intersection of the line $BE$ with the circle $c_1$ (the point $L$ lies on the segment $BE$) and the points of intersection of $K$ and $M$ of line $AF$ with circle $c_2$ (point $K$ is in section $AF$). Prove that $K LM N$ is a cyclic quadrilateral.

2002 USAMTS Problems, 3

Tags: analytic geometry

An integer lattice point in the Cartesian plane is a point $(x,y)$ where $x$ and $y$ are both integers. Suppose nine integer lattice points are chosen such that no three of them lie on the same line. Out of all 36 possible line segments between pairs of those nine points, some line segments may contain integer lattice points besides the original nine points. What is the minimum number of line segments that must contain an integer lattice point besides the original nine points? Prove your answer.

2006 Tournament of Towns, 4

Tags: pyramid , geometry , 3d geometry , prism

Is it possible to split a prism into disjoint set of pyramids so that each pyramid has its base on one base of the prism, while its vertex on another base of the prism ? (6)

1988 National High School Mathematics League, 3

Tags: analytic geometry , graphing lines , slope

On the coordinate plane, is there a line family of infinitely many lines $l_1,l_2,\cdots,l_n,\cdots$, satisfying the following? (1) Point$(1,1)\in l_n$ for all $n\in \mathbb{Z}_{+}$. (2) For all $n\in \mathbb{Z}_{+}$,$k_{n+1}=a_n-b_n$, where $k_{n+1}$ is the slope of $l_{n+1}$, $a_n,b_n$ are intercepts of $l_n$ on $x$-axis, $y$-axis. (3) $k_nk_{n+1}\geq0$ for all $n\in \mathbb{Z}_{+}$.

2001 Austria Beginners' Competition, 4

Tags: geometry , isosceles right triangle , parallelogram

Let $ABC$ be a triangle whose angles $\alpha=\angle CAB$ and $\beta=\angle CBA$ are greater than $45^{\circ}$. Above the side $AB$ a right isosceles triangle $ABR$ is constructed with $AB$ as the hypotenuse, such that $R$ is inside the triangle $ABC$. Analogously we construct above the sides $BC$ and $AC$ the right isosceles triangles $CBP$ and $ACQ$, right at $P$ and in $Q$, but with these outside the triangle $ABC$. Prove that $CQRP$ is a parallelogram.

2020 AMC 12/AHSME, 20

Tags: rotation , probability

Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance? $\textbf{(A)}\ \frac{9}{64} \qquad\textbf{(B)}\ \frac{289}{2048} \qquad\textbf{(C)}\ \frac{73}{512} \qquad\textbf{(D)}\ \frac{147}{1024} \qquad\textbf{(E)}\ \frac{589}{4096}$

2005 Cuba MO, 6

Tags: inequalities , algebra

All positive differences $a_i -a_j$ of five different positive integers $a_1$, $a_2$, $a_3$, $a_4$ and $a_5$ are all different. Let $A$ be the set formed by the largest elements of each group of $5$ elements that meet said condition. Determine the minimum element of $A$.

2024 JHMT HS, 12

Tags: algebra , floor function

Let $\{ a_n \}_{n=0}^{\infty}$, $\{ b_n \}_{n=0}^{\infty}$, and $\{ c_n \}_{n=0}^{\infty}$ be sequences of real numbers such that for all $k\geq 1$, \begin{align*} a_k&=\left\lfloor \sqrt{2}+\frac{k-1}{2024} \right\rfloor+a_{k-1} \\ b_k+c_k&=1 \\ a_{k-1}b_k&=a_kc_k. \end{align*} Suppose that $a_0=1$, $b_0=2$, and $c_0=3$. Given that $\sqrt2\approx1.4142$, compute \[ \sum_{k=1}^{2024}(a_kb_k-a_{k-1}c_k). \]

2015 Latvia Baltic Way TST, 5

Tags: geometry , parallel , circumcircle

$BE$ is the altitude of acute triangle $ABC$. The line $\ell$ touches the circumscribed circle of the triangle $ABC$ at point $B$. A perpendicular $CF$ is drawn from $C$ on line $\ell$. Prove that the lines $EF$ and $AB$ are parallel.