Olympiad problems

2003 Swedish Mathematical Competition, 6

Tags: combinatorics

Consider an infinite square board with an integer written in each square. Assume that for each square the integer in it is equal to the sum of its neighbor to the left and its neighbor above. Assume also that there exists a row $R_0$ in the board such that all numbers in $R_0$ are positive. Denote by $R_1$ the row below $R_0$ , by $R_2$ the row below $R_1$ etc. Show that for each $N \ge 1$ the row $R_N$ cannot contain more than $N$ zeroes.

2023 Purple Comet Problems, 7

Tags: algebra

Elijah went on a four-mile journey. He walked the first mile at $3$ miles per hour and the second mile at $4$ miles per hour. Then he ran the third mile at $5$ miles per hour and the fourth mile at $6$ miles per hour. Elijah’s average speed for this journey in miles per hour was $\frac{m}{n}$, where m and $n$ are relatively prime positive integers. Find $m + n$.

2016 Latvia Baltic Way TST, 2

Tags: number theory , inequalities , algebra

Given natural numbers $m, n$ and $X$ such that $X \ge m$ and $X \ge n$. Prove that one can find two integers $u$ and $v$ such that $|u| + |v| > 0$, $|u| \le \sqrt{X}$, $|v| \le \sqrt{X}$ and $$0 \le mu + nv \le 2 \sqrt{X}.$$

2005 China Team Selection Test, 2

Tags: perimeter , cyclic quadrilateral , pythagorean theorem , geometry

Cyclic quadrilateral $ABCD$ has positive integer side lengths $AB$, $BC$, $CA$, $AD$. It is known that $AD=2005$, $\angle{ABC}=\angle{ADC} = 90^o$, and $\max \{ AB,BC,CD \} < 2005$. Determine the maximum and minimum possible values for the perimeter of $ABCD$.

2012 AMC 10, 12

Tags: modular arithmetic , algorithm

A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$) or is divisible by $4$ but not by $100$ (such as $2012$). The $200\text{th}$ anniversary of the birth of novelist Charles Dickens was celebrated on February $7$, $2012$, a Tuesday. On what day of the week was Dickens born? $ \textbf{(A)}\ \text{Friday} \qquad\textbf{(B)}\ \text{Saturday} \qquad\textbf{(C)}\ \text{Sunday} \qquad\textbf{(D)}\ \text{Monday} \qquad\textbf{(E)}\ \text{Tuesday} $

CNCM Online Round 2, 5

Tags:

Consider a regular $n$-gon of side length 1. For each of its vertices, a circle of radius one is drawn centered at that vertex. The resulting figure, consisting of the polygon and the $n$ circles, partitions the plane into $f(n)$ finite, bounded regions. Find $$\sum_{n=3}^{25} f(n).$$ The first term corresponding to $i=3$ is shown; each of the various colors corresponds to a distinct region with $f(3)=10$. Note that the lines corresponding to the polygon are treated no differently than the arcs corresponding to the circles in counting regions. Proposed by Hari Desikan (HariDesikan)

Novosibirsk Oral Geo Oly VIII, 2017.7

Tags: geometry , angle

A car is driving along a straight highway at a speed of $60$ km per hour. Not far from the highway there is a parallel to him a $100$-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than $1100^o$

2013 Tuymaada Olympiad, 2

Tags: geometry , rhombus , geometric transformation , reflection , symmetry , trapezoid , trigonometry

Points $X$ and $Y$ inside the rhombus $ABCD$ are such that $Y$ is inside the convex quadrilateral $BXDC$ and $2\angle XBY = 2\angle XDY = \angle ABC$. Prove that the lines $AX$ and $CY$ are parallel. [i]S. Berlov[/i]

2024 Czech-Polish-Slovak Junior Match, 2

Tags: number theory , algebra

Among all triples $(a,b,c)$ of natural numbers satisfying \[(a+14\sqrt{3})(b-14c\sqrt{3})=2024,\] determine the one with the maximal value of $a$.

1980 IMO, 14

Tags: trigonometry , trapezoid , geometry

Let $A$ be a fixed point in the interior of a circle $\omega$ with center $O$ and radius $r$, where $0<OA<r$. Draw two perpendicular chords $BC,DE$ such that they pass through $A$. For which position of these cords does $BC+DE$ maximize?

May Olympiad L2 - geometry, 2022.5

Tags: combinatorics , winning strategy , game strategy

The vertices of a regular polygon with $N$ sides are marked on the blackboard. Ana and Beto play alternately, Ana begins. Each player, in turn, must do the following: $\bullet$ join two vertices with a segment, without cutting another already marked segment; or $\bullet$ delete a vertex that does not belong to any marked segment. The player who cannot take any action on his turn loses the game. Determine which of the two players can guarantee victory: a) if $N=28$ b) if $N=29$

1994 Irish Math Olympiad, 5

Tags: euler

If a square is partitioned into $ n$ convex polygons, determine the maximum possible number of edges in the obtained figure. (You may wish to use the following theorem of Euler: If a polygon is partitioned into $ n$ polygons with $ v$ vertices and $ e$ edges in the resulting figure, then $ v\minus{}e\plus{}n\equal{}1$.)

1955 Miklós Schweitzer, 2

Tags: real analysis , function

[b]2.[/b] Let $f_{1}(x), \dots , f_{n}(x)$ be Lebesgue integrable functions on $[0,1]$, with $\int_{0}^{1}f_{1}(x) dx= 0$ $ (i=1,\dots ,n)$. Show that, for every $\alpha \in (0,1)$, there existis a subset $E$ of $[0,1]$ with measure $\alpha$, such that $\int_{E}f_{i}(x)dx=0$. [b](R. 17)[/b]

MIPT Undergraduate Contest 2019, 2.2

Tags: inequalities , probability and stats , expectation , calculus , integration , probability , function

Petya and Vasya are playing the following game. Petya chooses a non-negative random value $\xi$ with expectation $\mathbb{E} [\xi ] = 1$, after which Vasya chooses his own value $\eta$ with expectation $\mathbb{E} [\eta ] = 1$ without reference to the value of $\xi$. For which maximal value $p$ can Petya choose a value $\xi$ in such a way that for any choice of Vasya's $\eta$, the inequality $\mathbb{P}[\eta \geq \xi ] \leq p$ holds?

2010 Contests, 2

Tags: circumcircle , rotation , analytic geometry , perpendicular bisector , lalala , geometry

$AB$ is a diameter of a circle with center $O$. Let $C$ and $D$ be two different points on the circle on the same side of $AB$, and the lines tangent to the circle at points $C$ and $D$ meet at $E$. Segments $AD$ and $BC$ meet at $F$. Lines $EF$ and $AB$ meet at $M$. Prove that $E,C,M$ and $D$ are concyclic.

2023 LMT Spring, 9

Tags: geometry

In $\vartriangle ABC$, $AB = 13$, $BC = 14,$ and $C A = 15$. Let $E$ and $F$ be the feet of the altitudes from $B$ onto $C A$, and $C$ onto $AB$, respectively. A line $\ell$ is parallel to $EF$ and tangent to the circumcircle of $ABC$ on minor arc $BC$. Let $X$ and $Y$ be the intersections of $\ell$ with $AB$ and $AC$ respectively. Find $X Y$ .

2019 Moldova Team Selection Test, 5

Tags: geometry

Point $H$ is the orthocenter of the scalene triangle $ABC.$ A line, which passes through point $H$, intersect the sides $AB$ and $AC$ at points $D$ and $E$, respectively, such that $AD=AE.$ Let $M$ be the midpoint of side $BC.$ Line $MH$ intersects the circumscribed circle of triangle $ABC$ at point $K$, which is on the smaller arc $AB$. Prove that Nibab can draw a circle through $A, D, E$ and $K.$

2000 Taiwan National Olympiad, 1

Tags: euler , function , modular arithmetic , quadratic , prime number , number theory

Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.

2013 AMC 10, 3

Tags:

On a particular January day, the high temperature in Lincoln, Nebraska, was 16 degrees higher than the low temperature, and the average of the high and low temperatures was $3^{\circ}$. In degrees, what was the low temperature in Lincoln that day? $\textbf{(A) }-13\qquad\textbf{(B) }-8\qquad\textbf{(C) }-5\qquad\textbf{(D) }3\qquad\textbf{(E) }11$

1998 Belarus Team Selection Test, 1

Tags: vector , combinatorics , combinatorial geometry , extremal combinatorics , geometry

For each finite set $ U$ of nonzero vectors in the plane we define $ l(U)$ to be the length of the vector that is the sum of all vectors in $ U.$ Given a finite set $ V$ of nonzero vectors in the plane, a subset $ B$ of $ V$ is said to be maximal if $ l(B)$ is greater than or equal to $ l(A)$ for each nonempty subset $ A$ of $ V.$ (a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively. (b) Show that, for any set $ V$ consisting of $ n \geq 1$ vectors the number of maximal subsets is less than or equal to $ 2n.$

2016 Korea Junior Math Olympiad, 2

Tags: geometry , incenter

A non-isosceles triangle $\triangle ABC$ has its incircle tangent to $BC, CA, AB$ at points $D, E, F$. Let the incenter be $I$. Say $AD$ hits the incircle again at $G$, at let the tangent to the incircle at $G$ hit $AC$ at $H$. Let $IH \cap AD = K$, and let the foot of the perpendicular from $I$ to $AD$ be $L$. Prove that $IE \cdot IK= IC \cdot IL$.

2009 AIME Problems, 14

Tags: linear algebra , matrix , modular arithmetic

For $ t \equal{} 1, 2, 3, 4$, define $ \displaystyle S_t \equal{} \sum_{i \equal{} 1}^{350}a_i^t$, where $ a_i \in \{1,2,3,4\}$. If $ S_1 \equal{} 513$ and $ S_4 \equal{} 4745$, find the minimum possible value for $ S_2$.

1957 AMC 12/AHSME, 29

Tags: inequalities

The relation $ x^2(x^2 \minus{} 1)\ge 0$ is true only for: $ \textbf{(A)}\ x \ge 1\qquad \textbf{(B)}\ \minus{} 1 \le x \le 1\qquad \textbf{(C)}\ x \equal{} 0,\, x \equal{} 1,\, x \equal{} \minus{} 1\qquad \\\textbf{(D)}\ x \equal{} 0,\, x \le \minus{} 1,\, x \ge 1\qquad \textbf{(E)}\ x \ge 0$

2012 Indonesia MO, 3

Tags: geometry , circumcircle , angle bisector , perpendicular

Given an acute triangle $ABC$ with $AB>AC$ that has circumcenter $O$. Line $BO$ and $CO$ meet the bisector of $\angle BAC$ at $P$ and $Q$, respectively. Moreover, line $BQ$ and $CP$ meet at $R$. Show that $AR$ is perpendicular to $BC$. [i]Proposer: Soewono and Fajar Yuliawan[/i]

2018 Hanoi Open Mathematics Competitions, 6

Tags: algebra

Three students $A, B$ and $C$ are traveling from a location on the National Highway No.$5$ on direction to Hanoi for participating the HOMC $2018$. At beginning, $A$ takes $B$ on the motocycle, and at the same time $C$ rides the bicycle. After one hour and a half, $B$ switches to a bicycle and immediately continues the trip to Hanoi, while $A$ returns to pick up $C$. Upon meeting, $C$ continues the travel on the motocycle to Hanoi with $A$. Finally, all three students arrive in Hanoi at the same time. Suppose that the average speed of the motocycle is $50$ km per hour and of the both bicycles are $10$ km per hour. Find the distance from the starting point to Hanoi.