Olympiad problems

2004 All-Russian Olympiad, 4

Tags: algorithm , combinatorics

A rectangular array has 9 rows and 2004 columns. In the 9 * 2004 cells of the table we place the numbers from 1 to 2004, each 9 times. And we do this in such a way that two numbers, which stand in exactly the same column in and differ around at most by 3. Find the smallest possible sum of all numbers in the first row.

2022 Rioplatense Mathematical Olympiad, 6

Tags: combinatorics

Let $N(a,b)$ be the number of ways to cover a table $a \times b$ with domino tiles. Let $M(a,2b+1)$ be the number of ways to cover a table $a \times 2b+1$ with domino tiles, such that there are [b]no[/b] vertical tile in the central column. Prove that $$M(2m,2n+1)=2^m \cdot N(2m,n)\cdot N(2m,n-1)$$

2008 Baltic Way, 12

Tags: function , modular arithmetic , divisibility tests , graph theory , combinatorics , number theory

In a school class with $ 3n$ children, any two children make a common present to exactly one other child. Prove that for all odd $ n$ it is possible that the following holds: For any three children $ A$, $ B$ and $ C$ in the class, if $ A$ and $ B$ make a present to $ C$ then $ A$ and $ C$ make a present to $ B$.

2019 Middle European Mathematical Olympiad, 6

Tags: geometry , circumcircle

Let $ABC$ be a right-angled triangle with the right angle at $B$ and circumcircle $c$. Denote by $D$ the midpoint of the shorter arc $AB$ of $c$. Let $P$ be the point on the side $AB$ such that $CP=CD$ and let $X$ and $Y$ be two distinct points on $c$ satisfying $AX=AY=PD$. Prove that $X, Y$ and $P$ are collinear. [i]Proposed by Dominik Burek, Poland[/i]

1998 Gauss, 10

Tags: gauss

At the waterpark, Bonnie and Wendy decided to race each other down a waterslide. Wendy won by $0.25$ seconds. If Bonnie’s time was exactly $7.80$ seconds, how long did it take for Wendy to go down the slide? $\textbf{(A)}\ 7.80~ \text{seconds} \qquad \textbf{(B)}\ 8.05~ \text{seconds} \qquad \textbf{(C)}\ 7.55~ \text{seconds} \qquad \textbf{(D)}\ 7.15~ \text{seconds} \qquad $ $\textbf{(E)}\ 7.50~ \text{seconds}$

1968 IMO Shortlist, 21

Tags: number theory , divisibility

Let $a_0, a_1, \ldots , a_k \ (k \geq 1)$ be positive integers. Find all positive integers $y$ such that \[a_0 | y, (a_0 + a_1) | (y + a1), \ldots , (a_0 + a_n) | (y + a_n).\]

2012 JHMT, 5

Tags: geometry

Let $ABC$ be an equilateral triangle with side length $1$. Draw three circles $O_a$, $O_b$, and $O_c$ with diameters BC, $CA$, and $AB$, respectively. Let $S_a$ denote the area of the region inside $O_a$ and outside of $O_b$ and $O_c$. Define $S_b$ and $S_c$ similarly, and let $S$ be the area of the region inside all three circles. Find $S_a + S_b + S_c - S$.

2005 Germany Team Selection Test, 1

Tags: number theory

[b](a)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ ends with the string $2004$, followed by a number of digits from the set $\left\{0;\;4\right\}$ ? [b](b)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ starts with the string $2004$ ?

1996 Tuymaada Olympiad, 2

Tags: algebra , set theory , real , set

Given a finite set of real numbers $A$, not containing $0$ and $1$ and possessing the property: if the number a belongs to $A$, then numbers $\frac{1}{a}$ and $1-a$ also belong to $A$. How many numbers are in the set $A$?

LMT Speed Rounds, 24

Tags: algebra

Evaluate $$2023 \cdot \frac{2023^6 +27}{(2023^2 +3)(2024^3 -1)}-2023^2.$$ [i]Proposed by Evin Liang[/i]

1978 Romania Team Selection Test, 4

Tags: perimeter , rhombus , pure geometry , geometry

Diagonals $ AC $ and $ BD $ of a convex quadrilateral $ ABCD $ intersect a point $ O. $ Prove that if triangles $ OAB,OBC,OCD $ and $ ODA $ have the same perimeter, then $ ABCD $ is a rhombus. What happens if $ O $ is some other point inside the quadrilateral?

2014 NIMO Problems, 14

Tags: geometry , circumcircle , incenter , geometric transformation , reflection , euler , ratio

Let $ABC$ be a triangle with circumcenter $O$ and let $X$, $Y$, $Z$ be the midpoints of arcs $BAC$, $ABC$, $ACB$ on its circumcircle. Let $G$ and $I$ denote the centroid of $\triangle XYZ$ and the incenter of $\triangle ABC$. Given that $AB = 13$, $BC = 14$, $CA = 15$, and $\frac {GO}{GI} = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. [i]Proposed by Evan Chen[/i]

1991 India Regional Mathematical Olympiad, 7

Tags: induction , modular arithmetic , number theory , prime number

Prove that $n^4 + 4^{n}$ is composite for all values of $n$ greater than $1$.

2004 India IMO Training Camp, 2

Tags: trigonometry , number theory

Show that the only solutions of te equation \[ p^{k} + 1 = q^{m} \], in positive integers $k,q,m > 1$ and prime $p$ are (i) $(p,k,q,m) = (2,3,3,2)$ (ii) $k=1 , q=2,$and $p$ is a prime of the form $2^{m} -1$, $m > 1 \in \mathbb{N}$

2020 Azerbaijan National Olympiad, 4

Tags: geometry

There is a non-equilateral triangle $ABC$.Let $ABC$'s Incentri $I$.Point $D$ is on the $BC$ side.The circle drawn outside the triangle $IBD$ and $ICD$ intersects the sides $AB$ and $AC$ at points $E$ and $F.$The circle drawn outside the triangle $DEF$ intersects the sides $AB$ and $AC$ at $N$ and $M$.Prove that $EM\parallel FN $.

2023 LMT Fall, 11

Tags: geometry

Let $LEX INGT_1ONMAT_2H$ be a regular $13$-gon. Find $\angle LMT_1$, in degrees. [i]Proposed by Edwin Zhao[/i]

2016 Auckland Mathematical Olympiad, 3

Tags: algebra

In two weeks three cows eat all the grass on two hectares of land, together with all the grass that regrows there during the two weeks. In four weeks, two cows eat all the grass on two hectares of land, together with all the grass that regrows there during the four weeks. How many cows will eat all the grass on six hectares of land in six weeks, together with all the grass that regrows there over the six weeks? (Assume: $\bullet$ the quantity of grass on each hectare is the same when the cows begin to graze, $\bullet$ the rate of growth of the grass is uniform during the time of grazing, $\bullet$ the cows eat the same amount of grass each week.)

2021 Kosovo National Mathematical Olympiad, 4

Tags: geometry , geometric inequality , inequalities

Let $M$ be the midpoint of segment $BC$ of $\triangle ABC$. Let $D$ be a point such that $AD=AB$, $AD\perp AB$ and points $C$ and $D$ are on different sides of $AB$. Prove that: $$\sqrt{AB\cdot AC+BC\cdot AM}\geq\frac{\sqrt{2}}{2}CD.$$

2023 Stanford Mathematics Tournament, 1

Tags: geometry

Let $\omega$ be a circle with radius $1$. Equilateral triangle $\vartriangle ABC$ is tangent to $\omega$ at the midpoint of side $BC$ and $\omega$ lies outside $\vartriangle ABC$. If line $AB$ is tangent to $\omega$ , compute the side length of $\vartriangle ABC$.

2006 Tournament of Towns, 6

Tags: number theory , decreasing , sequence

Let $1 + 1/2 + 1/3 +... + 1/n = a_n/b_n$, where $a_n$ and $b_n$ are relatively prime. Show that there exist infinitely many positive integers $n$, such that $b_{n+1} < b_n$. (8)

2014 German National Olympiad, 1

Tags: prime number , number theory

For which non-negative integers $n$ is \[K=5^{2n+3} + 3^{n+3} \cdot 2^n\] prime?

2003 District Olympiad, 1

Tags: 3d geometry , equilateral , plane , angle , geometry

Let $ABC$ be an equilateral triangle. On the plane $(ABC)$ rise the perpendiculars $AA'$ and $BB'$ on the same side of the plane, so that $AA' = AB$ and $BB' =\frac12 AB$. Determine the measure the angle between the planes $(ABC)$ and $(A'B'C')$.

2016 Germany Team Selection Test, 3

Tags: geometry

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

2002 China Team Selection Test, 2

Tags: number theory

Find all non-negative integers $m$ and $n$, such that $(2^n-1) \cdot (3^n-1)=m^2$.

2019 Math Prize for Girls Olympiad, 1

Tags:

Let $A_1$, $A_2$, $\ldots\,$, $A_n$ be finite sets. Prove that \[ \Bigl| \bigcup_{1 \le i \le n} A_i \Bigr| \ge \frac{1}{2} \sum_{1 \le i \le n} \left| A_i \right| - \frac{1}{6} \sum_{1 \le i < j \le n} \left| A_i \cap A_j \right| \, . \] Recall that if $S$ is a finite set, then its cardinality $|S|$ is the number of elements of $S$.