Olympiad problems

2017 Sharygin Geometry Olympiad, P12

Tags: geometry , circumcircle

Let $AA_1 , CC_1$ be the altitudes of triangle $ABC, B_0$ the common point of the altitude from $B$ and the circumcircle of $ABC$; and $Q$ the common point of the circumcircles of $ABC$ and $A_1C_1B_0$, distinct from $B_0$. Prove that $BQ$ is the symmedian of $ABC$. [i]Proposed by D.Shvetsov[/i]

2009 Nordic, 1

Tags: ratio , geometry

A point $P$ is chosen in an arbitrary triangle. Three lines are drawn through $P$ which are parallel to the sides of the triangle. The lines divide the triangle into three smaller triangles and three parallelograms. Let $f$ be the ratio between the total area of the three smaller triangles and the area of the given triangle. Prove that $f\ge\frac{1}{3}$ and determine those points $P$ for which $f =\frac{1}{3}$ .

2025 PErA, P1

Tags: combinatorial geometry , combinatorics

Let $S$ be a set of at least three points of the plane in general position. Prove that there exists a non-intersecting polygon whose vertices are exactly the points of $S$.

2017 IMO Shortlist, A4

Tags: function , algebra

A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation $$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$ Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.

Russian TST 2017, P2

Tags: inequalities , algebra

Let $a_1, a_2,...,a_n$ be positive real numbers, prove that $$\sum {\frac{a_{i+1}}{a_i}} \ge \sum{\sqrt{\frac{a_{i+1}^2+1}{a_i^2+1}}}$$ $a_{n+1}=a_1$

2018 Nepal National Olympiad, 4a

Tags: geometry

[b]Problem Section #4 a) There is a $6 * 6$ grid, each square filled with a grasshopper. After the bell rings, each grasshopper jumps to an adjacent square (A square that shares a side). What is the maximum number of empty squares possible?

1979 All Soviet Union Mathematical Olympiad, 277

Tags: square , geometry , combinatorial geometry , area

Given some square carpets with the total area $4$. Prove that they can fully cover the unit square.

1971 AMC 12/AHSME, 18

Tags: ratio , quadratic

The current in a river is flowing steadily at $3$ miles per hour. A motor boat which travels at a constant rate in still water goes downstream $4$ miles and then returns to its starting point. The trip takes one hour, excluding the time spent in turning the boat around. The ratio of the downstream to the upstream rate is $\textbf{(A) }4:3\qquad\textbf{(B) }3:2\qquad\textbf{(C) }5:3\qquad\textbf{(D) }2:1\qquad \textbf{(E) }5:2$

1963 IMO Shortlist, 6

Tags: permutation , combinatorics

Five students $ A, B, C, D, E$ took part in a contest. One prediction was that the contestants would finish in the order $ ABCDE$. This prediction was very poor. In fact, no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $ DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

2021 CCA Math Bonanza, L3.1

Tags:

A point is chosen uniformly at random from the interior of a unit square. Let $p$ be the probability that any circle centered at the point that intersects a diagonal of the square must also intersect a side of the square. Given that $p^2$ can be written as $m-\sqrt{n}$ for positive integers $m$ and $n$, what is $m+n$? [i]2021 CCA Math Bonanza Lightning Round #3.1[/i]

2002 Iran Team Selection Test, 5

Tags: combinatorics

A school has $n$ students and $k$ classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Prove that we can divide students into $n-k+1$ parts taht students in each part are not friends.

2012 Indonesia TST, 2

Tags: combinatorics

Let $P_1, P_2, \ldots, P_n$ be distinct $2$-element subsets of $\{1, 2, \ldots, n\}$. Suppose that for every $1 \le i < j \le n$, if $P_i \cap P_j \neq \emptyset$, then there is some $k$ such that $P_k = \{i, j\}$. Prove that if $a \in P_i$ for some $i$, then $a \in P_j$ for exactly one value of $j$ not equal to $i$.

2019 Saudi Arabia Pre-TST + Training Tests, 5.3

Tags: algebra , min , inequalities

Let $x, y, z, a,b, c$ are pairwise different integers from the set $\{1,2,3, 4,5,6\}$. Find the smallest possible value for expression $xyz + abc - ax - by - cz$.

2014 Putnam, 4

Tags: algebra , polynomial , calculus , inequalities

Show that for each positive integer $n,$ all the roots of the polynomial \[\sum_{k=0}^n 2^{k(n-k)}x^k\] are real numbers.

2003 Argentina National Olympiad, 5

Tags: algebra , combinatorics , number theory

Carlos and Yue play the following game: First Carlos writes a $+$ sign or a $-$ sign in front of each of the $50$ numbers $1,2,\cdots,50$. Then, in turns, each one chooses a number from the sequence obtained; Start by choosing Yue. If the absolute value of the sum of the $25$ numbers that Carlos chose is greater than or equal to the absolute value of the sum of the $25$ numbers that Yue chose, Carlos wins. In the other case, Yue wins. Determine which of the two players can develop a strategy that will ensure victory, no matter how well their opponent plays, and describe said strategy.

1971 Bundeswettbewerb Mathematik, 1

Tags: modular arithmetic , invariant , induction , arithmetic series

The numbers $1,2,...,1970$ are written on a board. One is allowed to remove $2$ numbers and to write down their difference instead. When repeated often enough, only one number remains. Show that this number is odd.

2003 Bulgaria Team Selection Test, 1

Tags: rectangle , geometry

Cut $2003$ disjoint rectangles from an acute-angled triangle $ABC$, such that any of them has a parallel side to $AB$ and the sum of their areas is maximal.

1996 French Mathematical Olympiad, Problem 4

Tags: inequality , optimization , inequalities

(a) A function $f$ is defined by $f(x)=x^x$ for all $x>0$. Find the minimum value of $f$. (b) If $x$ and $y$ are two positive real numbers, show that $x^y+y^x>1$.

2019 Pan-African Shortlist, G4

Tags: geometry , circumcenter , equilateral triangle , circumcircle

Let $ABC$ be a triangle, and $D$, $E$, $F$ points on the segments $BC$, $CA$, and $AB$ respectively such that $$ \frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}. $$ Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral triangle.

2014 AMC 8, 7

Tags: ratio

There are four more girls than boys in Ms. Raub's class of $28$ students. What is the ratio of number of girls to the number of boys in her class? $\textbf{(A) }3 : 4\qquad\textbf{(B) }4 : 3\qquad\textbf{(C) }3 : 2\qquad\textbf{(D) }7 : 4\qquad \textbf{(E) }2 : 1$

Mathematical Minds 2024, P3

Tags: combinatorics , olympiad combinatorics , board

On the screen of a computer there is an $2^n\times 2^n$ board. On each cell of the main diagonal there is a file. At each step, we may select some files and move them to the left, on their respective rows, by the same distance. What is the minimum number of necessary moves in order to put all files on the first column? [i]Proposed by Vlad Spătaru[/i]

2014 BMT Spring, 4

Tags: polynomial , algebra

The function $f(x)=x^5-20x^4+ax^3+bx^2+cx+24$ has the interesting property that its roots can be arranged to form an arithmetic sequence. Determine $f(8)$.

2019 Stars of Mathematics, 1

Tags: number theory

Determine all positive integers $n$ such that for every positive devisor $ d $ of $n$, $d+1$ is devisor of $n+1$.

2013 AMC 12/AHSME, 16

Tags: probability

$A$, $B$, $C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40$ pounds, the mean weight of the rocks in $B$ is $50$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $43$ pounds, and the mean weight of the rocks in the combined piles $A$ and $C$ is $44$ pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles $B$ and $C$? $ \textbf{(A)} \ 55 \qquad \textbf{(B)} \ 56 \qquad \textbf{(C)} \ 57 \qquad \textbf{(D)} \ 58 \qquad \textbf{(E)} \ 59$

2013 239 Open Mathematical Olympiad, 7

Tags:

Dima wrote several natural numbers on the blackboard and underlined some of them. Misha wants to erase several numbers (but not all) such that a multiple of three underlined numbers remain and the total amount of the remaining numbers would be divisible by $2013$; but after trying for a while he realizes that it's impossible to do this. What is the largest number of the numbers on the board?