Olympiad problems

1970 AMC 12/AHSME, 18

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$\sqrt{3+2\sqrt{2}}-\sqrt{3-2\sqrt{2}}$ is equal to $\textbf{(A) }2\qquad\textbf{(B) }2\sqrt{3}\qquad\textbf{(C) }4\sqrt{2}\qquad\textbf{(D) }\sqrt{6}\qquad \textbf{(E) }2\sqrt{2}$

2006 China Team Selection Test, 2

Tags: geometry , circumcircle , geometric transformation , reflection , parallelogram , ratio , analytic geometry

Let $\omega$ be the circumcircle of $\triangle{ABC}$. $P$ is an interior point of $\triangle{ABC}$. $A_{1}, B_{1}, C_{1}$ are the intersections of $AP, BP, CP$ respectively and $A_{2}, B_{2}, C_{2}$ are the symmetrical points of $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of side $BC, CA, AB$. Show that the circumcircle of $\triangle{A_{2}B_{2}C_{2}}$ passes through the orthocentre of $\triangle{ABC}$.

2010 Stanford Mathematics Tournament, 25

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There are balls that look identical, but their weights all dier by a little. We have a balance that can compare only two balls at a time. What is the minimum number of times, in the worst case, we have to use to balance to rank all balls by weight?

2004 Kazakhstan National Olympiad, 4

Tags: combinatorics

In some village there are $1000$ inhabitants. Every day, each of them shares the news they learned yesterday with all their friends. It is known that any news becomes known to all residents of the village. Prove that it is possible to select $90$ residents so that if you tell all of them at the same time some news, then in $10$ days it will become known to all residents of the village.

2010 Contests, 1

Tags: modular arithmetic , number theory

Let $n$ be a positive integer. Let $T_n$ be a set of positive integers such that: \[{T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}\] Find all $n$ for which there don't exist two distinct positive integers $a, b \in T_n$ such that $a\equiv b \pmod{110}$

2009 Romanian Master of Mathematics, 4

Tags: trigonometry , invariant , algebra

For a finite set $ X$ of positive integers, let $ \Sigma(X) \equal{} \sum_{x \in X} \arctan \frac{1}{x}.$ Given a finite set $ S$ of positive integers for which $ \Sigma(S) < \frac{\pi}{2},$ show that there exists at least one finite set $ T$ of positive integers for which $ S \subset T$ and $ \Sigma(S) \equal{} \frac{\pi}{2}.$ [i]Kevin Buzzard, United Kingdom[/i]

2007 AMC 10, 14

Tags: ratio , geometry

A triangle with side lengths in the ratio $ 3: 4: 5$ is inscribed in a circle of radius $ 3$. What is the area of the triangle? $ \textbf{(A)}\ 8.64 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 5\pi \qquad \textbf{(D)}\ 17.28 \qquad \textbf{(E)}\ 18$

Kyiv City MO Juniors Round2 2010+ geometry, 2021.7.4

Tags: geometry , concyclic , equilateral

The sides of the triangle $ABC$ are extended in both directions and on these extensions $6$ equal segments $AA_1 , AA_2, BB_1,BB_2, CC_1, CC_2$ are drawn (fig.). It turned out that all $6$ points $A_1,A_2,B_1,B_2,C_1, C_2$ lie on the same circle, is $\vartriangle ABC$ necessarily equilateral? (Bogdan Rublev) [img]https://cdn.artofproblemsolving.com/attachments/0/3/a499f6e6d978ce63d2ab40460dc73b62882863.png[/img]

2021 Argentina National Olympiad, 6

Tags: algebra

Decide if it is possible to choose $330$ points in the plane so that among all the distances that are formed between two of them there are at least $1700$ that are equal.

2007 Junior Balkan MO, 3

Tags: combinatorics , combinatorial geometry

Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.

2023 USA IMOTST, 1

Tags: floor function , algebra , function

Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rfloor$$ is even for every $k \ge 0$. Find all positive integers $a$ such that for any integers $x>y\ge 0$, we have \[ x\oplus ax \neq y \oplus ay. \] [i]Carl Schildkraut[/i]

1979 IMO Longlists, 80

Tags: algebra , functional equation

Prove that the functional equations \[f(x + y) = f(x) + f(y),\] \[ \text{and} \qquad f(x + y + xy) = f(x) + f(y) + f(xy) \quad (x, y \in \mathbb R)\] are equivalent.

2016 Puerto Rico Team Selection Test, 6

Tags: composition , function , algebra , recurrence

$N$ denotes the set of all natural numbers. Define a function $T: N \to N$ such that $T (2k) = k$ and $T (2k + 1) = 2k + 2$. We write $T^2 (n) = T (T (n))$ and in general $T^k (n) = T^{k-1} (T (n))$ for all $k> 1$. (a) Prove that for every $n \in N$, there exists $k$ such that $T^k (n) = 1$. (b) For $k \in N$, $c_k$ denotes the number of elements in the set $\{n: T^k (n) = 1\}$. Prove that $c_{k + 2} = c_{k + 1} + c_k$, for $1 \le k$.

2020 Moldova EGMO TST, 1

Tags: inequalities , number theory

Let[i] $a,b,c$[/i] be positive integers , such that $A=\frac{a^2+1}{bc}+\frac{b^2+1}{ca}+\frac{c^2+1}{ab}$ is, also, an integer. Proof that $\gcd( a, b, c)\leq\lfloor\sqrt[3]{a+ b+ c}\rfloor$.

2001 Swedish Mathematical Competition, 4

Tags: geometry , angle

$ABC$ is a triangle. A circle through $A$ touches the side $BC$ at $D$ and intersects the sides $AB$ and $AC$ again at $E, F$ respectively. $EF$ bisects $\angle AFD$ and $\angle ADC = 80^o$. Find $\angle ABC$.

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

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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]