Olympiad problems

1990 French Mathematical Olympiad, Problem 5

Tags: triangle , geometry

In a triangle $ABC$, $\Gamma$ denotes the excircle corresponding to $A$, $A',B',C'$ are the points of tangency of $\Gamma$ with $BC,CA,AB$ respectively, and $S(ABC)$ denotes the region of the plane determined by segments $AB',AC'$ and the arc $C'A'B'$ of $\Gamma$. Prove that there is a triangle $ABC$ of a given perimeter $p$ for which the area of $S(ABC)$ is maximal. For this triangle, give an approximate measure of the angle at $A$.

2017 Harvard-MIT Mathematics Tournament, 9

Tags: algebra

Find the minimum value of $\sqrt{58-42x}+\sqrt{149-140\sqrt{1-x^2}}$ where $-1 \le x \le 1$.

2016 Indonesia TST, 5

Tags: greatest common divisor , combinatorics

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

1964 AMC 12/AHSME, 12

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Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$? $ \textbf{(A)}\ \text{For all x}, x^2 < 0\qquad$ $\textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad$ $\textbf{(C)}\ \text{For no x}, x^2>0\qquad$ ${\textbf{(D)}\ \text{For some x}, x^2>0 }\qquad$ ${{\textbf{(E)}\ \text{For some x}, x^2 \le 0}} $

1992 Czech And Slovak Olympiad IIIA, 6

Tags: altitude , concyclic , circles , geometry

Let $ABC$ be an acute triangle. The altitude from $B$ meets the circle with diameter $AC$ at points $P,Q$, and the altitude from $C$ meets the circle with diameter $AB$ at $M,N$. Prove that the points $M,N,P,Q$ lie on a circle.

2024 Serbia National Math Olympiad, 5

Tags: algebra

Let $n \geq 3$ be a positive integer. Find all positive integers $k$, such that the function $f:\mathbb{R} \rightarrow \mathbb{R}$ defined by $$f(x)=\cos^k(x)+\cos^k(x+\frac{2\pi}{n})+\ldots +\cos^k(x+\frac{2(n-1)\pi}{n})$$ is constant.

2011 Laurențiu Duican, 2

Tags: linear algebra , nilpotence

Let be a field $ \mathbb{F} $ and two nonzero nilpotent matrices $ M,N\in\mathcal{M}_2\left( \mathbb{F} \right) $ that commute. Show that: [b]a)[/b] $ MN=0 $ [b]b)[/b] there exists a nonzero element $ f\in\mathbb{F} $ such that $ M=fN $ [i]Dorel Miheț[/i]

1994 Czech And Slovak Olympiad IIIA, 2

Tags: cube , polyhedron , volume , max , 3d geometry , projection , geometry

A cuboid of volume $V$ contains a convex polyhedron $M$. The orthogonal projection of $M$ onto each face of the cuboid covers the entire face. What is the smallest possible volume of polyhedron $M$?

2020 Dutch IMO TST, 2

Tags: combinatorics , game , winning strategy , game strategy

Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn. During their turn, a player must pick two numbers a and b on the sheet such that $gcd(a, b) = 1$, erase these numbers from the sheet, and write the number $a + b$ on the sheet. The first player who is not able to do so, loses. Determine which player can always win this game.

2016 Saudi Arabia IMO TST, 3

Tags: miscellaneous

Find the number of permutations $ ( a_1, a_2, . \ . \ , a_{2016}) $ of the first $ 2016 $ positive integers satisfying the following two conditions: 1. $ a_{i+1} - a_i \leq 1$ for all $i = 1, 2, . \ . \ . , 2015$, and 2. There are exactly two indices $ i < j $ with $ 1 \leq i < j \leq 2016 $ such that $ a_i = i $ and $ a_j = j$.

2012 Princeton University Math Competition, A1 / B2

Tags: combinatorics

If the probability that the sum of three distinct integers between $16$ and $30$ (inclusive) is even can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m + n$.

2019 Bundeswettbewerb Mathematik, 3

Tags: equal angles , incircle , geometry

Let $ABC$ be atriangle with $\overline{AC}> \overline{BC}$ and incircle $k$. Let $M,W,L$ be the intersections of the median, angle bisector and altitude from point $C$ respectively. The tangent to $k$ passing through $M$, that is different from $AB$, touch $k$ in $T$. Prove that the angles $\angle MTW$ and $\angle TLM$ are equal.

2014 Ukraine Team Selection Test, 3

Tags: geometry , complex number , hexagon

Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.

Russian TST 2016, P3

Tags: incircle , geometry

The scalene triangle $ABC$ has incenter $I{}$ and circumcenter $O{}$. The points $B_A$ and $C_A$ are the projections of the points $B{}$ and $C{}$ onto the line $AI$. A circle with a diameter $B_AC_A$ intersects the line $BC$ at the points $K_A$ and $L_A$. [list=i] [*]Prove that the circumcircle of the triangle $AK_AL_A$ touches the incircle of the triangle $ABC$ at some point $T_A$. [*]Define the points $T_B$ and $T_C$ analogously. Prove that the lines $AT_A,BT_B$ and $CT_C$ intersect on the line $OI$. [/list]

2010 Mexico National Olympiad, 2

Tags: circumcircle , power of a point , radical axis , geometry

Let $ABC$ be an acute triangle with $AB\neq AC$, $M$ be the median of $BC$, and $H$ be the orthocenter of $\triangle ABC$. The circumcircle of $B$, $H$, and $C$ intersects the median $AM$ at $N$. Show that $\angle ANH=90^\circ$.

2014 Math Prize For Girls Problems, 9

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Let $abc$ be a three-digit prime number whose digits satisfy $a < b < c$. The difference between every two of the digits is a prime number too. What is the sum of all the possible values of the three-digit number $abc$?

2009 National Olympiad First Round, 22

Tags: modular arithmetic , geometry , 3d geometry

$ (a_n)_{n \equal{} 0}^\infty$ is a sequence on integers. For every $ n \ge 0$, $ a_{n \plus{} 1} \equal{} a_n^3 \plus{} a_n^2$. The number of distinct residues of $ a_i$ in $ \pmod {11}$ can be at most? $\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$

1954 AMC 12/AHSME, 10

Tags: algebra , polynomial , binomial theorem

The sum of the numerical coefficients in the expansion of the binomial $ (a\plus{}b)^8$ is: $ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 7$

1985 Swedish Mathematical Competition, 1

Tags: inequalities , algebra , radical

If $a > b > 0$, prove the inequality $$\frac{(a-b)^2}{8a}< \frac{a+b}{2}- \sqrt{ab} < \frac{(a-b)^2}{8b}.$$

2007 AMC 10, 15

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The angles of quadrilateral $ ABCD$ satisfy $ \angle A \equal{} 2 \angle B \equal{} 3 \angle C \equal{} 4 \angle D$. What is the degree measure of $ \angle A$, rounded to the nearest whole number? $ \textbf{(A)}\ 125 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 153 \qquad \textbf{(D)}\ 173 \qquad \textbf{(E)}\ 180$

2022 CCA Math Bonanza, L1.4

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Jongol and Gongol are writing calculus questions and grading tests. They want to write 90 calculus problems and they have 120 tests to grade. Jongol can write 3 questions per minute or grade 4 tests per minute. Gongol can write 1 question per minute or grade 2 tests per minute. Evaluate the shortest possible time, in minutes, for them to complete the two tasks. [i]2022 CCA Math Bonanza Lightning Round 1.4[/i]

2017 Finnish National High School Mathematics Comp, 2

Tags: algebra , system of equations

Determine $x^2+y^2$ and $x^4+y^4$, when $x^3+y^3=2$ and $x+y=1$

2019 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory

Determine positive integers $a$ and $b$ co-prime such that $a^2+b = (a-b)^3$ .

2014 AMC 8, 12

Tags: probability

A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly? $\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{1}{6}\qquad\textbf{(C) }\frac{1}{4}\qquad\textbf{(D) }\frac{1}{3}\qquad \textbf{(E) }\frac{1}{2}$

2017 Azerbaijan Team Selection Test, 2

Tags: cyclic quadrilateral , geometry , incenter , reflection , inversion

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.