Olympiad problems

2001 Romania Team Selection Test, 1

Find all pairs $\left(m,n\right)$ of positive integers, with $m,n\geq2$, such that $a^n-1$ is divisible by $m$ for each $a\in \left\{1,2,3,\ldots,n\right\}$.

2013 Hanoi Open Mathematics Competitions, 9

Tags: algebra , system

Solve the following system in positive numbers $\begin{cases} x+y\le 1 \\ \frac{2}{xy} +\frac{1}{x^2+y^2}=10\end{cases}$

2010 Belarus Team Selection Test, 8.3

Tags: trigonometry , inradius , circumcircle , incenter , geometry

Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$. [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

2016 AMC 10, 23

Tags: area , geometry

In regular hexagon $ABCDEF$, points $W$, $X$, $Y$, and $Z$ are chosen on sides $\overline{BC}$, $\overline{CD}$, $\overline{EF}$, and $\overline{FA}$ respectively, so lines $AB$, $ZW$, $YX$, and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$? $\textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{10}{27}\qquad\textbf{(C)}\ \frac{11}{27}\qquad\textbf{(D)}\ \frac{4}{9}\qquad\textbf{(E)}\ \frac{13}{27}$

2020 BMT Fall, 9

Tags: algebra

A sequence $a_n$ is defined by $a_0 = 0$, and for all $n \ge 1$, $a_n = a_{n-1} + (-1)^n \cdot n^2$. Compute $a_{100}$

2024 AMC 10, 20

Tags: counting

Three different pairs of shoes are placed in a row so that no left shoe is next to a right shoe from a different pair. In how many ways can these six shoes be lined up? $ \textbf{(A) }60\qquad \textbf{(B) }72\qquad \textbf{(C) }90\qquad \textbf{(D) }108\qquad \textbf{(E) }120\qquad $

2019 Centers of Excellency of Suceava, 3

Tags: bijection , geometric sequence , permutation of the naturals , arithmetic sequence , algebra

Let $ \left( a_n \right)_{n\ge 1} $ be a non-constant arithmetic progression of positive numbers and $ \left( g_n \right)_{n\ge 1} $ be a non-constant geometric progression of positive numbers satisfying $ a_1=g_1 $ and $ a_{2019} =g_{2019} . $ Specify the set $ \left\{ k\in\mathbb{N} \big| a_k\le g_k \right\} $ and prove that it bijects the natural numbers. [i]Gheorghe Rotariu[/i]

2020 CMIMC Team, 12

Tags: team

Determine the maximum possible value of $$\sqrt{x}(2\sqrt{x}+\sqrt{1-x})(3\sqrt{x}+4\sqrt{1-x})$$ over all $x\in [0,1]$.

2015 BMT Spring, Tie 2

Tags: algebra

Let $S_n = 1 + 2 + ,,, + n$. Define $$T_n =\frac{S_2}{S_2- 1}\cdot \frac{S_3}{S_3 - 1}\cdot ... \cdot \frac{S_n}{S_n - 1}.$$ Find $T_{2015}.$

1983 IMO Shortlist, 7

Tags: sequence , recurrence relation , algebra , number theory

Let $a$ be a positive integer and let $\{a_n\}$ be defined by $a_0 = 0$ and \[a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ).\] Show that for each positive integer $n$, $a_n$ is a positive integer.

2022 Irish Math Olympiad, 2

Tags: geometry

2. Let [i]ABCD [/i]be a square and let $\Gamma$ denote the circle with diameter [i]CD[/i]. A tangent line is drawn to the circle $\Gamma$ from [b][i]B[/i][/b], meeting the circle $\Gamma$ at [i]E[/i] and intersecting the segment [i]AD[/i] at [i]K[/i]. Prove that |[i]AD[/i]| = 4 |[i]KD[/i]|.

2006 Korea - Final Round, 1

Tags: inequalities

Given three distinct real numbers $a_{1}, a_{2}, a_{3}$ , deﬁne $b_{j}= (1+\frac{a_{j}a_{i}}{a_{j}-a_{i}})(1+\frac{a_{j}a_{k}}{a_{j}-a_{k}})$, where $\{i, j, k\}= \{1, 2, 3\}$. Prove that $1+|a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}| \leq (1+|a_{1}|)(1+|a_{2}|)(1+|a_{3}|)$ and ﬁnd the cases of equality.

1997 AMC 8, 18

Tags: percent , percent decrease

At the grocery store last week, small boxes of facial tissue were priced at 4 boxes for \$5. This week they are on sale at 5 boxes for \$4. The percent decrease in the price per box during the sale was closest to $\textbf{(A)}\ 30\% \qquad \textbf{(B)}\ 35\% \qquad \textbf{(C)}\ 40\% \qquad \textbf{(D)}\ 45\% \qquad \textbf{(E)}\ 65\%$

2022 South Africa National Olympiad, 3

Tags: gcd , number theory

Let a, b, and c be nonzero integers. Show that there exists an integer k such that $$gcd\left(a+kb, c\right) = gcd\left(a, b, c\right)$$

1986 AIME Problems, 4

Tags: system of equations , algebra

Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below. \[ \begin{array}{l} 2x_1+x_2+x_3+x_4+x_5=6 \\ x_1+2x_2+x_3+x_4+x_5=12 \\ x_1+x_2+2x_3+x_4+x_5=24 \\ x_1+x_2+x_3+2x_4+x_5=48 \\ x_1+x_2+x_3+x_4+2x_5=96 \\ \end{array} \]

2000 Greece Junior Math Olympiad, 2

Tags:

A three-digit bumber $\overline{abc}$ (in decimal representation) is such that (i) its hundreds digit is equal to the sum of the other two digits, and (ii)$b(c+1)=52-4a$. Find all such numbers.

2021 Kyiv Mathematical Festival, 1

Tags: combinatorial geometry , arithmetic sequence , geometry

Is it possible to mark four points on the plane so that the distances between any point and three other points form an arithmetic progression? (V. Brayman)

2025 Bundeswettbewerb Mathematik, 3

Tags: touching circles , perpendicular lines , geometry , circles

Let $k$ be a semicircle with diameter $AB$ and midpoint $M$. Let $P$ be a point on $k$ different from $A$ and $B$. The circle $k_A$ touches $k$ in a point $C$, the segment $MA$ in a point $D$, and additionally the segment $MP$. The circle $k_B$ touches $k$ in a point $E$ and additionally the segments $MB$ and $MP$. Show that the lines $AE$ and $CD$ are perpendicular.

2015 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: combinatorics

On competition there were $67$ students. They were solving $6$ problems. Student who solves $k$th problem gets $k$ points, while student who solves incorrectly $k$th problem gets $-k$ points. $a)$ Prove that there exist two students with exactly the same answers to problems $b)$ Prove that there exist at least $4$ students with same number of points

2001 SNSB Admission, 3

Tags: positive definite , matrix , lebesgue integral , real analysis

Let be an $ n\times n $ positive-definite symmetric real matrix $ A. $ Prove the following equality. $$ \tiny\int_{\mathbb{R}^n} \exp\left( -\begin{pmatrix} x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}^\intercal A\begin{pmatrix} x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}\right) dx_1dx_2\cdots dx_n=\normalsize\frac{\pi^{n/2}}{\sqrt{\det A} } $$

2022 Purple Comet Problems, 4

Tags:

Of $450$ students assembled for a concert, $40$ percent were boys. After a bus containing an equal number of boys and girls brought more students to the concert, $41$ percent of the students at the concert were boys. Find the number of students on the bus.

2004 Austrian-Polish Competition, 10

Tags: polynomial , algebra

For each polynomial $Q(x)$ let $M(Q)$ be the set of non-negative integers $x$ with $0 < Q(x) < 2004.$ We consider polynomials $P_n(x)$ of the form \[P_n(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_{n-1} \cdot x + 1\] with coefficients $a_i \in \{ \pm1\}$ for $i = 1, 2, \ldots, n-1.$ For each $n = 3^k, k > 0$ determine: a.) $m_n$ which represents the maximum of elements in $M(P_n)$ for all such polynomials $P_n(x)$ b.) all polynomials $P_n(x)$ for which $|M(P_n)| = m_n.$

1940 Putnam, B6

Tags: matrix , matrix determinant , linear algebra

Prove that the determinant of the matrix $$\begin{pmatrix} a_{1}^{2}+k & a_1 a_2 & a_1 a_3 &\ldots & a_1 a_n\\ a_2 a_1 & a_{2}^{2}+k & a_2 a_3 &\ldots & a_2 a_n\\ \ldots & \ldots & \ldots & \ldots & \ldots \\ a_n a_1& a_n a_2 & a_n a_3 & \ldots & a_{n}^{2}+k \end{pmatrix}$$ is divisible by $k^{n-1}$ and find its other factor.

2019 Centers of Excellency of Suceava, 3

Tags: differentiability , real analysis , function

For two real intervals $ I,J, $ we say that two functions $ f,g:I\longrightarrow J $ have property $ \mathcal{P} $ if they are differentiable and $ (fg)'=f'g'. $ [b]a)[/b] Provide example of two nonconstant functions $ a,b:\mathbb{R}\longrightarrow\mathbb{R} $ that have property $ \mathcal{P} . $ [b]b)[/b] Find the functions $ \lambda :(2019,\infty )\longrightarrow (0,\infty ) $ having the property that $ \lambda $ along with $ \theta :(2019,\infty )\longrightarrow (0,\infty ), \theta (x)=x^{2019} $ have property $ \mathcal{P} . $ [i]Dan Nedeianu[/i]

2019 Abels Math Contest (Norwegian MO) Final, 1

Tags: square grid , combinatorics

You have an $n \times n$ grid of empty squares. You place a cross in all the squares, one at a time. When you place a cross in an empty square, you receive $i+j$ points if there were $i$ crosses in the same row and $j$ crosses in the same column before you placed the new cross. Which are the possible total scores you can get?