This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2006 Vietnam National Olympiad, 4
Tags: derivative , calculus , limit , algebra , function
Given is the function $f(x)=-x+\sqrt{(x+a)(x+b)}$, where $a$, $b$ are distinct given positive real numbers. Prove that for all real numbers $s\in (0,1)$ there exist only one positive real number $\alpha$ such that \[ f(\alpha)=\sqrt [s]{\frac{a^s+b^s}{2}} . \]

2005 MOP Homework, 6
Tags: geometry , circumcircle , exterior angle , angle bisector , incenter
A circle which is tangent to sides $AB$ and $BC$ of triangle $ABC$ is also tangent to its circumcircle at point $T$. If $I$ in the incenter of triangle $ABC$, show that $\angle ATI=\angle CTI$.

2025 Kosovo National Mathematical Olympiad`, P1
Tags: nice , system of equations , algebra
Find all real numbers $a$, $b$ and $c$ that satisfy the following system of equations: $$\begin{cases} ab-c = 3 \\ a+bc = 4 \\ a^2+c^2 = 5\end{cases}$$

1997 Moldova Team Selection Test, 5
Tags: polynomial , algebra
Let $P(x)\in\mathbb{Z}[x]$ with deg $P=2015$. Let $Q(x)=(P(x))^2-9$. Prove that: the number of distinct roots of $Q(x)$ can not bigger than $2015$

2024/2025 TOURNAMENT OF TOWNS, P5
Tags: geometry
Given a circle ${\omega }_{1}$ , and a circle ${\omega }_{2}$ inside it. An arbitrary circle ${\omega }_{3}$ is chosen which is tangent to the two latter circles and both tangencies are internal. The tangency points are linked by a segment. A tangent line to ${\omega }_{2}$ is drawn through the meet point of this segment and the circle ${\omega }_{2}$ . Thus a chord of the circle ${\omega }_{3}$ is obtained. Prove that the ends of all such chords (obtained by all possible choices of ${\omega }_{3}$ ) belong to a fixed circle. Pavel Kozhevnikov

2024 Indonesia TST, 1
Tags: geometry
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$. Prove that line $AO$ passes through the midpoint of segment $BE$.

2007 Denmark MO - Mohr Contest, 4
Tags: geometry , angle , circles , tangent circle
The figure shows a $60^o$ angle in which are placed $2007$ numbered circles (only the first three are shown in the figure). The circles are numbered according to size. The circles are tangent to the sides of the angle and to each other as shown. Circle number one has radius $1$. Determine the radius of circle number $2007$. [img]https://1.bp.blogspot.com/-1bsLIXZpol4/Xzb-Nk6ospI/AAAAAAAAMWk/jrx1zVYKbNELTWlDQ3zL9qc_22b2IJF6QCLcBGAsYHQ/s0/2007%2BMohr%2Bp4.png[/img]

2015 AIME Problems, 3
Tags: prime number , factoring polynomials , number theory
There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.

2000 Tuymaada Olympiad, 7
Tags: combinatorial geometry , geometry , pentagon , regular polygon , combinatorics
Every two of five regular pentagons on the plane have a common point. Is it true that some of these pentagons have a common point?

Kyiv City MO Juniors 2003+ geometry, 2021.9.5
Tags: geometry , median , equal segments
Let $BM$ be the median of the triangle $ABC$, in which $AB> BC$. Point $P$ is chosen so that $AB \parallel PC$ and$ PM \perp BM$. The point $Q$ is chosen on the line $BP$ so that $\angle AQC = 90^o$, and the points $B$ and $Q$ lie on opposite sides of the line $AC$. Prove that $AB = BQ$. (Mikhail Standenko)

2003 Croatia Team Selection Test, 1
Tags: perfect square , number theory
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.

2005 Slovenia National Olympiad, Problem 1
Tags: system of equations , algebra
Find all real numbers $x,y$ such that $x^3-y^3=7(x-y)$ and $x^3+y^3=5(x+y)$.

2003 Greece JBMO TST, 4
Tags: area of a triangle , geometry , area , circles , ratio
Given are two points $B,C$. Consider point $A$ not lying on the line $BC$ and draw the circles $C_1(K_1,R_1)$ (with center $K_1$ and radius $R_1$) and $C_2(K_2,R_2)$ with chord $AB, AC$ respectively such that their centers lie on the interior of the triangle $ABC$ and also $R_1 \cdot AC= R_2 \cdot AB$. Let $T$ be the intersection point of the two circles, different from $A$, and M be a random pointof line $AT$, prove that $TC \cdot S_{(MBT)}=TB \cdot S_{(MCT)}$

2012 NIMO Summer Contest, 11
Tags:
Let $a$ and $b$ be two positive integers satisfying the equation \[ 20\sqrt{12} = a\sqrt{b}. \] Compute the sum of all possible distinct products $ab$. [i]Proposed by Lewis Chen[/i]

1993 Hungary-Israel Binational, 2
Tags: algebra , polynomial
Determine all polynomials $f (x)$ with real coeffcients that satisfy \[f (x^{2}-2x) = f^{2}(x-2)\] for all $x.$

2016 MMATHS, 4
Tags: algebra , inequalities
For real numbers $a, b, c$ with $a + b + c = 3$, prove that $$a^2(b - c)^2 + b^2(c - a)^2 + c^2(a - b)^2 \ge \frac9 2 abc(1 - abc)$$ and state when equality is reached.

1994 Moldova Team Selection Test, 3
Tags: circumcircle , geometry
Triangles $MAB{}$ and $MA_1B_1{}$ are similar and have the same orientation. Prove that the circumcircles of these triangles cointain the intersection point of lines $AA_1{}$ and $BB_1{}$.

1998 Spain Mathematical Olympiad, 1
Tags: algebra , search
Find the tangents of the angles of a triangle knowing that they are positive integers.

2019 Belarusian National Olympiad, 11.6
Tags: geometry , inequalities
The diagonals of the inscribed quadrilateral $ABCD$ intersect at the point $O$. The points $P$, $Q$, $R$, and $S$ are the feet of the perpendiculars from $O$ to the sides $AB$, $BC$, $CD$, and $DA$, respectively. Prove the inequality $BD\ge SP+QR$. [i](A. Naradzetski)[/i]

2010 China Team Selection Test, 2
Tags: inequalities , triangle inequality , floor function , polynomial , algebra
Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$, and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.

2018 Taiwan TST Round 2, 2
Tags: combinatorics
Let $n > 1$ be a given integer. An $n \times n \times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one colour. For each $n \times n \times 1$ box consisting of $n^2$ unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get $3n$ sets of colours, split into three groups according to the orientation. It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of $n$, the maximal possible number of colours that are present.

2007 Stanford Mathematics Tournament, 16
Tags: symmetry , geometry
Find the area of a square inscribed in an equilateral triangle, with one edge of the square on an edge of the triangle, if the side length of the triangle is $ 2\plus{}\sqrt{3}$.

2017 Moldova EGMO TST, 4
Tags:
The points $P$ and $Q$ are placed in the interior of the triangle $\Delta ABC$ such that $m(\angle PAB)=m(\angle QAC)<\frac{1}{2}m(\angle BAC)$ and similarly for the other $2$ vertices($P$ and $Q$ are isogonal conjugates). Let $P_{A}$ and $Q_{A}$ be the intersection points of $AP$ and $AQ$ with the circumcircle of $CPB$, respectively $CQB$. Similarly the pairs of points $(P_{B},Q_{B})$ and $(P_{C},Q_{C})$ are defined. Let $PQ_{A}\cap QP_{A}=\{M_{A}\}$, $PQ_{B}\cap QP_{B}=\{M_{B}\}$, $PQ_{C}\cap QP_{C}=\{M_{C}\}$. Prove the following statements: $1.$ Lines $AM_{A}$, $BM_{B}$, $CM_{C}$ concur. $2. $ $M_{A}\in BC$, $M_{B}\in CA$, $M_{C}\in AB$

2023 Stanford Mathematics Tournament, 9
Tags:
Let $x,y,z$ be nonzero numbers, not necessarily real, such that \[(x-y)^2+(y-z)^2+(z-x)^2=24yz\] and \[\tfrac{x^2}{yz}+\tfrac{y^2}{zx}+\tfrac{z^2}{xy}=3.\] Compute $\tfrac{x^2}{yz}$.

2002 Estonia Team Selection Test, 2
Tags: trigonometry , isosceles , angle bisector , geometry
Consider an isosceles triangle $KL_1L_2$ with $|KL_1|=|KL_2|$ and let $KA, L_1B_1,L_2B_2$ be its angle bisectors. Prove that $\cos \angle B_1AB_2 < \frac35$